Zbigniew Palmowski

Martijn Pistorius
email: m.pistorius@imperial.ac.uk

Cramér asymptotics for finite time first passage probabilities of general Lévy processes

.

Introduction

The study of boundary crossing probabilities of Lévy processes has applications in many fields, including ruin theory (see e.g. Rolski et al. [START_REF] Rolski | Stochastic Processes for Insurance and Finance[END_REF] and Asmussen [START_REF] Asmussen | Ruin Probabilities[END_REF]), queueing theory (see e.g. Borovkov [START_REF] Borovkov | Stochastic processes in queueing theory[END_REF] and Prabhu [START_REF] Prabhu | Insurance, queues[END_REF]), statistics (see e.g. Siegmund [START_REF] Siegmund | Sequential Analysis Test and Confidence Intervals[END_REF]) and mathematical finance (see e.g. Roberts and Shortland [START_REF] Roberts | [END_REF]).

As in many cases closed form expressions for (finite time) first passage probabilities are either not available or intractable, a good deal of the literature has been devoted to logarithmic or exact asymptotics for first passage probabilities, using different techniques. Martin-Löf [START_REF] Martin-Löf | Entropy estimates for the first passage time of a random ralk to a time dependent barrier[END_REF] and Collamore [START_REF] Collamore | Large deviations for first passage times[END_REF] derived large deviation results for first passage probabilities of a general class of processes. Employing two-dimensional renewal theory and asymptotic properties of ladder processes, respectively, Höglund [START_REF] Höglund | An asymptotic expression for the probability of ruin within finite time[END_REF] and von Bahr [START_REF] Von Bahr | Ruin probabilities expressed in terms of ladder height distributions[END_REF] obtained exact asymptotics for ruin probabilities of the classical risk process (see also Asmussen [START_REF] Asmussen | Ruin Probabilities[END_REF]). Bertoin and Doney [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF] generalised the classical Cramér-Lundberg approximation (of the perpetual ruin probability of a classical risk process) to general Lévy processes.

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In this paper we obtain the exact asymptotics of the finite time ruin probability P (τ (x) ≤ t), where τ (x) = inf{t ≥ 0 : X(t) > x}, for a general Lévy process X(t) (X(0) = 0), if x and t jointly tend to infinity in fixed proportion, generalising Arfwedson [START_REF] Arfwedson | Research in collective risk theory[END_REF] and Höglund [START_REF] Höglund | An asymptotic expression for the probability of ruin within finite time[END_REF] who treated the case of a classical risk process. The proof is based on an embedding of the ladder process of X and a two-dimensional renewal theorem of Höglund [START_REF] Höglund | An asymptotic expression for the probability of ruin within finite time[END_REF].

The remainder of the paper is organized as follows. In Section 2 the main result is presented, and its proof is given in Section 3.

Main result

Let X be a Lévy process with non-monotone paths that satisfies E[e α0X (1) ] < ∞ for some α 0 > 0, (

and denote by τ (x) = inf{t ≥ 0 : X(t) > x} the first crossing time of x. We exclude the case that X is a compound Poisson process with non-positive infinitesimal drift, as this corresponds to the random walk case which has already been treated in the literature. The law of X is determined by its Laplace exponent ψ(θ) = log E[e θX (1) ] that is well defined on the maximal domain Θ = {θ ∈ R : ψ(θ) < ∞}. Restricted to the interior Θ o , the map θ → ψ(θ) is convex and differentiable, with derivative ψ ′ (θ). 1

Moreover, ψ ′ (0+) = E[X(1)] if E[|X(1)|] < ∞.
By the strict convexity of ψ, it follows that ψ ′ is strictly increasing on (0, ∞) and we denote by Γ : ψ ′ (0, ∞) → (0, ∞) its right-inverse function.

Associated to the measure P is the exponential family of measures {P (c) : c ∈ Θ} defined by their Radon-Nikodym derivatives dP (c)

dP Ft = exp (cX(t) -ψ(c)t) . (2.2)
It is well known that under this change of measure X is still a Lévy processes and its new Laplace exponent satisfies

ψ (c) (α) = ψ(α + c) -ψ(c). (2.3)
Related to X and its running supremum are the local time L of X at its supremum, its right-continuous inverse L -1 and the upcrossing ladder process H respectively. The Laplace exponent κ of the bivariate (possibly killed) subordinator (L -1 , H),

e -κ(α,β)t = E[e -αL -1 t -βHt 1 (L -1 t <∞) ], (2.4)
is related to ψ via the Wiener-Hopf factorisation identity

u -ψ(θ) = kκ(u, -θ) κ(u, θ), u ≥ 0, θ ∈ Θ o , (2.5)
for some constant k > 0 where κ is the Laplace exponent of the dual ladder process. Refer to Bertoin [4, Ch. VI] for further background on the fluctuation theory of Lévy processes. Bertoin and Doney [START_REF] Bertoin | Cramér's estimate for Lévy processes[END_REF] showed that, if the Cramér condition holds, that is γ > 0, where

γ := sup{θ ∈ Θ : ψ(θ) = 0}, (2.6)
the Cramér-Lundberg approximation remains valid for a general Lévy process: lim

x→∞ e γx P [τ (x) < ∞] = C γ , (2.7)
where C γ ≥ 0 is positive if and only if E[e γX (1) |X(1)|] < ∞ and is then given by

C γ = β γ /[γm γ ],
where

β γ = -log P [H 1 < ∞], m γ = E[e γH1 H 1 1 (H1<∞) ].
Further, Doob's optional stopping theorem implies the following bound:

e γx P (τ (x) < ∞) = E (γ) [e -γ(X(τ (x))-x) 1 (τ (x)<∞) ] ≤ 1. (2.8)
The result below concerns the asymptotics of the finite time ruin probability P (τ (x) ≤ t) when x, t jointly tend to infinity in fixed proportion. For a given proportion v the rate of decay is either equal to γvt or to ψ * (v)t, where ψ * is the convex conjugate of ψ:

ψ * (u) = sup α∈R (αu -ψ(α)).
We restrict ourselves to Lévy processes satisfying the following condition σ > 0 or the Lévy measure is non-lattice, (H)

where σ denotes the Gaussian coefficient of X. Recall that a measure is called non-lattice if its support is not contained in a set of the form {a + bh, h ∈ Z}, for some a, b > 0. Note that (H) is satisfied by any Lévy process whose Lévy measure has infinite mass. We write f ∼ g if lim x,t→∞,x=vt+o(t 1/2 ) f (x, t)/g(x, t) = 1.

Theorem 1 Assume that (H) holds. Suppose that 0 < ψ ′ (γ) < ∞ and that there exists a

Γ(v) ∈ Θ • such that ψ ′ (Γ(v)) = v. If x and t tend to infinity such that x = vt + o(t 1/2 ) then P (τ (x) ≤ t) ∼ C γ e -γx , if 0 < v < ψ ′ (γ), D v t -1/2 e -ψ * (v)t , if v > ψ ′ (γ),
with C 0 = 1 and D v given by

D v = -v log E[e -ηv L -1 1 1 (L -1 1 <∞) ] η v E[e Γ(v)H1-ηvL -1 1 H 1 1 (L -1 1 <∞) ] × 1 Γ(v) 2πψ ′′ (Γ(v)) ,
where

η v = ψ(Γ(v)).
Remark 1 (a) For a spectrally negative Lévy process the joint exponent of the ladder process is given by κ(α, β) = β + Φ(α) (α, β ≥ 0), where Φ(α) is the largest root of ψ(θ) = α, and thus

D v = D v := v ψ(Γ(v)) 2πψ ′′ (Γ(v)) , C γ ≡ 1.
(2.9) Indeed,

D v = D v × κ(η v , 0) Γ(v) ∂ ∂β κ(η v , β) |β=-Γ(v) exp{-κ(η v , -Γ(v))} = D v × 1 exp{-Φ(η v ) + Γ(v)} = D v since Φ(η v ) = Γ(v). (b) If X is spectrally positive, κ(α, β) = [α -ψ(-β)]/[ Φ(α) -β]
(see e.g. [4, Thm VII.4]), where Φ(α) is the largest root of ψ(-θ) = α and we find that

D v = Γ(v) + Γ(v) Γ(v) Γ(v) 1 2πψ ′′ (Γ(v)) , C γ = ψ ′ (0) ψ ′ (γ) ,
where Γ(v) = sup{θ : ψ(-θ) = ψ(Γ(v))}, recovering formulas that can be found in Arfwedson [START_REF] Arfwedson | Research in collective risk theory[END_REF] and Feller [START_REF] Feller | An introduction to probability theory and its applications[END_REF] respectively, for the case of a classical risk process.

Remark 2 Heuristically, in the case v > ψ ′ (γ), the asymptotics in Thm.

1 can be regarded as a consequence of the central limit theorem, that is, under the tilted measure P Γ(v) , asymptotically

τ (x) -x/v ω √ x
follows a standard normal distribution, where by (2.3) and choice of Γ(v),

ω 2 = Var (Γ(v)) [X 1 ] E (Γ(v)) [X 1 ] 3 = ψ (Γ(v))′′ (0) ψ (Γ(v))′ (0) 3 = ψ ′′ (Γ(v)) v 3 .
This explains why the asymptotics remain valid if x deviates o(x 1/2 ) = o(t 1/2 ) from the line vt.

In the boundary case v = ψ ′ (γ), in which case E (Γ(v)) [τ (x)] = t, the exact asymptotics of P (τ (x) ≤ t) may depend on the way in which x/t tends to v. Note that this case is excluded from Theorem 1.

Remark 3 In the case 0 < v < ψ ′ (γ), the asymptotics in Theorem 1 are a consequence of the law of large numbers. To see why this is the case, note that e γx P (τ (x) ≤ t) = e γx P (τ (x) < ∞) -e γx P (t < τ (x) < ∞), where the first term tends to C γ in view of (2.7), while for the second term the Markov property and (2.8) imply that e γx P (t < τ (x) < ∞) = x -∞ P (τ (x) > t, X(t) ∈ dy)e γy e γ(x-y) P (τ (xy) < ∞) ≤ x -∞ P (X(t) ∈ dy)e γy = P (γ) (X(t) ≤ x), which tends to 0 as t tends to infinity in view of the law of large numbers since E (γ) [X(t)] = tψ ′ (γ) > x. The proof below deals with the case that v > ψ ′ (γ).

Proof of Theorem 1

The idea of the proof is to lift asymptotic results that have been established for random walks by Höglund [START_REF] Höglund | An asymptotic expression for the probability of ruin within finite time[END_REF] and Arfwedson [START_REF] Arfwedson | Research in collective risk theory[END_REF] to the setting of Lévy processes by considering suitable random walks embedded in the Lévy process (more precisely, in its ladder process). We first briefly recall these results following the Höglund [START_REF] Höglund | An asymptotic expression for the probability of ruin within finite time[END_REF] formulation.

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Review of Höglund's random walk asymptotics

Let (S, R) = {(S i , R i ), i = 1, 2, . . .} be a (possibly killed) random walk starting from (0, 0) whose components S and R have non-negative increments, and consider the crossing probabilities

G a,b (x, y) = P (N (x) < ∞, S N (x) > x + a, R N (x) ≤ y + b), K a,b (x, y) = P (N (x) < ∞, S N (x) > x + a, R N (x) ≥ y + b),
where a ≥ 0, b ∈ R and N (x) = min{n : S n > x}. Let F denote the (possibly defective) distribution function of the increments of the random walk with joint Laplace transform φ and set F (u,v) (dx, dy) = e -ux-vy F (dx, dy)/φ(u, v).

Let V (ζ) = E ζ [(R 1 E ζ [S 1 ] -S 1 E ζ [R 1 ]) 2 ]/E ζ [S 1 ] 3 for ζ = (ξ, η) where E ζ denotes the expectation w.r.t. F ζ .
For our purposes it will suffice to consider random walks that satisfy the following non-lattice assumption (the analogue of the non-lattice assumption in one dimension):

The additive group spanned by the support of F contains R 2 + .

(G) Specialised to our setting Prop. 3.2 in [START_REF] Höglund | An asymptotic expression for the probability of ruin within finite time[END_REF] jointly with the remark given on p. 380 therein read as follows:

Proposition 1 Assume that (G) holds, and that there exists a ζ = (ξ, η)

with φ(ζ) = 1 such that v = E ζ [S 1 ]/E ζ [R 1 ]
, where φ is finite in a neighbourhood of ζ and (0, η). If x, y tend to infinity such that x = vy + o(y 1/2 ) > 0 then it holds that

G a,b (x, y) ∼ D(a, b)x -1/2 e xξ+yη if η > 0, K a,b (x, y) ∼ D(a, b)x -1/2 e xξ+yη if η < 0, for a ≥ 0, b ∈ R, where D(a, b) = C(a, b) • (2πV (ζ)) -1/2 , with V (ζ) > 0 and C(a, b) = 1 |η|E ζ [S 1 ] e bη ∞ a P ζ (S 1 ≥ x)e ξx dx.

Embedded random walk

Denote by e 1 , e 2 , . . . a sequence of independent exp(q) distributed random variables and by σ n = n i=1 e i , with σ 0 = 0, the corresponding partial sums, and consider the two-dimensional (killed) random walk {(S i , R i ), i = 1, 2 . . .} starting from (0, 0) with step-sizes distributed according to

F (q) (dt, dx) = P (H σ1 ∈ dx, L -1 σ1 ∈ dt)
, and write G (q) for the corresponding crossing probability

G (q) (x, y) = G 0,0 (x, y) = P (N (x) < ∞, R N (x) ≤ y).
Note that F (q) is a probability measure that is defective precisely if X drifts to -∞, with Laplace transform φ given by φ(u, v) = e -ut-vx F (q) (dt, dx) = q qκ(u, v) .

The key step in the proof is to derive bounds for P (τ (x) ≤ t) in terms of crossing probabilities involving the random walk (S, R):

Lemma 1 Let M, q > 0. For x, t > 0 it holds that G (q) (x, t) ≤ P (τ (x) ≤ t) ≤ G (q) (x, t + M )/h(0-, M ), (3.1)
where h(0-, M ) = lim x↑0 h(x, M ), with h(x, t) := P (H σ1 > x, L -1 σ1 ≤ t). Proof: Let T (x) = inf{t ≥ 0 : H t > x} and note that τ (x) = L -1 T (x) . By applying the Markov property it follows that

P (τ (x) ≤ t) = P (T (x) < ∞, L -1 T (x) ≤ t) = ∞ n=1 P (σ n-1 ≤ T (x) < σ n , L -1 T (x) ≤ t) (3.2) = ∞ n=1 P (H σn-1 ≤ x, H σn > x, L -1 T (x) ≤ t) = ∞ n=1 P (H σn-1 ∈ dy, L -1 σn-1 ∈ ds) × P (H σ1 > x -y, L -1 T (x-y) ≤ t -s) (3.3) = ∞ n=0 F (q)⋆n ⋆ f (x, t) = (U ⋆ f )(x, t), (3.4)
where U = ∞ n=0 F (q)⋆n , f (x, t) = P (H σ1 > x, L -1 T (x) ≤ t) and ⋆ denotes convolution. Following a similar reasoning it can be checked that G (q) (x, t) = U ⋆ h(x, t).

(3.5)

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In view of (3.4) and (3.5), the lower bound in (3.1) follows since

f (x, t) ≥ h(x, t),
taking note of the fact that H σ1 > x precisely if T (x) < σ 1 , while the upper bound in (3.1) follows by observing that for fixed M > 0,

h(x, t + M ) ≥ P (H σ1 > x, L -1 T (x) ≤ t, L -1 σ1 -L -1 T (x) ≤ M ) = P (H σ1 > x, L -1 T (x) ≤ t)P (L -1 σ1 ≤ M ) = f (x, t)h(0-, M ),
where we used the strong Markov property of L -1 and the lack of memory property of σ 1 . Applying Höglund's asymptotics in Proposition 1 yields the following result: Lemma 2 Let the assumptions of Proposition 1 hold true. If x, t → ∞ such that for v > ψ ′ (γ) we have x = vt + o(t 1/2 ) then

G (q) (x, t + M ) ∼ D q,M t -1/2 e -ψ * (v)t , M ≥ 0, where D q,M = v √ 2πψ ′′ (Γ(v)) C q,M with C q,M = e ψ(Γ(v))M κ(ψ(Γ(v)), 0) c v ψ(Γ(v))Γ(v) q q + κ(ψ(Γ(v)), 0)
,

where c v = E[e Γ(v)H1 -ψ(Γ(v))L -1 1 H 1 1 (L -1 1 <∞)
]. Lemma 2 is a consequence of the following auxiliary identities:

Lemma 3 Let u > γ, u ∈ Θ o . φ(z, -u) = 1 iff κ(z, -u) = 0 iff ψ(u) = z (3.6) ψ ′ (u) = E (u) [X(1)] = E (u) [H σ1 ] • (E (u) [L -1 σ1 ]) -1 (3.7) ψ ′′ (u) = E (u) [(H σ1 -ψ ′ (u)L -1 σ1 ) 2 ] • (E (u) [L -1 σ1 ]) -1 = ψ ′ (u)E (u) [(H σ1 -ψ ′ (u)L -1 σ1 ) 2 ] • (E (u) [H σ1 ]) -1 (3.8) ψ * (v) = vΓ(v) -ψ(Γ(v)) for v > 0 with Γ(v) ∈ Θ o . (3.9)
Proof: Eq (3.6): Note that for u, z > 0 it holds that κ(z, u) > 0. In view of the identity (2.5) the statement follows.

Proof: The assertion is a consequence of the following identity between measures on (0, ∞) 2 (which is itself a consequence of the Wiener-Hopf factorisation, see e.g. Bertoin [4, Cor VI.10])

P (X t ∈ dx)dt = t ∞ 0 P (L -1 u ∈ dt, H u ∈ dx)u -1 du. ( 3.11)
Fix (y, v) ∈ (0, ∞) 2 in the support of µ X (dt, dx) = P (X t ∈ dx)dt and let B be an arbitrary open ball around (y, v). Then µ X (B) > 0; in view of the identity (3.11) it follows that there exists a set A with positive Lebesgue measure such that P ((L -1 u , H u ) ∈ B) > 0 for all u ∈ A and thus P ((L -1 σ1 , H σ1 ) ∈ B) > 0. Since B was arbitrary we conclude that (y, v) lies in the support of F (q) . To complete the proof we next verify that if a Lévy process X satisfies (H) then µ X satisfies (G). To this end, let X satisfy (H). Suppose first that its Lévy measure ν has infinite mass or σ > 0. Then P (X t = x) = 0 for any t > 0 and x ∈ R, according to Sato [START_REF] Sato | Lévy processes and infinitely divisible distributions[END_REF]Thm. 27.4 ]. Thus, the support of P (X t ∈ dx) is uncountable for any t > 0, so that µ X satisfies (G). If ν has finite mass then it is straightforward to verify that P (X t ∈ dx) is non-lattice for any t > 0 if ν is, and that then µ X satisfies (G).

Proof of Theorem 1: Suppose that v > ψ ′ (γ) (the case v < ψ ′ (γ) was shown in Remark 3). Writing l(t, x) = t 1/2 e ψ * (v)t P (τ (x) ≤ t), Lemmas 1, 2 and 3 imply that s = lim sup

x,t→∞,x=tv+o(t 1/2 ) l(t, x) ≤ D q,M /h(0-, M ), i = lim inf

x,t→∞,x=tv+o(t 1/2 ) l(t, x) ≥ D q,0 .

By definition of h and D q,M it directly follows that, as q → ∞, D q,0 → D v , D q,M → D v e ψ(Γ(v))M and h(0-, M ) = P (L -1 σ1 ≤ M ) → 1. Letting M ↓ 0 yields that s = i = D v , and the proof is complete.

For θ ∈ Θ\Θ o , ψ ′ (θ) is understood to be limη→θ,η∈Θo ψ ′ (η).

Acknowledgments

We thank Florin Avram for helpful advice and useful comments. We are grateful to the referee for supplying the short direct proof (Remark 3) for the case v < ψ ′ (γ). This work is partially supported by EPSRC grant EP/D039053/1 and the Ministry of Science and Higher Education of Poland under the grant N N2014079 33 (2007)(2008)(2009).

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Eq (3.7): Note that if u > γ then by the fact that ψ(0) = ψ(γ) = 0 and the strict convexity of ψ it follows that ψ(u) > 0. In view of (2.5) it follows then that κ(ψ(u), -u) = 0 for u ∈ Θ o , u > γ. Differentiating with respect to u shows that

(3.10) Also, note that

] and -u). Eq (3.8) follows as a matter of calculus, by differentiation of (3.10) with respect to u. Finally, Eq. (3.9) follows from the definition of ψ * .

Proof of Lemma 2 The proof follows by an application of Prop. 1 to

Note that, by (3.6) with u = Γ(v), φ(ζ) = 1, and that

and Eqs. (3.9), (3.7),(3.8) imply that ξx + ηt = -ψ * (v)t and

To complete the proof we are left to verify the form of the constants. The calculation of the C q,M = C(0, 0)e ηM goes as follows:

, in view of the definition (2.4) of κ. Combining all results completes the proof.

As final preparation for the proof of Theorem 1 we show that the nonlattice condition holds: Lemma 4 Suppose that (H) holds true. Then F (q) satisfies (G).


Hongsheng Dai

Yanchun Bao

An inverse probability weighted estimator for the bivariate distribution function under right censoring

Keywords: Bivariate survival function, bivariate censored observations, consistency, correlated failure times, inverse probability weighted estimator, matched pairs

An inverse probability weighted estimator is proposed for the joint distribution function of bivariate random vectors under right censoring. The new estimator is based on the idea of transformation of bivariate survival functions and bivariate random vectors to univariate survival functions and univariate random variables. The estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function. Numerical studies show that the new estimator is more efficient than some existing inverse probability weighted estimators.

Introduction

In some experiments each unit consists of a pair of components and life times for each component are recorded. We use (T (1) , T (2) ) to denote the pair of life times. Both times are subject to random right censoring at the observed censoring time (C (1) , C (2) ). Examples include twin studies, eye studies and matched pair studies, where censoring is due to units are removed from the study before failure has been observed. In such studies, the joint distribution of bivariate times need to be estimated. Nonparametric estimators of bivariate distributions under right censoring have been proposed by [START_REF] Campbell | Nonparametric bivariate estimation with randomly censored data[END_REF] ;Tsai et.al. (1986); [START_REF] Burke | Estimation of a bivariate distribution function under random censorship[END_REF]; [START_REF] Dabrowska | Kaplan-Meier estimate on the plane[END_REF]; [START_REF] Prentice | Covariance and survivor function estimation using censored multivariate failure time data[END_REF]; [START_REF] Lin | A simple nonparametric estimator of the bivariate survival function under univariate censoring[END_REF]; [START_REF] Van Der Laan | Efficient estimation in the bivariate censoring model and repairing NPMLE[END_REF]; [START_REF] Wang | Nonparametric estimators of the bivariate survival function under simplied censoring conditions[END_REF]; [START_REF] Akritas | Estimation of bivariate and marginal distributions with censored data[END_REF] and [START_REF] Prentice | Hazard-based nonparametric survivor function estimation[END_REF]. The Non-parametric MLE (MPMLE) in [START_REF] Campbell | Nonparametric bivariate estimation with randomly censored data[END_REF] is not unique and is computationally intractable. The repaired NPMLE in van der Laan (1996) deals with the nonuniqueness of NPMLE based on reduced data, but it is sensitive to the choice of bandwidth. The repaired NPMLE can be viewed as a special case of a class of estimators in [START_REF] Prentice | Hazard-based nonparametric survivor function estimation[END_REF]. [START_REF] Moodie | An adjustment to improve the bivariate survivor function repaired NPMLE[END_REF] improve the repaired NPMLE to make it more robust to the choice of bandwidth. [START_REF] Burke | Estimation of a bivariate distribution function under random censorship[END_REF] introduces two inverse probability weighted (IPW) estimators which do not depend on any smoothing parameter or bandwidth, required by the repaired NPMLE and kernel estimates in Tsai et.al. (1986); [START_REF] Akritas | Estimation of bivariate and marginal distributions with censored data[END_REF]. IPW estimators induce nonnegative probability mass and satisfy the monotonicity requirements of a distribution function.

Monotonicity of a bivariate distribution estimator is very important, since the non-monotone estimators [START_REF] Dabrowska | Kaplan-Meier estimate on the plane[END_REF][START_REF] Prentice | Covariance and survivor function estimation using censored multivariate failure time data[END_REF]) may result in negative conditional probabilities in application. Thus this paper focuses on IPW estimators and we propose a new IPW estimator of the bivariate distribution function in the presence of right censoring. The novelty of our method is the variable transformation, which enables us to transfer the bivariate estimation problem to a univariate estimation problem and prove in a simple way that the estimator converges weakly to a zero-mean Gaussian process with an easily estimated covariance function.

Numerical studies show that the new estimator performs more efficiently than the estimators in [START_REF] Burke | Estimation of a bivariate distribution function under random censorship[END_REF] and the covariance function estimator also performs remarkably well.

Preliminaries

Let (T (1) , T (2) ) be a pair of nonnegative random variables defined on a probability space (Ω, F, P ).

The bivariate cumulative distribution function and bivariate survival function of this random vector are

F (t 1 , t 2 ) = P (T (1) ≤ t 1 , T (2) ≤ t 2 ) and S(t 1 , t 2 ) = P (T (1) > t 1 , T (2) > t 2 ) re-
spectively. The pair of censoring times is (C (1) , C (2) ) which has survival function G(t 1 , t 2 ) = P (C (1) > t 1 , C (2) > t 2 ). For simplicity, throughout this paper we assume that S and G are continuous functions. When S and G are discrete our results also hold.

The observable random variables are given by (X (1) , X (2) ) and (δ (1) , δ (2) ) where

X (k) = min(T (k) , C (k) ) and δ (k) = I[T (k) ≤ C (k) ], k = 1, 2. Let H(t 1 , t 2 ) = P (X (1) > t 1 , X (2) > t 2 ) (1)
be the survival function of (X (1) , X (2) ).

Throughout this paper we assume that the following assumption holds.

(A): (T (1) , T (2) ) and (C (1) , C (2) ) are independent.

Under assumption (A) we have that

H(t 1 , t 2 ) = S(t 1 , t 2 )G(t 1 , t 2 ). Let F * (t 1 , t 2 ) = P (X (1) ≤ t 1 , δ (1) = 1, X (2) ≤ t 2 , δ (2) = 1
). Then it can be derived that

F * (t 1 , t 2 ) = t1 0 t2 0 G(v 1 -, v 2 -)F (dv 1 , dv 2 ). It follows immediately that F (t 1 , t 2 ) = t1 0 t2 0 1 G(v 1 -, v 2 -) F * (dv 1 , dv 2 ). ( 2
)
Suppose that {(T

(1) i , T (2)
i , C

i , C

(2) i ), i = 1, • • • , n} are i.i.d. samples of the random vector (T (1) , T (2) , C (1) , C (2) ). The observed data are {(X

(1) i , X (2) i , δ (1) i , δ (2) i ), i = 1, • • • , n} where X (k) i = min{T (k) i , C (k) i } and δ (k) i = I[T (k) i ≤ C (k) i ], k = 0, 1. Suppose that Ĝ(t 1 , t 2
) is an estimator of G(t 1 , t 2 ) based on the observed data. Then an IPW estimator for F (t 1 , t 2 ) is given by

F (t 1 , t 2 ) = t1 0 t2 0 F * n (dv 1 , dv 2 ) Ĝ(v 1 -, v 2 -) (3)
where [START_REF] Campbell | Large sample properties of nonparametric bivariate estimators with censored data[END_REF], using (3), [START_REF] Burke | Estimation of a bivariate distribution function under random censorship[END_REF] constructs two IPW estimators. However, [START_REF] Burke | Estimation of a bivariate distribution function under random censorship[END_REF] does not derive their asymptotic distributions.

F * n (t 1 , t 2 ) = n i=1 I[X (1) i ≤ t 1 , δ (1) i = 1, X (2) i ≤ t 1 , δ (2)
3 Variable transformation and methodology 3.1 Transformation of S(t 1 , t 2 ) and G(t 1 , t 2 ) to univariate functions

Define transformation from (t 1 , t 2 ) to (α, z) as α = t 2 /t 1 and z = t 2 1 + t 2 2 . Given θ ∈ [0, π/2], such that cos θ = 1/ √ 1 + α 2 or sin θ = 1/ √ 1 + α -2
, we have t 1 = z cos θ and t 2 = z sin θ.

Obviously (z, θ) is the polar coordinates of (t 1 , t 2 ). If (t 1 , t 2 ) is given, then α is fixed and S(t 1 , t 2 ) can be transformed to a univariate function, S(z|α), according to the following formula,

S(t 1 , t 2 ) = P (T (1) > t 1 , T (2) > t 2 ) = P min T (1) t 1 , T (2) t 2 > 1 = P   min    T (1) 1 + t 2 t 1 2 , T (2) 1 + t 2 t 1 -2    > t 2 1 + t 2 2   = P (Z(α) > z) := S(z|α), (4)
where

Z(α) = min{T (1) √ 1 + α 2 , T (2) √ 1 + α -2 }.
The above transformation can be explained by Figure 1. Points p 1 : (T (1) , T (1) α) and p 2 :

(T (2) α -1 , T (2)
) correspond to vertical and horizontal projection of (T (1) , T (2) ) onto the line c:

v 2 = αv 1 . The value of Z(α)
is the minimum of the distances of these two points from the origin. Therefore S(z|α) = P (Z(α) > z) means the survival function of Z(α) on line c. Note 1) . Therefore the above transformation exists for all (t

that if t 1 = 0, t 2 > 0 then Z(α) = T (2) and if t 1 > 0, t 2 = 0 then Z(α) = T (
1 , t 2 ) ∈ [0, ∞) × [0, ∞). v 1 v ) , ( 2 1 t t ) , ( : ) 2 ( 1 ) 2 ( 2 T T - Ρ α ) , ( ) 2 ( ) 1 ( T T ) , ( : ) 1 ( ) 1 ( 1 α T T Ρ 1 2 t t = α 1 2 v v α = Figure 1: (T (2) α -1 , T (2)
) is obtained by horizontal projection; (T (1) , T (1) α) is obtained by vertical projection.

Similarly we have

G(t 1 , t 2 ) = P (C (1) > t 1 , C (2) > t 2 ) = P (Z ′ (α) > z) := G(z|α), (5)
where

Z ′ (α) = min{C (1) √ 1 + α 2 , C (2) √ 1 + α -2 }.
Note that S(z|α) and G(z|α) are not conditional survival functions. They are univariate survival functions if α is fixed and they are bivariate functions if α is not fixed.

The IPW estimator based on transformation for censored data

In practice, due to the censorship of (T (1) , T (2) ) and (C (1) , C (2) ), the values of Z(α) and Z ′ (α)

may not be obtained. We can only obtain

X(1) = X (1) 1 + α 2 , X(2) = X (2) 1 + α -2 . (6) Let Z(α) = min{ X(1) , X(2) }, δ ′ (α) = (1 -δ (1) )I[ X(1) < X(2) ] + (1 -δ (2) )I[ X(1) > X(2) ] +(1 -min(δ (1) , δ (2) ))I[ X(1) = X(2) ]. (7)
Let

H(z|α) = P ( Z(α) > z). (8)
Then from ( 6), ( 7) and ( 1) we have

H(z|α) = P (X (1) > t 1 , X (2) > t 2 ) = H(t 1 , t 2 ). ( 9
)
We also have Z(α) and δ ′ (α) are the censored value and censoring indicator for Z ′ (α), since if

δ ′ (α) = 1 then Z ′ (α) = Z(α) is observed and if δ ′ (α) = 0 then Z(α) = Z(α) < Z ′ (α) which means that Z ′ (α) is censored. Thus based on observations { Zi (α), δ ′ i (α), i = 1, • • • , n} obtained from transformation in (7), if we define N ′ (s|α) = n i=1 I[ Zi (α) ≤ s, δ ′ i (α) = 1]/n, H n (s|α) = n i=1 I[ Zi (α) > s]/n, (10)
then the KM estimator for G(z|α

) = P (Z ′ (α) > z) is ĜKM (z|α) = s≤z 1 -∆ N ′ (s|α) Hn(s-|α) , where ∆ N ′ (s|α) = N ′ (s|α) -N ′ (s -|α). When α is fixed, ĜKM (z|α) is a univariate KM estimator.
Since G(t 1 , t 2 ) = G(z|α), ĜKM (z|α) must also be a consistent estimator of G(t 1 , t 2 ).

According to (3) and ĜKM (z|α), we can construct a monotone IPW estimator of F (t 1 , t 2 ),

F (t 1 , t 2 ) = t1 0 t2 0 F * n (dv 1 , dv 2 ) Ĝ(v 1 -, v 2 -) := t1 0 t2 0 F * n (dv 1 , dv 2 ) ĜKM (s -|α v ) where s = v 2 1 + v 2 2 , α v = v 2 /v 1 .
Note that the proposed IPW estimator can be written as

F (t 1 , t 2 ) = 1 n n i=1 I[X (1) i ≤ t 1 , X (2) i ≤ t 2 ]δ (1) i δ (2) i ĜKM X (1) i 2 + X (2) i 2 -|α i = 1 n n i=1 I[X (1) i ≤ t 1 , X (2) i ≤ t 2 ]δ (1) i δ (2) i ĜKM ( Zi (α i ) -|α i ) ,
where α i = X

(2)

i /X

(1) i and the second equality sign is due to

X (1) i 2 + X (2) i 2 = Zi (α i ).
We can see that the IPW estimator F (t 1 , t 2 ) can be calculated by summing up all probability mass on points (X

i , X

(2)

i ) satisfying X (1) i ≤ t 1 , X (2) i ≤ t 2 and δ (1) i = 1, δ (2)
i , δ

(1) i = 1, δ (2) i
= 1). The probability mass on each doubly-observed point is 1/ n ĜKM ( Zi (α i ) -|α i ) , which can be estimated by the following steps.

(1) Calculate α i = X

(2)

i /X

(1)

i . (2) Project all {X (1) j , X (2) j , j = 1, • • • , n} vertically or horizontally on the line v 2 = α i v 1 and calculate the corresponding values { Zj (α i ), δ ′ j (α i ), j = 1, • • • , n}. (3) Calculate the univariate KM estimator ĜKM (z -|α i ) based on { Zj (α i ), δ ′ j (α i ), j = 1, • • • , n} and 1/ n ĜKM ( Zi (α i ) -|α i ) .

The large sample properties

Based on the following Lemma (representing ĜKM (z|α) and 1/ ĜKM (z|α) as sum of i.i.d. random variables) we can show the large sample properties for F (t 1 , t 2 ).

Lemma 3.1. Let H 0 (s|α) = P ( Z(α) ≤ s, δ ′ (α) = 1), H 0n (s|α) = n j=1 I[ Zj (α) ≤ s, δ ′ j (α) = 1]/n, (11)
and ξ i (z|α) =

δ ′ i (α)I[ Zi (α)≤z] H( Zi (α)|α) - z 0 I[s≤ Zi (α)]
H 2 (s|α) dH 0 (s|α). Put ς n = n -3/4 (log n) 3/4 . Let τ be such that

(τ / √ 1 + α 2 , τ / √ 1 + α -2 ) > 0 and S(τ / √ 1 + α 2 , τ / √ 1 + α -2 ) > 0 for any α ∈ [0, ∞]. Then we have (1) ĜKM (z|α) = G(z|α) n n i=1 [1 -ξ i (z|α)] + R n (z, α) (2) 1 ĜKM (z|α) = 1 nG(z|α) n i=1 [1 + ξ i (z|α)] + R n (z, α),
and

sup z∈[0,τ ],α∈[0,∞] |R n (z, α)| = O(ς n ), a.s.. Proof. See appendix A.
The following theorem provides the asymptotic distribution of F (t 1 , t 2 ).

Theorem 3.1. Let

η i = I[X (1) i ≤ t 1 , X (2) i ≤ t 2 ]δ (1) i δ (2) i G( Zi (α i )|α i ) , µ i = t1 0 t2 0 [ξ i (s -|α v )] F * (dv 1 , dv 2 ) G(v 1 , v 2 ) . ( 12
)
For any (t 1 , t 2 ) such that t 2 1 + t 2 2 < τ , where τ is given in Lemma 3.1, we have

√ n( F (t 1 , t 2 ) - F (t 1 , t 2 )) ⇒ N (0, σ 2 (t 1 , t 2 ))
, where σ 2 (t 1 , t 2 ) = V ar(η 1 ) + V ar(µ 1 ) + 2Cov(η 1 , µ 1 ).

Proof. See Appendix B.

Then it follows immediately that F (t 1 , t 2 ) converges to F (t 1 , t 2 ) in probability.

Let ηi =

I[(X (1) i ,X (2)
i )≤(t1,t2)]δ

(1)

i δ (2) i ĜKM ( Zi (α i )|α i ) , ξi (z|α) = δ ′ i (α)I[ Zi (α)≤z]
Hn( Zi (α)|α) -

z 0 I[s≤ Zi (α)] H 2
n (s|α) dH 0n (s|α), and μi =

(t1,t2) (0,0) ξi (s -|α v ) F * n (dv1,dv2) ĜKM (s-|αv)
. We then obtain that Cov(η 1 , µ 1 ) := i ηi μi /n and V ar(µ 1 ) := i μ2

i /n are consistent estimators for Cov(η 1 , µ 1 ) and V ar(µ 1 ). In addition, according to

V ar(η 1 ) = t1 0 t2 0 F * (dv 1 , dv 2 ) G 2 (v 1 , v 2 ) -F (t 1 , t 2 ) 2 , a consistent estimator for V ar(η 1 ) is V ar(η 1 ) = t1 0 t2 0 F * n (dv 1 , dv 2 ) [ ĜKM (s -|α v )] 2 - t1 0 t2 0 F * n (dv 1 , dv 2 ) ĜKM (s -|α v ) 2 . Thus a consistent estimator for σ 2 (t 1 , t 2 ) is σ2 (t 1 , t 2 ) = V ar(η 1 ) + V ar(µ 1 ) + 2 Cov(η 1 , µ 1 ).

Simulation studies

In this section we study the properties of the proposed estimator via sets of 200 simulations under two different scenarios.

Scenario 1: We choose the well-known bivariate parametric model in [START_REF] Clayton | A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence[END_REF]. The joint distribution of (T (1) , T (2) ) is F (t 1 , t 2 ) = (F 1 (t 1 ) -φ + F 2 (t 2 ) -φ -1) -φ -1 with φ = 4. The marginal distributions F i (t i ), i = 1, 2 are specified as unit exponential and C (1) , C (2) are independent and identically distributed as exp(β). Simulation studies are carried out on different censoring percentages.

We compare the mean squared errors (MSE), 200 k=1 ( Fk -F ) 2 /200, of Burke's estimators, the estimator in [START_REF] Dabrowska | Kaplan-Meier estimate on the plane[END_REF] and our proposed estimator. For simplicity the mean estimates and empirical variances of Burke's estimators and Dabrowska's estimator are not provided.

Comparing the MSEs of the estimators in Table 1, we can see that the proposed estimator is more efficient (has smaller MSE) than Burke's estimators, at the tail of distribution functions and under high censoring. The proposed estimator is slightly less efficient than Dabrowska's estimator. The simulation also show that the variance estimators perform very well.

Scenario 2: Data are generated from, T (1) = 0.9τ 1 + 0.1τ 2 and T (2) = 0.2τ 1 + 0.8τ 2 , where τ 1 ∼ Gamma(3, 0.3) and τ 2 ∼ Gamma(2, 0.3). The distributions of independent censoring Table 1: Simulation studies: (a): theoretical survival probabilities, (b): empirical means of F (x, y), (c): empirical variances of F (x, y), (d): empirical means of variance estimates for F (x, y), (M): MSEs of our proposed estimator, (M1) and (M2): MSEs of Burke's two estimators, (M3): MSEs of the estimator in [START_REF] Dabrowska | Kaplan-Meier estimate on the plane[END_REF]. (i) β = 4.0 corresponds to 12% censoring for T (1) and T (2) respectively, (ii) β = 3.0 corresponds to 25% censoring, (iii) β = 1.0 corresponds to 50% censoring. (2) are both chosen to be exp(β). Such models arise in many reliability problems, for example in many systems which are composed of components, the system failure time can be expressed as a sum or a linear combination of component life times. Under this scenario the proposed estimator also performs much better than Burke's estimators but is slightly less efficient than Dabrowska's estimator. For simplicity the simulation results are not shown here.

n = 80 (x, y) (0.5, 0.5) (1.0, 0.5) (1.5, 0.5) (a) 0.3060 0.3834 0.3926 (i) (ii) (iii) (i) (ii) (iii) (i) (ii) (iii) (b) 0.

Discussion

This paper proposed a new IPW estimator for bivariate distribution function under right censoring. Its large sample properties are proved and it is more efficient than some existing IPW estimators. Comparing the proposed IPW estimator with other estimators, such as NPMLE and repaired NPMLE, is left as a future research work.

The Kaplan-Meier estimate for univariate data is known to be uniformly consistent over the

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT ¡ ¢ £ ¤ - ¥ ¦ § © θ α - θ α = + ! θ " # $ % θ α & ' ( ) 0 1 2 θ α + 3 4 5 6 7 7 α = Figure 2: For all points (t 1 , t 2 ) (with polar coordinates (z i,α , θ)) in C i , the function H(t 1 , t 2 ) = H(z i,α |α = tan θ) = u i .
entire support of the censored data distribution. This guarantees the univariate Kaplan-Meier estimate is reliable over the entire support of the censored data distribution. Another future research work is to study the global consistency for the bivariate estimator.

A Proof of Lemma 3.1

The idea of proving Lemma 3.1 follows from [START_REF] Lo | The product-limit estimator and the bootstrap: some asymptotic representations[END_REF]. We first need the following lemma.

Lemma A.1. Following the definitions of H n (z|α) in ( 10), H 0n (z|α) in ( 11) and ς n = n -3/4 (log n) 3/4 , we have

sup α∈[0,∞] sup z∈[0,τ ] z 0 H -1 n (s|α) -H -1 (s|α) d H -1 0n (s|α) -H -1 0 (s|α) = O(ς n ) a.s. Proof. Let H n (t 1 , t 2 ) = n i=1 I[X (1) i > t 1 , X (2)
i > t 1 ]/n. Obviously we have H n (t 1 , t 2 ) = H n (z|α), where z = t 2 1 + t 2 2 , α = t 2 /t 1 . We divide [0, 1] into subintervals [u i+1 , u i ], i = 0, • • • , k n , where k n = O( n/ log n) and the sequence 1 = u 0 > u 1 > • • • > u kn = 0 are such that |u i -u i-1 | ≤ O( log n/n), i = 1, • • • , k n . Let C i = {(v 1 , v 2 ) : such that H(v 1 , v 2 ) = u i }. Since H(v 1 , v 2 ) is continuous, C i is a continuous curve.
Given α, the line v 2 = αv 1 can be partitioned by points with polar coordinates (z i,α , θ = arctan α), i = 1, • • • , k n , where z i,α is such that H(z i,α |α) = u i . Note that (z i,α , θ) is the polar coordinates of the intersection point of line v 2 = αv 1 and C i . See Figure 2 for details.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Following from the proof of Lemma 2 in [START_REF] Lo | The product-limit estimator and the bootstrap: some asymptotic representations[END_REF], we have that for any α ∈ [0, ∞],

z 0 H -1 n (s|α) -H -1 (s|α) d H -1 0n (s|α) -H -1 0 (s|α) ≤ k n sup 0≤z≤τ |H -1 n (s|α) -H -1 (s|α)| max 0≤i≤kn -1 |H 0n (z i+1,α |α) -H 0n (z i,α |α) -H 0 (z i+1,α |α) + H 0 (z i,α |α)| +2 max 0≤i≤kn-1 sup s∈[zi,α,zi+1,α] |H -1 n (s|α) -H -1 n (z i,α |α) -H -1 (s|α) + H -1 (z i,α |α)| := A(α) + B(α). (13)
Now we prove sup α B(α

) = O(ς n ). We further partition [u i+1 , u i ] into subintervals [u i(j+1) , u ij ], j = 0, • • • , a n -1, such that |u i(j+1) -u ij | = O(n -3/4 (log n) 3/4 ) uniformly in i, j and a n = O(n 1/4 (log n) -1/4 ). Define C ij = {(v 1 , v 2 ) : such that H(v 1 , v 2 ) = u ij } and z ij,α is such that H(z ij,α |α) = u ij . Note that (z ij,α , θ) is the polar coordinates of the intersection point of line v 2 = αv 1 and C ij . Since sup s,α |H n (s|α) -H(s|α)| 2 = sup v1,v2 |H n (v 1 , v 2 ) -H(v 1 , v 2 )| 2 = O((log n)/n), similarly as
the results in [START_REF] Lo | The product-limit estimator and the bootstrap: some asymptotic representations[END_REF], we have that for a given value of α,

sup s∈[z i,α ,z i+1,α ] |H -1 n (s|α) -H -1 n (z i,α |α) -H -1 (s|α) + H -1 (z i,α |α)| ≤ ρ sup s∈[z i,α ,z i+1,α ] |H n (s|α) -H n (z i,α |α) -H(s|α) + H(z i,α |α)| + O log n n , a.s.,
where ρ is a constant and does not depend on α.

We partition [0, π/2] into subintervals 0 = θ 0 < θ 1 < • • • < θ bn = π/2 and let α l = tan θ l .

Let point p l,ij with coordinates (t 1,l,ij , t 2,l,ij ) be the intersection point of line v 2 = α l v 1 and curve C ij . A rectangle R l,ij is given by points p l,ij , p l+1,ij and points p * l,ij := (t 1,l,ij , t 2,l+1,ij ), p * l+1,ij := (t 1,l+1,ij , t 2,l,ij ). See Figure 3 for details. If b n is large enough, we can choose the sequence θ

l , l = 0, • • • , b n such that u i(j-1) > H(t 1,l+1,ij , t 2,l,ij ) > u ij u ij > H(t 1,l,ij , t 2,l+1,ij ) > u i(j+1) , ∀i, j, l, (14)
which means that points p * l,ij and p * l+1,ij are between C i(j+1) and C i(j-1) . Thus any two points (t 1 , t 2 ) and (t . Lines v 2 = α l+1 v 1 and v 2 = α l v 1 intersect with C i at points p l+1,i and p l,i , which gives a rectangle R l,i . The two lines intersect with C ij at points p l+1,ij and p l,ij , which gives a rectangle R l,ij .

* 1 , t * 2 ) within the rectangle R l,ij are such that |H(t 1 , t 2 ) -H(t * 1 , t * 2 )| ≤ |u i(j+1) - u i(j-1) | = O(ς n ), uniformly in i, j, l. A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 8 9 @ A B C D D θ α EF G = H I P Q R S + T U V W θ X + Y θ à b a - -c d e f g h i - p q r s t u v w xy + + d e f g h i j k + l m n o p q
With such partitions and following the results in [START_REF] Lo | The product-limit estimator and the bootstrap: some asymptotic representations[END_REF], we have

  i = 1. In other words, A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT the estimator has positive probability mass only on doubly-observed points (X

Figure 3 :

 3 Figure 3: Partition. The lower three curves are C i-1(an-1) , C i , C i1 and the upper three curves are Ci(j-1) , C ij , C i(j+1) . Lines v 2 = α l+1 v 1 and v 2 = α l v 1 intersectwith C i at points p l+1,i and p l,i , which gives a rectangle R l,i . The two lines intersect with C ij at points p l+1,ij and p l,ij , which gives a rectangle R l,ij .

  ,α,zi+1,α] |H n (s|α) -H n (z i,α |α) -H(s|α) + H(z i,α |α)| + O |H n (z ij,α |α) -H n (z i,α |α) -H(z ij,α |α) + H(z i,α |α)| + O(ς n ), a.s..

  For points p := (t 1 , t 2 ), we denote H n (t 1 , t 2 ) and H(t 1 , t 2 ) as H n (p) and H(p) for simplicity.According to the monotonicity of H n (t 1 , t 2 ) and H(t 1 , t 2 ) we haveH n (p * l,ij ) ≤ H n (z ij,α |α) ≤ H n (p * l+1,ij ) and H(p * l,ij ) ≤ H(z ij,α |α) ≤ H(p * l+1,ij ), for all α ∈ [α l , α l+1 ]. Then we have sup α∈[α l ,α l+1 ] |H n (z ij,α |α) -H n (z i,α |α) -H(z ij,α |α) + H(z i,α |α)| ≤ min |H n (p * l+1,ij ) -H n (p * l,i ) -H(p * l,ij ) + H(p * l+1,i )|, |H n (p * l,ij ) -H n (p * l+1,i ) -H(p * l+1,ij ) + H(p * l,i )| ≤ min |H n (p * l+1,ij ) -H n (p * l,i ) -H(p * l+1,ij ) + H(p * l,i )|, |H n (p * l,ij ) -H n (p * l+1,i ) -H(p * l,ij ) + H(p * l+1,i )| + O(ς n ), a.s..

		(16)
	Using similar methods as that in Lo and Singh (1985), we can also prove
	min |H n (p * l+1,ij ) -H n (p * l,i ) -H(p * l+1,ij ) + H(p * l,i )|,
	|H n (p * l,ij ) -H n (p * l+1,i ) -H(p * l,ij ) + H(p * l+1,i )| = O(ς n ), a.s..	(17)

Department of Mathematics and Statistics, Fylde College, University of Lancaster, Lancaster, LA1 4YF, UK; E-mail: h.dai@lancaster.ac.uk;

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

From ( 15), ( 16) and ( 17), we have sup α |B(α)| = O(ς n ). Similarly we can prove sup α A(α) = O(ς n ). The lemma then follows from (13).

Proof of Lemma 3.1. We have log ĜKM (z|α) -log G(z|α)

Following from the proof of Theorem 1 in [START_REF] Lo | The product-limit estimator and the bootstrap: some asymptotic representations[END_REF] and Lemma A.1, we have

Now we show that sup

To see this note that for any value of α,

where * extends over all i such that Z(i) (α) ≤ z and δ ′

ǫ, for all large n a.s.. Then (20) follows using two term Taylor's expansion for log(1 + w) with

Following from ( 18), ( 19) and ( 20), the first result of the lemma is proved by using two term

Taylor's expansion for log ĜKMlog G and the second result of the lemma is proved by using two term Taylor's expansion for log ŵlog w with ŵ = 1/ ĜKM , w = 1/G.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

B Proof of Theorem 3.1

Proof. According to Lemma 3.1 we can write

,

i /X

(1)

i . For simplicity we use Zi to denote Zi (α i ). Then

From the definition of η i in (12), we have

Note that U n is a U-statistic. From [START_REF] Serfling | Approximation theorems of Mathematical Statistics[END_REF] and EU n = 0, we have

k , δ

k , δ

k ) + o(n -1 (log n) γ ), for some γ > 0.

We also obtain that for i = j, E(η i ξ j ( Zi

i , δ

i , δ

i ) = 0 and

k , δ

k , δ

k , δ

k , δ

k ]/n 2 = (n-1)µ k /n 2 . Therefore we have F (t 1 , t 2 )-F (t 1 , t 2 ) = 1 n n i=1 [η i -F (t 1 , t 2 )]+ 1 n n k=1 µ k +O(ς n ). It follows that √ n( F (t 1 , t 2 ) -F (t 1 , t 2 )) ⇒ N (0, σ 2 (t 1 , t 2 )) with σ 2 (t 1 , t 2 ) = V ar(η 1 ) + V ar(µ 1 ) + 2Cov(η 1 , µ 1 ).


Jérôme Maucourant

POLANYI ON INSTITUTIONS AND MONEY -AN INTERPRETATION SUGGESTED BY A READING OF COMMONS

Introduction

The purpose of this contribution is to suggest an interpretation of the substantivist thought initiated by Karl Polanyi. An approach such as this is concerned with « historical economics », in the sense that substantive concepts are the result not of a retrospective projection of modern bias but of an attempt to apprehend universal categories of economic analysis in order to clarify the true particularities of different economic systems1. The point is to show that the coherence of institutionalist thought founded by economists such as Mitchell, Commons and Veblen are in agreement the reflexions of Karl Polanyi who thinks of money as an institution that embeds the economy into social relationships. In this analytical framework, stripping money of its institutional character and trying to reduce it to a commodity is bound to create chaos.

It is important to note that, rather than focusing on facts, the paper concerns the history of ideas. Certainly, Polanyi"s writings on ancient economy can no longer be wholly accepted, if the evolution of the work of historians is seriously taken into account. But this is not the real issue according to Duncan, Tandy (1994 : 23)2. The history of ideas is the history of problems, which can suggest new interpretations that do not depend substantially on facts. It is interesting to remark that Ian Morris"s (1994 : 54) attempt to combine Finley"s thesis and Polanyi"s insights constitutes a recent attempt to escape from the fallacy of the primitivism/modernism dichotomy3; such a solution has surely been an objective of Polanyi.

I shall, in the first part of this analysis, give an account of Polanyi"s demonstration that economy owes its very existence to institutionalised processes which produce those inflexibilities necessary for both collective and individual action. This view does not, however, requires a holistic approach and an objectivist theory of value. In that sense, classical institutionalist analysis4 is a way to go beyond subjectivism and objectivism. I shall try to show in what respect the ideas of Commons, Mitchell and Veblen are interesting complements of the work of Polanyi. This being so, it is possible to point out certain interesting consequences for the history of money : over and above institutional diversity, this institution sets the rules for the quantification and extinction of debts. Money is not the result of market necessity; the independence of money in relation to market may be the characteristic of numerous non-modern societies.

Bearing this in mind, in the second part of this analysis, I shall stress that it is impossible to presuppose the unity of the « catallactic triad » i.e. external trade, money and markets. This idea has significant consequences. Unlike modern money, archaic money is characterised by a separation of functions which obliges us to use the terms money uses. As a result, it is more appropriate to speak in terms of accounting and payment uses rather than use the word money.

Nevertheless, in spite of their differences, both ancient and modern monies politically produce social integration. This is obviously quite different from the empiricist language often used by Polanyi. For instance, he disagrees with « pseudo-philosophies » of money (Polanyi, 1968 : 175) despite the fact that his social philosophy takes the money issue seriously into account.

Part One -Institution as a concept : the case of money. §1/Karl Polanyi's method : a point of view Karl Polanyi defines formal economics as the pure realm of calculating reason in the context of scarcity of goods (when the ends are given). His interest is in studying power relations and values insofar as they institute a certain number of economic practices. Polanyi thus presupposes that economic activity is not isolated and that institutional analysis remains a way of carrying out economic reflections.

But it should be noted that there is no reason to deny the hypothesis that individual rationality has a wide-ranging field of effectiveness, nor, as a result, to imagine the existence of a whole that would be radically independent from its component parts. Polanyi does not neglect the determinisms resulting from individual interest, a factor which once again brings him closer to Commons : Neale (1990 : 147) remarks that, according to both authors, political authority intervenes as a result of the reaction of individual interests that are similarly threatened. Taking into consideration the case of tribal society, which is largely founded on gift and counter-gift, Polanyi (1944 : 46) further remarks that "the premium set on generosity is so great when measured in terms of social prestige as to make any other behaviour than utter self-forgetfulness simply not pay". Polanyi (1968 : 65) claims that modern anthropology of his time invalidates fantasies that are connected to the "communist psychlogy of savages"5 ; man has always aimed at maximising his social interests : "Aristotle was right : man is not an economic, but a social being. He does not aim at safeguarding his individual interest in the acquisition of material possessions but rather at ensuring social good will, social status, social assets. Man's economy is, as a rule, submerged in his social relations" (65).

More generally, the hypothesis that institutional constraints should structure behaviour does not contradict the idea that self-interest is the driving force behind action, provided that one acknowledges the nature of the ends assigned by the institution. As a result, individual interests are admittedly not autonomous. So Polanyi"s position does not seem to be either classical holism or « institutional individualism » defined by Biddle (1975) ; rather his conclusions seem to belong to a real institutional method that differs both from micro and from macro foundations of economic analysis.

The embeddedness hypothesis therefore calls for a redefinition of economics. Indeed, what has been surreptitiously introduced is the impossibility of fencing off the economic field, a factor which implicitly amounts to a definition of economy as a social subsystem. In fact, a science of pure rational action could, theoretically, bring us to describe all behaviour as economic. Therefore the issue at hand involves restricting the possible scope of economics while introducing a degree of anthropological complexity. §2 / Substantive Economy and the Question of Value Polanyi "s (1947 : 99) claim that "the truth is that man has been very much the same all through history"6 allows us to advocate a definition of economy7 that incorporates the institutional varieties to be found throughout history8. Thus Polanyi (1977 : 27) suggests the famous definition that is substantive to economy, focusing upon the idea that the livelihood of man is not strictly determined by scarcity, insofar as custom or politics eliminate the choice of agents. Only a specific institutional situation, that of price-making markets, can completely reduce the substantive sense of economy to a formal sense of economy9.

It can therefore be said that Polanyi, following Aristotle, rejects the theory of utilityscarcity value. Indeed, Polanyi emancipates himself from the common postulate of an "unquestioning acceptance of the sovereignty of individual wants" , according to Hodgson (1992 : 436) taking into account that the real human needs that Aristotle names as natural.

Actually, Aristotle understands the social nature of scarcity that must be distinguished from the obviousness of "the finite state of things" 10.

It is clear that Polanyi, in his concern for justice (in its modern and democratic form), national sovereignty and the permanence of social ties, is just as anxious as Aristotle may have been : developments of the market contradicts the idea that the economy should be the embodiment of a human purpose. Thus, liberal doctrine, which states that the role of the law must be to regulate the means of action, not the ends, is the antithesis of Polanyi"s socialism [START_REF] Mendell | Une analyse économique de la redistribution est-elle possible ?-Elements de comparaison entre la "new institutional economics" et l'approche substantive[END_REF]. In fact, in this type of socialism, social ends are determined by democratic debates. Polanyi is equally close to Veblen. Both authors agree with the Aristotelian ethic that rejects the continual pursuit of ever changing ends.

However, in spite of the well-known feature of Polanyian analysis that I have just mentioned here, there"s no direct link between Aristotle, Marx and Polanyi. The latter even wishes to emancipate himself from the paradigm of value in order to participate in a new paradigm based on the recognition of institutions as creators of economic values.

For instance, even if the real needs of man are changing, a democratic deliberation on social needs will decide on the kind of technical progress it intends to privilege. This is indeed

Polanyi"s point of view in the 1922 model of federal socialism, which leaves no place for the labor theory of value. In the second place, even if the ends of social activity should not, according to Polanyi, be left to a logic that is purely economic, it does not follow that every type of economic spontaneity should be abolished. He acknowledges the impossibility of central planning in the case of a complex division of labor 11. Consequently, this means that the evaluation of economic cost is not fundamentally a matter of statistical evaluation, but the result of an intersubjective process, even if the subject concerned is a selfmanaged group. The local and concrete activities of collectives of associated workers are the only means of examining the disutility of labour, according to [START_REF] Polanyi | Sozialistische Rechnungslegung[END_REF]Polanyi ( , 1925)). Elsewhere, central planning, which makes human economy a technique, is a non capitalist form of alienation, since by taking responsibility away from its producers, it denies the fact that ethics should be associated with labour. Polanyi (1922 : 93) writes "Humanity will not be free until it knows the cost of its ideals" .

As for the issue of money institution, it is hardly surprising that Polanyi (1944 : 25) writes : "the war between heaven and hell ignored the money issue, leaving capitalists and socialists miraculously united". Thus, it would be wrong to think that Poyani supports a radically objectivist conception of value. Once again, Polanyi does not base economy on value but on institutional theory, which enables to return to Commons"s question of reasonable value.

Beyond the objectivity of Marx and the subjectivity of Mises, Polanyi lies within institutionalism. The illusion that value can exist outside an institutional vestment stems from a totally utilitarian idea, common to both marxists and liberals ; this is the same illusion which states that economy can exist autonomously12. §3. Towards a concept of institution As a result, Polanyi develops an institutionalist conception of institution. To classify Karl

Polanyi"s work within the framework of institutionalism would mean the loss of his contributions for the contemporary economist : the idea that there are economic laws that are radically independent from the rest of the social field is in fact widespread. Nevertheless, it is clear that due to certain recent and commonly accepted developments in economic analysis, a certain number of doubts have arisen, thanks to the "undersocialised view"13 of economic analysis14.

Therefore, if one acknowledges that economy is to a certain extent embedded in social relationships, one must spot in the economy the mark of the logic belonging to the political area and value system that characterize a culture. Thus, institution in Polanyi"s thought could be a category referring to the combination of different fields of society and revealing, as Commons often says, a human purpose. The laws that govern the property rights, be they exclusive (in the case of private property) or relative to functions (in the case of public property in the ancient era) make up an essential part of an institutional analysis.

In this framework, Karl Polanyi considers the two-sided nature of economy : it is partly the process of interaction between man and nature, and partly an institutionalised process insofar as the necessary livelihood of man requires the institution of certain rigidities15. Polanyi thus defines empirical regularities which are fundamental to all social systems and which allow the integration of economy, therefore ensuring its permanence. This collective behaviour, which is now well known, is called form of integration16, and fits into the supporting structures of society. It is important to note here that Polanyi"s emphasis on behavioural rigidities can be enriched in a non-contradictory way by a reading of Commons.

It is relevant at this point to summarize Commons"s main theses (Maucourant, 1993b).

According to him, institution of social practices is not an arbitrary procedure arising from pure will. Institutions express the empirical regularities of social life while taking into account causes which can be economic. Commons (1934, p. 802) makes this very point when he asserts that the "law of supply and demand must not be abolished", but rather utilized. Similarly, Polanyi (1944, p. 252) contends that even if the Great Transformation soundly constitutes the "end of market society", the new reality does not constitute in any way the absence of markets : "Markets ensure freedom of consumers to indicate shifting in demand, to influence producers' income, and to serve as an instrument of accountancy, whilst ceasing altogether to be an organ of selfregulation".

Like Polanyi, Commons does not confuse the arbitrary nature of a will that does not institute anything durable other than chaos, and the device of collective action that fits into history17. Actually, institutions are collective mental entities made rigid by history ; the very importance of this historical process does not allow an inclusion of Commons"s approach in institutional individualism, in spite of an interesting paper by [START_REF] Biddle | Purpose and evolution in Commons's institutionalism[END_REF] on this topic. As a result, regularities of collective behaviour -institutions -which create individual expectations, are the matrix of social order : this point is crucial in Commons"s analysis of "futurity" and efficiency of monetary policy according to [START_REF] Whalen | Saving capitalism by making it good[END_REF]. More generally, collective action considered as a social constraint which controls individual actions broadens the sphere of individual action, according to Commons, by creating new opportunities, as if the ties that bring individuals together could not exist without the social mediation of institutions. §4 Money as an institution Polanyi (1977 : 14) claims that : "In primitive societies, credit, through which debt is formalised, is provided originally by the reciprocity practised within clan and neighborhood".

He assimilates the original category of debt to that of gift, which implies that there is no such thing as free gift. Douglass (1989 : 111) writes that the institution of gift constitutes the "Invisible Hand of primitive societies". In the same way, Commons (1934 : 474) makes the interesting point that "Primitive societies often have the institution of gift which is their method of creating debt, and they are even known to have set up a money of account". It is therefore possible to take the comparison between Commons and Polanyi further.

From the birth of the utopia of the Big Market, the domination of money as a means of exchange entails concomitant advancements, according to Polanyi, of the commoditisation of society. But this is not always the case as money is linked to requirements prior to the One Big Market. This is why Polanyi (1944 : 257) wishes to break up the "catallactic triad"18: "Catallactically, trade, money and market form an indivisible whole (...) such an approach must induce a more or less tacit acceptance of the heuristic principle according to which, where trade is in evidence, markets should be assumed". As a result, Polanyi (1977 : LIII) remarks : "debts and obligations are primitive phenomena that antedate the existence of markets, and the storage economics of antiquity practised large scale financial planning and accountancy long before the use of money as a means of exchange gained importance".

When Polanyi disassociates the question of money uses -i.e. the modes of cancelling out and accounting debts -from the question of the market, Commons (1934 : 474) asserts : "yet historical investigations show (...) that Hawtrey's logical foundation, Debt, is also historically the starting point of an economic history which is not a romance. (...) It needs only Knapp's distinction of unreleasable and releasable debts (...) negotiability, and legal tender, to bring about an economic theory which not only unifies production with credit but also history with logic".

According to Commons, money is the institution defined by the rules which fix the conditions of the issuing, evaluation, circulation and releasing of debts. There is no exchange, but rather payment, insofar as a social constraint aims at releasing debts. Money as an institution is a device that bring a public purpose into economy. Commons considers McLeod to be the initiator of institutional economics because of his penetrating analysis of the commoditisation of debt, and above all because of the idea that the central bank must eventually go against private banks. It is more interesting to notice that Commons (1923 : 241) also rejects the classical view of money as "simply nominal value, containing nothing more in itself than a yardstick or an empty basket" because "there is a public purpose in every system of money, even the most primitive" ( 244). Smith"s view ignores the concrete history of civil society that is embodied in the development of Common Law. Commons contends that (242) : "Money originated, indeed, out of the habits and customs of individuals in their transactions". The difference between the classical concept of exchange and the institutional concept of transaction is crucial.

Thus, when taking into account the case of Common Law, it is important to note that the practice of the judge is enacted within the frame of a changing collective purpose. Such a purpose could originally have been civil peace, followed by fiscal income acquisition, and finally "Adam Smith's big wheel of trade" (244). The collective purpose embodied in the system is no longer Locke"s commodity-money concept, but a stable price level that guarantees a healthy working capitalism based on the debitor/creditor opposition. Polanyi"s (1933) analysis of the Great Depression of the 1930"s a recognition of the institutional nature of the payment process [START_REF] Mendell | Une analyse économique de la redistribution est-elle possible ?-Elements de comparaison entre la "new institutional economics" et l'approche substantive[END_REF].

Consequently, it seems that Commons"s fruitful approach is different from the modern transaction cost approach. If Mitchell"s study of the progress of money economy in England is now considered, the importance of the State and of the power relations in this process must be stressed (Rutherford in [START_REF] Mitchell | Money economy and modern civilization", edited and with an introduction by M. Rutherford[END_REF]. Actually, it is far from evident that the lowering of transaction costs, which is made possible by the growing use of money, is spontaneous [START_REF] Servet | Nomismata -Etats et origines de la monnaie, Lyon, PUL[END_REF], as proved by anthropological and historical data. In other respects, Veblen also illustrates the institutionalist conception of institution. His point is not to know exactly what money, according to the pattern which represents human nature in a way that is as abstract as it is utilitarian, should be. Undoubtedly, it is not so much exchange which creates money, as the development of money as a way of understanding the world which makes more and more exchanges possible19. As a result, Veblen (1898 : 382-383) rejects the "conjectural history" : "As instances of the use of this ceremonial canon of knowledge may be cited the "conjectural history" that plays so large a part in the classical treatment of economic institutions, such as the normalised accounts of the beginnings of barter in the transaction of the putative hunter, fisherman, and boat-builder, or the man with the plane and the two planks, or the two men with the basket of apples and the basket of nuts. Of a similar import is the characterization of money as « the great wheel of circulation » or as « the medium of exchange ». Money is here discussed in terms of the end which, « in the normal case », should work out according to the given writer's ideal of economic life, rather than in terms of causal relation".

Part Two -Money in History : a Hypothesis

To summarise the previous argument it may be said that money, as a rule, is independent from the market; money should not be regarded as a commodity but as the institution which governs the issuing, evaluation and cancelling out of debts. Monetary order is a means of organising men into a hierarchy in that it determines the modes of measuring and cancellling out debts i.e. social obligations20. This is why social integration, which money helps to fulfil, has, from its origins, been closely linked to power relations.

With the publication of the Great Transformation, Polanyi continuously highlights the function of social integration21 fulfilled by money throughout history. Certainly, it does not follow that the forms and meanings of money are unchanging. The question of social integration from the point of view of money will be examined in the period before and after the Big Market.

But, before discussing this issue, it seems useful to emphasize the specificity of archaic money uses. §1/ The specificity of Archaic Monetary Institutions Before examining these methods of politically integrating economy, it is relevant to examine here the archaic functions of money. For Polanyi, money should be divided into "special purpose" and "all purpose" money. This division corresponds to the contrast between the two states of money, the first appearing before the One Big Market, the other afterwards. In this respect Polanyi succeeds in breaking up the unity of a phenomenon that the catallactic approach had believed in. It may very well be that, in ancient societies, one type of money object could fulfil the accounting function, whilst another object fulfilled that of payment22.

Sometimes, even the possible field of payment or account is strictly specified. The fact that money uses are instituted separately proves the very particular nature of non-modern money.

Money"s submission to the singular imperatives of exchange is nothing other than a characteristic of the One Big Market institution. Two examples will show that the natural aim of money is not to create an integrative exchange. Better still, money contributes23 to the avoidance of true money-market uses.

The absence of organised markets and coined money characterised the institutional pattern of Ancient Mesopotamia24. In this case, the requirements of trade and bartering necessitate the use of money both as a unit of account and as a means of recording debts which are the consequences of transactions. Banking operations are therefore a functional substitute necessary because of the absence of universally recognised means of exchange. The function of the banker is therefore to organise clearing and to testify to the existence of debt in an era when the spoken oath was fundamental. Polanyi (1977 : 142) remarks : "In the end, there is not much to show that in Mesopotamia, banking did actually prove itself an avenue to the changeover from administered trade to market trade". Archaic banking therefore renders modern forms of money useless.

It is also puzzling to note that even when the Greeks introduce the market and coined money, banking does not develop by conforming to exchange logic. No doubt this is due to the fact that the agora is only a "market element" where production goods are not exchanged. In any case, banking is reduced because of the custom of paying cash25. In reality, the institutional pattern of the bank of ancient Greece seems desperately rudimentary, as though the specifically Greek invention -the link of trade and money to the market (253) -did not create the natural impetus of banking, despite the notable growth in the market trade. One of the main roles of the trapezite banker is to change currencies26. Elsewhere, the city controls the activity of this banker who is often a slave. Polanyi (1968 : 315) stresses that the bank is a public, not a private affair.

Even more fundamentally, the bank within this institutional pattern does not create money, as no bank loan can be made without the order of the bank depositor. Otherwise, the bank loan can be a capital investment of the bank (Polanyi, 1977 : 265).

Finally, at no time is public deficit covered by bank loans. Such a phenomenon does not develop until the end of the Middle Ages. This is most certainly due to a way of thinking which, in principle, excluded deficit, be it individual or public, as illustrated by the precepts of Xenophon27 ; if a deficit ever occurred, sumptuary laws were used as an expedient, as in Rome at a later date (270). Another possibility for financing was to reduce metal content of money, which made recourse to banking useless.

It is useful to show how political authority is at the heart of the value of coins28, which means that money is in essence fiat money. One example, among others quoted by Polanyi (260, quoting Pseudo-Aristotle), shows to what extent the regularities of market economy are difficult to bring to the fore in ancient Greece. In the same way, a city in Asia Minor, experiencing a shortage of resources and having to reimburse large sums of money to mercenaries, found a strategy which alleviated the weight of the debt. The rich citizens of the town were forced to lend to the city in precious metals while they received money made of iron in return. The iron money was declared to have the same value as the money made of silver. Thanks to this procedure, the mercenaires were paid back in cash. One observes that the money supply, internal to the City, did not rise. Thus the hypothesis stating that a rise in global monetary circulations would be inflationary is simply anachronistic, insofar as one implicitly assumes the existence of an interconnected market system. This is not the case, and can explain the rigidities of the ancient price system. Consequently, it is remarkable that the citizens are still be subjected to tribal prerogatives29 as Cities can easily play on the fineness of monies and regulate prices. §2. Money as a political means of social integration in ancient societies Following Tönnies, Polanyi makes a sharp distinction between community and society.

Money, as a pure means of economic exchange is, according to Polanyi, associated to a tendency to "fluidity and instability" (118). This is the mark of a society ideally based on the levelling of conditions. On the contrary, the organic community of premodern times is characterised by its stability. Thus a reversal of perspectives is suggested (lii) : "Contrary to all assumptions, the origin of fluctuating prices, not of fixed prices, is the problem for the historian of antiquity".

Polanyi (120) suggests a political hypothesis on the nature of non-modern money which takes into account these rigidities : "The variety and often minute articulations of money institution thus help to achieve integration and stabilize status priviliges without the use of open force (underlined by us)".

In non-modern societies, monetary stability, which is at times multi-secular, is the rule, whilst very openly the differentiation of money is associated to social status (117). In this perspectives, in the Mali empire at around 1352, there existed "a poor man's money" (a fine thread of copper) and a rich man"s money (a thick thread of copper). The poor could only buy rudimentary consumption goods, while the rich could buy not only these goods, but naturally elite goods (such as slaves and horses). This fact has an important consequence (120) : "lower class status (...) is maintained by restricting living standards to the coarse food and bare necessities that native money is allowed to purchase". So the ostentatious standards of consumption are made rigid by rules that are relative to payment and make it difficult to transform the eventual wealth of the poor person into prestige and power. This stated, one can put forward an hypothesis related to a strange ancient rating of money interests. Strangely enough, Mesopotamian temples practised two different rates of interest. For barley and silver, the rates were 33.3% and 20%. At the same time, the loans agreed on by the temples were particular as they were made in barley for peasants and silver for citizens.

Polanyi notes that if we acknowledge that, in the African manner, monies (here barley and silver) are related to status (118), it is perhaps not surprising that the rate of interest of the low status money should be higher than that of the high status money 29 bis.

As a result, these customary rates, associated with the differentiation of money uses, contribute to the establishment of strong social stratification. Interesting as it may be, Polanyi does not investigate this point further, as he is sometimes more attached to forms of social integration than to forms of oppression. In fact, the crises of Mesopotamian cities from the oldest age are due to the impossible reimbursement of debt by the dominated levels of society. One may think that the higher levels of interest rates play a part in perpetuating this state of events and that they are a source of political troubles, moratory institutions, etc. In other words, if ancient money greatly strenghens social bonds, while introducing a rigidity that has no need of bureaucratic control, as Polanyi claims, it would be right to remember that political breakdowns are closely connected with this rigidity.

The crises of debt in early antiquity make it easier to understand Commons"s remarks (1934 : 390) : "Historically, it is more accurate to say that the bulk of mankind lived in a state of unreleasable debts, and that liberty came by gradually sustituting releasable debts". On the contrary, with modernity, the institution of credit money, which is nothing other than the effective consecration of the principle of free circulation of debt, allows us at least possible questioning30 of age-old hierarchies (Commons, 1992, p. 390)

. §3. The money of One Big Market as an institution

Therefore, according to Polanyi, non-modern money was by nature linked quite directly to a political system. Society"s submission to a self-adjusting market would not be possible without the fusion of money uses into one single use : all-purpose money. This modern and contingent idea of all-purpose money is the mental product of the commodity-money ideology, which is magnificently illustrated by the reign of the gold-standard. This original metamorphosis is merely one part of the first transformation leading to the selfadjusting market. But the Big Market is only a utopia31 that entails a paradoxical movement of liberation and regulation, the « double movement ». It is necessary to reconsider this utopia.

As for the nostalgia for the gold-standard which held sway in the 1920"s, Polanyi (1944 : 25) observes a fact that was noticed earlier : "Whether the gold itself has value for the reason that it embodies labor, as the socialists held, or for the reason that it is useful and scarce, as the orthodox doctrine ran, for once made no difference. The war between heaven and hell ignored the money issue, leaving capitalists and socialists miraculously united (...) and Das Kapital implied the commodity theory of money, in its Ricardian form". Polanyi, refusing to consider money as a commodity, suggests an analysis which corresponds to the institutionalist conception of the institution. The modern money institution, once more, exemplifies the impossibility of fencing off the economy. Despite the role of private institutes in the creation of modern money, Polanyi (196) observes the importance of its validation by the State 32: "Now the institutional separation of the political and the economical sphere has never been complete, and it was precisely in the matter of currency that it was necessarily incomplete; the state, whose Mint seemed merely to certify the weight of the coins, was in fact the guarantee of the value of the token money, which it accepted for the payment of taxes and otherwise. This money was not (underlined by Polanyi) a means of payment, it was a means of exchange".

According to Polanyi, even all-purpose money cannot exist without the institutional forms which are conditions of social integration. The proof of this paradox is that the monetary policy of central banks renders the so-called automatism of the gold-standard meaningless.The market economy is not possible with a currency that is purely a commodity. The untimely scarcity of this money is so catastrophic that a token money becomes absolutely necessary (192)(193). Therefore, Polanyi (1968 : 196) agrees with a non-essentialist view of money, especially with the Keynesian view of money : "In the Keynesian system the role of money is purely pragmatic (...) the presence of money is here taken for granted, not conceptually deduced".

Thus, the fact that money practices can be carried out simply with symbols is not a distortion of the "true" monetary system which results from pure commodity exchanges (176-177).

As for the monetary system of the One Big Market, it can be stated that it functions on two levels : a) An international level, where money is a full commodity : b) A national level, where money is the mediation between the realm of commodities money and the domain of token or fiat money that is connected to domestic markets. The central bank is this mediation which makes possible the coexistence of the international gold-standard and of a national credit money.

Thus the Central Bank is where the two logics meet : the one is purely economic as it rests on the means of exchange; the other is political, because it embodies the collective identity on which the means of payment is based. The birth of the Federal Reserve System could be interpreted as the expression of a counter-movement protecting society and working towards the construction of a national identity. Paradoxically, according to Polanyi (204) during the era of the gold-standard which lasted from 1879 to 1929 the monetary system was the strongest among the economic forces integrating the nation. By centralising the supply of credit (194), the Central Bank makes possible the institution of market by money. Therefore the Bank can partly protect the national economy against deflationary forces created by the functioning of the gold-standard33. Open market policy organises a fall in price while raising the exportable surplus and liquidating marginal firms. From there, an element of rationality in the Big Market reduces the "automatism of gold standard to a mere pretence" (195).

At the very moment that one institutes markets with the aim of establishing a pure economic regulation of society, society is protecting itself against the devastating effects of international money-commodity by monetary interventionism. This counter-movement of social protection, which leads towards the creation of the double movement so characteristic of liberal society, shows to what extent money is a "fictitious commodity", even during the reign of the gold-standard.

As a result, institutions of money, like other institutions of societies, are mediations insofar as they create links between the different dimensions of society in order to establish the behavioral regularities that are the primary expression of institutionalised processes.

Conclusion

Polanyi claims that the money institution is substantial in founding social orders, be they primitive, archaic, capitalist or socialist. Actually, it is possible to extend Polanyi"s anthropology : man, as a social being -zoon politicon -necessarily knows money as the institution that strengthens social relationships, thanks to standardising and standardised devices [START_REF] Servet | Nomismata -Etats et origines de la monnaie, Lyon, PUL[END_REF].

Thus it is impossible to confirm, as Hodgson does, that Polanyi could have maintained that money existed only on the periphery of non-modern economies. Too many of Polanyi"s citations regarding money testify to the contrary. Moreover, Polanyi never ceases to maintain that market as money also has an important place in society, even if market, unlike money, is a younger institution of human history. The only danger is reducing money to a pure commodity and thinking that markets could be self-adjusting.

It has to be said that his Polanyi"s approach is embedded in a clearly socialist reflection, a point which is sometimes ignored by some recent commentators of his work. As a result, it has recently been claimed (Hodgson, 1992, p. 437) that Polanyi"s writings justified the creation of a single currency in Europe and the birth of a federal state, insofar as domestic markets are now thoroughly integrated.

Admittedly, such a suggestion could fit into the analytical device of Polanyi"s project because the decrease in the number of national currencies in the global financial market could be analysed as a means of decreasing the commoditisation process. It seems reasonable to think that a central European bank, centralising further the supply of credit, would control the market better. But the political aspect of this new institution has been ignored. In a letter dating back to 1943, Polanyi noticed that (quoted by Litvan, 1991, p. 260) "the financial powers intervened in the internal affairs of every State, because in the era of the gold standard co-operation was only 7Carl Menger supplied Polanyi with a solution to this problem. In the 1923 German edition of his Principles, Menger sets an economising direction -a characteristic of civilised societies -against a technical direction which only results from physical particulariteis of production. He does so to safeguard his Principles, which were to be challenged by later anthropological discoveries (Polanyi, 1977 : 22-23).

8One should note Polanyi"s display of interest for the marginalist school of thought. Elsewhere, the influence on Polanyi of the Austrian school of marginal utility is certain in the socialist economic calculation debate. The future author of the Great Transformation in 1922 is looking for an alternative to central planning, which he considers impossible -as would his brother Mickael later (Polanyi, 1951 :148-149) -and pure market economy, that he judges to be a source of wastage. A presentation of this non-Marxist criticism of capitalism can be found in Rosner (1990) and[START_REF] Mendell | Une analyse économique de la redistribution est-elle possible ?-Elements de comparaison entre la "new institutional economics" et l'approche substantive[END_REF]. A translation into French of 1922 Polanyi"s text of is available in Polanyi (1994) ; it seems that a translation into English does not exist.

9In this case, a general state of scarcity exists, as economic agents are able to make an infinite number of choices provided by markets.

10 See Berthoud (1986: 67). According to Aristotle, human choice and desires are not limitless because they have to be submitted to the social imperative of an honourable life. The essence of life, according to Aristotle (Polanyi, 1977 : 30), i.e. theatre, trials, politics and the contemplation of good, prohibits any type of effective economic accumulation. A contrario, the free display of pecuniary activity is contradictory to the natural order of the City which is based on principles : self-sufficiency, community (resting on the value of living together, the phylia) and justice (in the sense that everyone receives that which is due to his nature (Polanyi, 1957 a : 79). The fact that Aristotle has a substantive conception of economy explains his fear that economy could actually become removed from society. It is this fear that highlights his importance as an economic thinker. Schumpeter, convinced that the distinctive nature of economic analysis is formalising scarcity mechanisms, cannot accept the importance of Aristotle"s economic thought.

11This is why any form of socialism that denies the necessity of monetary order is not viable. It is worth noting that, in the case of the decentralised socialist society desired by Polanyi, special-purpose monies are a means of embedding economy into society in a humanist way, as contradictory evaluations of wills are able to express themselves. This is necessary to avoid the dangerous dream of a natural economy which is technically impossible when neither custom nor tradition strictly determines collective and individual actions. Weber (1921 : 110-111) possibly shortly before Mises, remarks upon this technical feature of money that Polanyi never ignores.

12To be precise, the nineteenth century transformation which aimed at instituting a watertight "closure" of the economy through the establishment of the Big Market was, in fact, illusory.

13Oversocialised view is unbearable as only a superior mind could understand the logic of the whole, which is contradictio in adjecto. see [START_REF] Hayek | Scientisme et Sciences Sociales[END_REF]) 14Granovetter (1994 : 86) highlights the social networks that create market transactions. This legitimates A. Orlean"s claim about another contribution by Granovetter. A joint economic and sociological study of the history of a system should make it possible to remove the indeterminate nature of the final equilibrium (Orlean, 1994 : 31).

15Polanyi (1977 : 31) writes : "If the material survival of men were the result of a mere fleeting chain of causation -possessing neither definite location in time or space (that is unity and stability), nor permanent point of reference (that is structure), or definite mode of action in regard to the whole (that is function), nor ways of being influenced by societal goals (that is policy relevance) -it could never have attained the dignity and importance of human economy".

16Polanyi"s taxinomy changes even if his intentions do not. It is the unity of an idea which is interesting here. For the sake of detail, one can note that in 1944 redistribution is called "principle of behaviour", whereas in the 1977 text, it is called "forms of integration"; in the same way, centricity is first called "institutional model" but becomes "structures" after. It is clear that these semantic variations make no difference. I suggest calling institutional model a special set of institutions which includes their relations to culture and historical singularities that an economic analysis must, in the first instance, overshadow. Thus the Assyrian institutional model differs from the Egyptian one, even if redistribution is a main feature of these archaic systems.

17Gautier"s (1992 : 42-52) account concerning Hume, notably on p. 43, is interesting in this respect. 18Also creating an alternative to the conjectural history of institutions already denounced by Veblen. 19This view and the deliberate use of this extensive meaning of money was developed by [START_REF] Mitchell | Money economy and modern civilization", edited and with an introduction by M. Rutherford[END_REF][START_REF] Mitchell | Money economy and modern civilization", edited and with an introduction by M. Rutherford[END_REF].

20Modern history of facts seems to agree with such hypotheses. See [START_REF] Depeyrot | Histoire de la monnaie -des origines au 18°sciècle[END_REF] and [START_REF] Hollard | La crise de la monnaie dans l'Empire romain au 3°siècle après J. C. -Synthèse de recherches et résultats nouveaux[END_REF]. 21The integration referred to here can naturally cause political oppressions. This is not sufficiently emphasised by Polanyi (see [START_REF] Godelier | Introduction[END_REF].

22The money in question can be totally abstract. It is this that Polanyi names ideal unit. However the situation can be even more complex. For example, a slave could have served as a unit of account and an external means of payment, the great local values being expressed in slaves. This means that a type of ideal slave was developed. Keynes had, in the same way, quoted in his Treatise the case of the average goat as a unit of account in Africa (Keynes, 1930 : 11-12).

23This fact can also be the result of redistribution requirements. See [START_REF] Gentet | La question de la monnaie en Egypte ancienne[END_REF], a, b). 24Polanyi (1957, c, 125). Silver"s use (1983 and[START_REF] Descat | L'économie antique et la cité grecque -un modèle en question[END_REF] of new institutional economics is not convincing. For instance, the hypothesis related to equilibrium value in ancient barter processes has no solid foundation. See [START_REF] Mendell | Une analyse économique de la redistribution est-elle possible ?-Elements de comparaison entre la "new institutional economics" et l'approche substantive[END_REF].

25By the nomismata. See. Polanyi (1977, p. 316). 26The exchange table was called a trapetza just like the table named banca from which the term bank originated.

27 See Polanyi (1977 : 268) and Nicolet (1983 : 203). It is true that financial conceptions, linked to money that is strictly metallic, forbid any other practice than the adjustment of spending and resources. Government deficit would not be a possibility in England and France until the XVIIth and XVIIIth centuries. This weak development of banking in its modern sense corresponds to an immersion of economy into society. C. Nicolet once again observes : "one mentions principally economic phenomena considered from the point of view of their features or fiscal consequences" (204). In a more general way, it does not appear that the conclusions of C. Nicolet invalidate the substantive approach to which de facto, a certain heuristic value (15 and 30-37) is attributed. Certain hypotheses are corroborated by C. Nicolet"s own research about the meaning of the generosity of lenders who are not anxious because of the absence of repayments. In fact, moneylending solidifies social bonds. Even if this interpretation is "biased and partial", according to C. Nicolet, it is thus useful to note that high banking is a means of superior prestige because it creates social debtor. Only then can money become prestige, which is the important fact in Rome.

28In order to show the particularity of ancient mentalities, Janssen (1975 : 545, fn. 13) recalls the extreme lack of precision of intrinsic value of the first nomismata.

29Polanyi (1977 : 169). As a result, the fact that the counterfeiting of money causes a rise in price cannot be considered as evident.

29 bis Certainly, the difficulty of converting the interest of the debt expressed in a quantity of inferior goods into superior goods is a part of the explanation of the difference between rates. 30This is possible but not necessary. This will be the object of later reflections by Commons on monetary policy. The absence or incoherence of these will only abolish the emancipatory possibilities of pure credit money.

3L. Dumont, in his introduction to Polanyi (1983) and [START_REF] Servet | Nomismata -Etats et origines de la monnaie, Lyon, PUL[END_REF] share the same opinion. 32 [START_REF] Guttmann | The properties of international money and their consequences for the world economy[END_REF] emphasises this fact in his analysis of the Great Transformation 33Thus, according to Polanyi, the creation of money is in part exogenous in the gold standard regime.

possible if their internal systems were similar." After the New Deal, a contrario :"there's no need to force all States of the world into the procrustean bed of federation, because it is enough for their governments to co-operate freely". As a consequence of the globalisation of markets, which are naturally short-term oriented, the "Procust bed" of federalism seems to be inevitable again. Admittedly, it may be better to share a collective sovereignty than to have an illusory national sovereignty. It is even said that national identities are meaningless because the states which support these identities cannot organise the accumulation of capital.

But the very withering of the Nation-states goes along, in Europe, with the return to extreme orthodox practices. Like the gold standard, the independence of the central bank dogma is reducing labour to a pure commodity. This fact would certainly have frightened Polanyi himself! The real problem is not the planned abolition of monetary forms that can no longer produce, as they used to, real political autonomy; the point is to find an appropriate pace for this abolition, and to choose freely the type of monetary policy to be implemented afterwards.

The purpose of this paper is not to deny the fact that nationalisms do create nations (Gellner, 1994 : 86), a point which Polanyi might well be unaware of. Nevertheless, one question remains unanswered : are not the new federalist ambitions a new hidden form of the Veblenian predatory instincts that aim at reversing the Great Transformation process? NOTES 0 The author is member of the CREUSET (Saint-Etienne, France) and of the Centre Walras (Lyon, France) ; these institutions have given financial support for the translation of the paper which was done by Sue Hillion and Laurence Collaud. The author wishes to thank Claude Gautier, Jean-Michel Servet and Bruno Tinel (Centre.Walras@mrash.fr) for their helpful suggestions and comments on the French version, i. e. Maucourant (1998 : 328/359).

1 Admittedly, since Polanyi"s latest works, the debate has been enriched by a series of contributions (see for instance, [START_REF] North | L'essor du monde occidental[END_REF], [START_REF] Silver | Karl Polanyi and markets in the ancient near east : the challenge of the evidence[END_REF][START_REF] Descat | L'économie antique et la cité grecque -un modèle en question[END_REF], Mayhew et alii (1985) and Janssen (1975 : 560) amongst others). The richness of Polanyi"s approach is taken into account beyond strict heterodoxical works (Orlean, 1992).

2The authors justly write : "Polanyi's importance in the study of economy of ancient Greece does not derive from his essays and chapters on Greece". We have tried, in the case of ancient Egypt, to check on Polanyi"s heuristic theses. See Gentet, Maucourant (1991 a, b).

3In their textbook, Austin and Vidal Naquet (1992 : 17) assert that Weber and Hasebroek have supplied the tools to put an end to the controversy resulting from the Bucher hypothesis. But such a controversy seems well over (Amouretti, 1996 : 244 andSablé, 1996 : 294) in a "post-Finley" world (Descat, 1995 : 988) i.e. we are not in a "post-Rostovtseff" world [START_REF] Andreau | la cité romaine dans ses rapports à l'échange et au monde de l'échange" in Les échanges dans l'Antiquité : Le rôle de l'Etat[END_REF]1995 : p. 949-950).

4Even if Neale (1990 :145) thinks that there exists in the writings of Karl Polanyi and American institutionalists a "common propensity for holistic analyses", he nonetheless points out that the analysis of social change by Commons and Polanyi does not actually rest on a "reification" of society (p. 147).

5 Of course the same applies to Smith"s myth of savage devoted to barter (Polanyi, 1944 : 43-44) in which it is possible to find an interpretation in [START_REF] Servet | Nomismata -Etats et origines de la monnaie, Lyon, PUL[END_REF].

6Berthoud also justly highlights the hypothesis that the constancy of human nature entails a hypertophy of the interest in institutions and the cultural factor. But this antropological constancy is an interesting fact : humanity never constructs its society in a lasting fashion using methods that are totally individualist or collectivist. Polanyi rightly observes that, in fact, since greatest Antiquity, man has always known how to elude ideological dichotomies ; liberty vs bureaucracy and total planning vs market. On this subject, he cites the examples of a Mesopotamian trading corporation, Athenian customs of the classic age, and the kingdom of Dahomey in the 18th century (Polanyi, 1977 : xli)


E. Yu. Nikolskii, * 1 A. S. Denikin, * 2, * 3 A. A. Korsheninnikov Previously, we have reported on the measurement of the excitation energy spectrum of 7 He nucleus in the one-neutron d( 8 He,t) 7 He transfer reaction at the RIPS facility. The reaction was studied at forward laboratory angles θ lab ≈ (11 • -22 • ) using the 8 He 42 MeV/u beam and a deuteron target. 1) In the spectrum of tritons, a strong peak corresponding to the ground state (g.s.) of 7 He was observed. Figure 1 shows the angular distribution for the 7 He g.s. state extracted from the experimental data. The differential cross sections were analyzed with the DWBA approach using the code DWUCK5. 2) The initial parameters of Woods-Saxon optical model potentials (OP) were obtained in two steps: (i) we performed optical model fitting of the elastic scattering data from the 8 He(p, p) reaction at E lab = 15, 26, 32.5, 66, and 72 MeV/u 3) and obtained OP parameters for E lab = 42 MeV/u by linear interpolation; (ii) by analyzing the 6 Li(p, p), [START_REF] Skaza | [END_REF] Li(d, d) and 6 Li( 3 He, 3 He) scattering data 4) at 25 MeV/u, we determined the tendency of changes in OP under projectile variation and applied it to the case of the deuteron-induced reaction. For the final t-7 He channel, the OP for the 6 Li( 3 He, 3 He) system was used. Taking the spectroscopic factors (SF ) SF [t = n + d] = 1.5 [START_REF] Nemets | Nucleon Associations in Atomic Nuclei and Many-Nucleon Transfer Reactions[END_REF] and SF [ 8 He = n + 7 He] = 4 (assuming 4 neutrons in p 3/2 state, see also [START_REF] Skaza | [END_REF] ), we obtained a good description of the angular distribution with the normalization factor of 1.4-1.5. We found that no normalization was needed when the parameters for the imaginary part in the exit channel were slightly varied (within less than 10%) to fit to the absolute cross section. The result of the corresponding calculation is shown in Fig. 1 by a solid line, and the optical parameters are given in Table 1. The obtained OP were used to estimate the DWBA cross section for the d( 8 He, 3 He) 7 H reaction which was simultaneously measured in this experiment.

  (d,t) 7 He g.s.

Fig. 1 .

 1 Fig. 1. The angular distribution for the 8 He(d, t) 7 Heg.s. reaction. The error bars are statistical only. The curve is the DWBA calculation.

  , H. Otsu, H. Suzuki, * 4 K. Yoneda, H. Baba, K. Yamada, Y. Kondo, * 5 N. Aoi, M. S. Golovkov, * 2 A. S. Fomichev, * 2 S. A. Krupko, * 2 M. Kurokawa, E. A. Kuzmin, * 6 I. Martel, * 7 W. Mittig, * 8 T. Motobayashi, T. Nakamura, * 4 M. Niikura, * 9 S. Nishimura, A. A. Ogloblin, * 6 P. Roussel-Chomaz, * 8 A. Sanchez-Benitez, * 7 Y. Satou, * 5 S. I. Sidorchuk, * 2 T. Suda, S. Takeuchi, K. Tanaka, G. M. Ter-Akopian, * 2 Y. Togano, * 10 and M. Yamaguchi � Nuclear reactions, d( 8 He,t) 7 He, unstable nuclei �

Table 1 .

 1 7) Optical potential parameters.

	U (r) = -V0f (r, rV , aV ) + 4aW WD d dr f (r, rW , aW ), f (r, ri, ai) = {1 + exp[(r -riA 1/3 )/ai]} -1
		V0	rV	aV	WD	rW	aW
		(MeV) (fm)	(fm)	(MeV) (fm) (fm)
	d+ 8 He	97.2	1.11 0.817	10.5	2.33 0.45
	t+ 7 He	89.7	1.03 0.790	8.8	1.50 0.70
	n+ 7 He	*)	1.30 0.750	-	-	-
	n + d	*)	1.25 0.600	-	-	-
	*) varied to reproduce the experimental separation energy.