Rainer Dahlhaus
email: dahlhaus@statlab.uni-heidelberg.de.

Local inference for locally stationary time series based on the empirical spectral measure

Keywords: Empirical spectral measure, asymptotic normality, locally stationary processes, nonstationary time series

The time varying empirical spectral measure plays a major role in the treatment of inference problems for locally stationary processes. The properties of the empirical spectral measure and related statistics are studied -both when its index function is fixed or dependent on the sample size. In particular we prove a general central limit theorem. Several applications and examples are given including semiparametric Whittle estimation, local least squares estimation and spectral density estimation.

Introduction

In recent years several methods have been derived for locally stationary time series models, that is for models which can locally be approximated by stationary processes. Out of the large literature we mention the work of [START_REF] Priestley | Evolutionary spectra and non-stationary processes[END_REF] on oscillatory processes, [START_REF] Robinson | Nonparametric estimation of time-varying parameters[END_REF] on time varying regression models, [START_REF] Dahlhaus | Fitting time series models to nonstationary processes[END_REF] on locally stationary processes, [START_REF] Neumann | Wavelet thresholding in anisotropic function classes and applications to adaptive estimation of evolutionary spectra[END_REF] on wavelet estimation of evolutionary spectra, [START_REF] Nason | Wavelet processes and adaptive estimation of evolutionary wavelet spectra[END_REF] on a wavelet-based model of evolutionary spectra, and more recent work such as [START_REF] Davis | Structural Break Estimation for Nonstationary Time Series Models[END_REF] on piecewise stationary processes, [START_REF] Fryzlewicz | A Haar-Fisz technique for locally stationary volatility estimation[END_REF] on locally stationary volatility estimation, [START_REF] Moulines | On recursive estimation for locally stationary time varying autoregressive processes[END_REF] on recursive estimation for time varying autoregressive processes and [START_REF] Sakiyama | Discriminant analysis for locally stationary processes[END_REF] on discriminant analysis for locally stationary processes.

In this paper we emphasize the relevance of the empirical spectral measure for locally stationary time series. Here a generalized form of the empirical spectral measure is used which depends on arbitrary index functions that may even depend on the sample size n. The empirical spectral measure for fixed index function has been studied in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] where a functional limit theorem and an exponential inequality have been established. For Gaussian processes some (sometimes stronger) results are contained in [START_REF] Dahlhaus | Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes[END_REF]. In this paper we investigate the asymptotic properties of the empirical spectral measure with index functions depending on n and some related statistics.

In Section 2 we introduce the empirical spectral measure and the empirical spectral process indexed by functions and give a central limit theorem with fixed index functions. In Section 3 we prove the corresponding result with index functions φ n depending on n. In Section 4 related estimates such as sum estimates and shifted esimates are studied.

The paper contains many applications and smaller examples. An overview is given in Example 2.3 where also additional references may be found.

All proofs are delegated without further reference to Section 5.

The time varying empirical spectral measure

For completeness and for a deeper understanding we first present in this section the results on the time varying empirical spectral measure with fixed index functions (c.f. [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF]. The presentation includes all necessary assumptions since we will use them in later sections.

Locally stationary processes were introduced in Dahlhaus (1997) by using a time varying spectral representation. In contrast to this we use in this paper a time varying MA(∞)representation and formulate the assumptions in the time domain. The assumptions are much weaker than for example in [START_REF] Dahlhaus | Fitting time series models to nonstationary processes[END_REF] since the parameter curves are allowed to have jumps (see Remark 2.8 below). As in nonparametric regression we rescale the functions in time to the unit interval in order to achieve a meaningful asymptotic theory.

Let

V (g) = sup 0≤x 0 <...<xm≤1, m∈N m k=1 g(x k ) -g(x k-1 ) (1)
be the total variation of a function g on [0, 1], and for some κ > 0 let

(j) := 1, |j| ≤ 1 |j| log 1+κ |j|, |j| > 1.
(2)

Assumption 2.1 (Locally stationary processes) The sequence of stochastic processes X t,n (t = 1, . . . , n) is called a locally stationary process if X t,n has a representation

X t,n = ∞ j=-∞ a t,n (j) ε t-j (3)
satisfying the following conditions:

sup t,n |a t,n (j)| ≤ K (j) (with K not depending on n), (4)
and there exist functions a(•, j) : (0, 1] → R with

sup u |a(u, j)| ≤ K (j) , ( 5
) sup j n t=1 |a t,n (j) -a( t n , j)| ≤ K, (6)
V (a(•, j)) ≤ K (j) . ( 7
)
The ε t are assumed to be independent and identically distributed with Eε t ≡ 0 and Eε 2 t ≡ 1. In addition we assume in this paper that all moments of ε t exist. We set κ 4 := cum 4 (ε t ).

The existence of all moments is assumed for the central limit theorems and the exponential inequality which both are proved via the method of cumulants. Of course it would be very worthwhile to relax this assumption.

Let

f (u, λ) := 1 2π |A(u, λ)| 2 with A(u, λ) := ∞ j=-∞ a(u, j) exp(-iλj)
is the time varying spectral density, and

c(u, k) := π -π f (u, λ) exp(iλk) dλ = ∞ j=-∞ a(u, k + j) a(u, j) (8)
is the time varying covariance of lag k at rescaled time u. Usually cov(X t,n X t+k,n ) = c( t n , k) + O(n -1 ) (see Remark 2.8 below). A simple example of a process X t,n which fulfills the above assumptions is X t,n = φ( t n )Y t where Y t = Σ j a(j)ε t-j is stationary with |a(j)| ≤ K/ (j) and φ is of bounded variation. Another example are time varying ARMA (tvARMA) processes whose coefficient functions are of bounded variation (see Dahlhaus and Polonik, 2008, Proposition 2.3). Since we frequently use the tvAR -case in our examples below we state it explicitly.

Example 2.2 (tvAR-processes) Consider the system of difference equations

p j=0 α j ( t n )X t-j,n = σ( t n ) ε t (9)
where ε t are iid with Eε t = 0, E|ε t | < ∞, α 0 (u) ≡ 1 and α j (u) = α j (0) for u < 0. If all α j (•) as well as σ 2 (•) are of bounded variation and p j=0 α j (u)z j = 0 for all u and all 0 < |z| ≤ 1 + δ for some δ > 0 then there exists a solution of the form

X t,n = ∞ j=0 a t,n (j) ε t-j
which fulfills (4)-( 7) of Assumption 2.1. The time varying spectral density is given by Dahlhaus and Polonik, 2008, Proposition 2.3).

f (u, λ) = σ 2 (u) 2π 1 | p j=0 α j (u) exp(iλj)| 2 . (see
The empirical spectral process is defined by

E n (φ) := √ n F n (φ) -F (φ)
where

F (φ) := 1 0 π -π φ(u, λ) f (u, λ) dλ du (10)
is the generalized spectral measure and

F n (φ) := 1 n n t=1 π -π φ( t n , λ) J n ( t n , λ) dλ (11)
the empirical spectral measure with the pre-periodogram

J n t n , λ := 1 2π k:1≤[t+1/2±k/2]≤n X [t+1/2+k/2],n X [t+1/2-k/2],n exp(-iλk). ( 12
) If X [t+1/2+k/2],n X [t+1/2-k/2],n is regarded as a (raw-) estimate of c( t n , k) then J n ( t n , λ
) can be regarded as a (raw-) estimate of f ( t n , λ) -however, in order to become consistent J n ( t n , λ) needs to be smoothed in time and frequency direction. The pre-periodogram J n was first defined by [START_REF] Neumann | Wavelet thresholding in anisotropic function classes and applications to adaptive estimation of evolutionary spectra[END_REF].

Example 2.3 Many statistics occurring in the analysis of locally stationary time series are of the form F n (φ). We discuss several examples in this paper. Here we give an overview (K n always denotes a kernel function). [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] More complex examples are nonparametric maximum likelihood estimation under shape restrictions [START_REF] Dahlhaus | Nonparametric quasi-maximum likelihood estimation for Gaussian locally stationary processes[END_REF], model selection with a sieve estimator (Van Bellegem and Dahlhaus, 2006) and wavelet estimates [START_REF] Dahlhaus | Locally adaptive fitting of semiparametric models to nonstationary time series[END_REF]. Moreover F n (φ) occurs with local polynomial fits [START_REF] Kim | Nonparametric kernel estimation of evolutionary autoregressive processes[END_REF][START_REF] Jentsch | Asymptotik eines nicht-parametrischen Kernschätzers für zeitvariable autoregressive Prozesse[END_REF] and several statistics suitable for goodness of fit testing (a goodness-of-fit test based on a different statistic can be found in [START_REF] Sergides | Frequency domain tests of semiparametric hypotheses for locally stationary processes[END_REF]. These applications are quite involved.

1. φ(u, λ) = K n (u 0 -u) cos(λk) local covariance estimator Example 3.5 2. φ(u, λ) = K n (u 0 -u) K n (λ 0 -λ) spectral density estimator Example 4.2 3. φ(u, λ) = K n (u 0 -u) ∇f θ (u, λ) -1 local Whittle estimator Example 3.6 4. φ(u, λ) ≈ K n (u 0 -u) ∇f θ (u, λ) -1 local least squares Example 4.4 5. φ(u, λ) = ∇f θ (u, λ) -1 parametric Whittle estimator Example 3.7 in
6. φ(u, λ) = I [0,u 0 ] (u)-u 0 I [0,λ 0 ] (λ)
We mention that examples 1. -4. are examples with index functions φ n depending on n.

The asymptotic properties of F n (φ n ) in this case is the main topic of this paper.

We now state a central limit theorem for E n (φ) with index functions φ that do not vary with n. We use the assumption of bounded variation in both components of φ(u, λ). Besides the definition in (1) we need a definition in two dimensions. Let

V 2 (φ) = sup ,m j,k=1 |φ(u j , λ k ) -φ(u j-1 , λ k ) -φ(u j , λ k-1 ) + φ(u j-1 , λ k-1 )| : 0 ≤ u 0 < . . . < u ≤ 1; -π ≤ λ 0 < . . . < λ m ≤ π; , m ∈ N .
For simplicity we set

φ ∞,V := sup u V φ(u, •) , φ V,∞ := sup λ V φ(•, λ) , φ V,V := V 2 (φ) and φ ∞,∞ := sup u,λ |φ(u, λ)|.
Theorem 2.4 Suppose Assumption 2.1 holds and let φ 1 , . . . , φ d be functions with φ j ∞,V , φ j V,∞ , φ j V,V and φ j ∞,∞ being finite (j = 1, . . . , d). Then

E n (φ j ) j=1,...,d D → E(φ j ) j=1,...,d
where E(φ j ) j=1,...,d is a Gaussian random vector with mean 0 and

cov E(φ j ), E(φ k ) = 2π 1 0 π -π φ j (u, λ) [φ k (u, λ) + φ k (u, -λ)] f 2 (u, λ) dλ du (13) + κ 4 1 0 π -π φ j (u, λ 1 )f (u, λ 1 ) dλ 1 π -π φ k (u, λ 2 )f (u, λ 2 ) dλ 2 du.
This result is proved in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF], Theorem 2.5.

Remark 2.5 (Stationary processes/Model misspecification by stationary models)

The classical central limit theorem for the weighted periodogram in the stationary case can be obtained as a corollary: If φ(u, λ) = φ(λ) is time-invariant then

F n (φ) = π -π φ(λ) 1 n n t=1 J n ( t n , λ) dλ.
We have

1 n n t=1 J n ( t n , λ) = 1 n n t=1 1 2π k 1≤[t+0.5+k/2],[t+0.5-k/2]≤n X [t+0.5+k/2],n X [t+0.5-k/2],n exp(-iλk) = 1 2π n-1 k=-(n-1) 1 n n-|k| t=1 X t X t+|k| exp(-iλk) = 1 2πn n s=1 X s exp(-iλs) 2 = I n (λ) ( 14
)
where I n (λ) is the classical periodogram. Therefore F n (φ) is the classical spectral measure in the stationary case with the following applications:

(i) φ(u, λ) = φ(λ) = cos λk is the empirical covariance estimator of lag k; (ii) φ(u, λ) = φ(λ) = 1 4π ∇f -1 θ (λ)
is the score function of the Whittle-likelihood.

Theorem 2.4 gives the asymptotic distribution for these examples -both in the stationary case and in the misspecified case where the true underlying process is only locally stationary. If φ(u, λ) = φ(λ) is a kernel we obtain an estimate of the spectral density whose asymptotic distribution is a special case of Theorem 3.2 below (also in the misspecified case).

We now briefly mention an exponential inequality. Since this is a non-asymptotic result it holds regardless whether φ depends on n. Let

ρ 2,n (φ) := 1 n n t=1 π -π φ( t n , λ) 2 dλ 1/2 . ( 15
)
Theorem 2.6 (Exponential inequality) Suppose Assumption 2.1 holds with E|ε t | k ≤ C k ε for all k ∈ N for the ε t . Then we have for all η > 0

P | √ n F n (φ) -EF n (φ) | ≥ η ≤ c 1 exp -c 2 η ρ 2,n (φ) ( 16
)
with some constants c 1 , c 2 > 0 independent of n.

This result is proved in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF], Theorem 2.7. There exist several variants of this result -for example in the Gaussian case it is possible to omit the √ • in ( 16) or to prove a Bernstein-type inequality which is even stronger (cf. Dahlhaus and Polonik, 2006, Theorem 4.1).

Furthermore, a maximal inequality may be proved under conditions on the metric entropy of the corresponding function class. We refer to [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF], Theorem 2.9. Applications of the maximal inequality are for example uniform rates of convergence for different type of estimates. We omit details.

By using a maximal inequality tightness of the empirical spectral process may be proved leading to functional central limit theorem for the empirical spectral process where the process is indexed by a function class (cf. Dahlhaus and Polonik, 2008, Theorem 2.11). Furthermore a Glivenko Cantelli type result for the empirical spectral process can be obtained (Theorem 2.12).

Example 2.7 (Testing for stationarity) The idea for a test of stationarity is to test whether the time varying spectral density f (u, λ) is constant in u. This is for example achieved by the test statistic

√ n sup u∈[0,1] sup λ∈[0,π] F n (u, λ) -u F n (1, λ)
where

F n (u, λ) := 1 n [un] t=1 λ 0 J n ( t n , µ) dµ
is an estimate of the integrated time frequency spectral density F (u, λ) :=

u 0 λ 0 f (v, µ) dµdv, and
u F n (1, λ) = u λ 0 I n (µ) dµ
is the corresponding estimate of F (u, λ) under the hypothesis of stationarity where f (v, µ) = f (µ). Under the hypothesis of stationarity we have

F (u, λ) -u F (1, λ) = 1 0 λ 0 I [0,u] (v) -u f (µ) dµ dv = 0 and therefore √ n F n (u, λ) -u F n (1, λ) = E n (φ u,λ ) with φ u,λ (v, µ) = I [0,u] (v) -u I [0,λ] (µ). We now need functional convergence of E n (φ u,λ ).
Convergence of the finite dimensional distributions follows from Theorem 2.4 above. Tightness and therefore the functional convergence follows from Theorem 2.11 of [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] (it is straightforward to check the conditions of this theorem on the metric entropy).

As a consequence we obtain under the null hypothesis

√ n F n (u, λ) -u F n (1, λ) u∈[0,1],λ∈[0,π] D → E u, λ u∈[0,1],λ∈[0,π]
where E u, λ is a Gaussian process with mean 0 and covariance structure cov

E(u 1 , λ 1 ), E(u 2 , λ 2 ) = 2π 1 0 I [0,u 1 ] (v) -u 1 I [0,u 2 ] (v) -u 2 min{λ 1 ,λ 2 } 0 f 2 (µ) dµ dv + κ 4 1 0 I [0,u 1 ] (v) -u 1 I [0,u 2 ] (v) -u 2 λ 1 0 f (µ 1 ) dµ 1 λ 2 0 f (µ 2 ) dµ 2 dv = min{u 1 , u 2 } -u 1 u 2 2π min{λ 1 ,λ 2 } 0 f 2 (µ) dµ + κ 4 λ 1 0 f (µ 1 ) dµ 1 λ 2 0 f (µ 2 ) dµ 2 .
If κ 4 = 0 (Gaussian case) and f (µ) = c this is the Kiefer-Müller process. However, for general f it is a difficult and unsolved task to calculate or estimate the limit distribution and in particular the distribution in (2.7). This may be done by transformations (like U por T p -type transforms) and/or by finding an adequate bootstrap method.

We conjecture that the same result can be obtained if we use instead of F n (u, λ) the more realistic sum statistic

F Σ n (u, λ) := 2π n 2 [un] t=1 [λn] j=1 J n ( t n , λ j )
with λ j = 2πj n . In this case convergence of finite dimensional distributions follows from Theorem 4.1 below while the derivation of tightness still is an open problem.

We mention that [START_REF] Paparoditis | Testing temporal constancy of the spectral structure of a time series[END_REF] has given a complete solution of this testing problem by using spectral density type statistics.

Remark 2.8 (The bounded variation assumption) The assumption of bounded variation in time direction (7) of Assumption 2.1 allows for jumps in the parameter curves. We recall that a function is of bounded variation if it is the difference of two bounded monotone functions. This implies that the function may have countably many jumps with absolutely summable jump-sizes. In between the function is continuous. Assumption 2.1 implies bounded variation for many other curves of interest. For example it has been proved in Proposition 5.4 of [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] that the covariances c(u, k) are of bounded variation in u with

V c(•, k) ≤ K (k) .
It can also be shown that f (u, λ) is of bounded variation in u for each λ. Furthermore, Proposition 5.4 of [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] implies

n t=1 cov X t,n X t+k,n -c t n , k ≤ K. ( 17
)
This result also reflects bounded variation. Heuristically it is due to the following:

cov X t,n X t+k,n -c t n , k =    O( 1 n ), if c(u, k) is continuous in t n , t+k n ; O(1), if c(u, k) has a jump in t n , t+k n . More precisely, it can e.g. be shown that cov X t,n X t+k,n -c t n , k = O(1/n) uniformly in k if all a j fulfill the Lipschitz-continuity assumption a j (u) -a j (v) ≤ K |u -v| (j) for all u, v ∈ t n , t + k n (cf. Assumption 2.1). The result (17) also holds for cov X [t+1/2+k/2],n , X [t+1/2-k/2],n .
We do not take full advantage of the bounded variation assumptions in the present paper.

The main results Theorems 2.4, 3.2, 4.1 and 4.3 are proved under these weak assumptions while nearly all examples deal with local kernel-type-estimates which require additional smoothness assumptions. Possible applications are estimation of jump-locations and jumpsizes, testing for breakpoints (in particular the behaviour of tests under the alternative) etc. These applications need to be investigated in future work.

A central limit theorem with index functions depending on n

We now prove a central limit theorem for F n (φ n )-F (φ n ) with index functions φ n depending on n. In addition we extend the definitions of Section 2 to tapered data

X (hn) t,n := h n t n • X t,n
where

h n : (0, 1] → [0, ∞) is a data taper (with h n (•) = I (0,1] (•)
being the nontapered case of Section 2). The main reason for introducing data-tapers is to include segment estimates -see the discussion below.

As before the empirical spectral measure is defined by

F n (φ) = F (h n ) n (φ) := 1 n n t=1 π -π φ( t n , λ) J (h n ) n ( t n , λ) dλ (18)
now with the tapered pre-periodogram

J (hn) n t n , λ = 1 2π k:1≤[t+1/2±k/2]≤n X (h n ) [t+1/2+k/2],n X (h n ) [t+1/2-k/2],n exp(-iλk) (19)
we mention that in some cases a rescaling may be necessary for J

(h n ) n (u, λ) to become a pre-estimate of f (u, λ) -an obvious example for this is h n (u) = (1/2) I (0,1] (u) . F (φ) is the theoretical counterpart of F n (φ) F (φ) = F (h n ) (φ) := 1 0 h 2 n (u) π -π φ(u, λ)f (u, λ) dλ du. ( 20
)
We stress that for h n (u) = I (0,1] (u) all definitions are the same as in Section 2. This is one important special case.

Another case of importance is h

(u 0 ) n ( t n ) := k u 0 -t/n b n with k having compact support on [-1 2 , 1 2 ] and b n → 0. If in addition φ(u, λ) = ψ(λ)
we obtain in this case as in ( 14)

F n (φ) = π -π ψ(λ) 1 n n t=1 J (h n ) n t n , λ dλ = π -π ψ(λ) 1 2πn n s=1 X (h n ) s exp(-iλs) 2 dλ = H 2,n n π -π ψ(λ) I (h n ) n (λ) dλ (21)
with the tapered periodogram

I (hn) n (λ) := 1 2πH 2,n n s=1 X (hn) s exp(-iλs) 2 (22)
and H 2,n := n t=1 h n (t/n) 2 (which is in this example proportional to b n n). This means that F n is a weighted mean of the tapered periodogram on the segment (u

0 -b n /2)n ≤ t ≤ (u 0 + b n /2)n. For k(•) = I [-1 2 , 1 2 ] (•) we have the classical nontapered periodogram.
The last example suggests to use 1 H 2,n instead of 1 n in (18) as a norming constant. However, this is not always the right choice (as can be seen from case (ii) in Remark 3.4).

We need the following assumption.

Assumption 3.1 The data taper h n : (0, 1] → [0, ∞) fulfills sup n V (h n ) ≤ C and sup u,n h n (u) ≤ C for some C < ∞. log h n (•) is assumed to be concave.
The assumption that log h n (•) is concave is very mild (note that even log(x m ) is concave).

It turns out that in the above situation the rate of converge of the empirical spectral measure becomes √ n/ρ

(h n ) 2 (φ n ) where ρ (h n ) 2 (φ) := 1 0 h 4 n (u) π -π φ(u, λ) 2 dλ du 1/2
.

Therefore we can embed this case into the situation treated in the last section by studying the convergence of

E (h n ) n φ n ρ (h n ) 2 (φ n ) = √ n ρ (h n ) 2 (φ n ) F n (φ n ) -F (h n ) (φ n ) . Furthermore, let c (h n ) E (φ j , φ k ) := 2π 1 0 h 4 n (u) π -π φ j (u, λ) φ k (u, λ) + φ k (u, -λ) f 2 (u, λ) dλ du (23) + κ 4 1 0 h 4 n (u) π -π φ j (u, λ 1 )f (u, λ 1 )dλ 1 π -π φ k (u, λ 2 )f (u, λ 2 )dλ 2 du.
Theorem 3.2 Suppose Assumptions 2.1 and 3.1 hold. Furthermore, let φ n1 , . . . , φ nd be functions with

1 + φ n ∞,∞ + (log n) φ n ∞,V + φ n V,∞ + (log n) φ n V,V √ n ρ (hn) 2 (φ n ) = o(1). ( 24
)
If the limit

Σ j,k := lim n→∞ c (hn) E (φ nj , φ nk ) ρ (hn) 2 (φ nj ) ρ (hn) 2 (φ nk ) (25)
exists for all j, k = 1, . . . , d then

√ n ρ (h n ) 2 (φ nj ) F n (φ nj ) -F (h n ) (φ nj ) j=1,...,d D → N (0, Σ). ( 26
)
Remark 3.3 (i) (Bias) In addition there exists the bias term √ n ρ

(h n ) 2 (φ n ) F (h n ) (φ n ) -lim n→∞ F (h n ) (φ n )
(see the examples below). The magnitude of this bias depends on the smoothness of the time varying spectral density. In this paper we usually require conditions such that this bias is of lower order. Of course it is an important task to study the relation between bias and variance and to investigate in that context the problem of adaptation, e.g. the choice of the bandwidth parameters in the examples below. However, this is beyond the scope of the present paper.

(ii) (Fixed index functions) Theorem 2.4 for fixed index functions φ j follows from the above result as a corollary.

(iii) (Stationary case) With φ n (u, λ) = ψn (λ) the result also covers the stationary casee.g. for kernel estimates of the spectral density (see ( 21)).

Remark 3.4 (Typical applications) A typical application of this result is the case of (kernel type) local estimators which can be constructed by using kernels, data-tapers or a combination of both: We now show how Theorem 3.2 leads to the asymptotic distribution for these estimates:

(i) φ n (u, λ) = 1 bn K( u 0 -u bn ) ψ(λ) h n (•) = I (0,1] (•) (ii) φ n (u, λ) = 1 b n K( u 0 -u b n ) ψ(λ) h n (u) = I [u 0 -b n /2,u 0 +b n /2] (u) (iii) φ n (u, λ) = ψ(λ) h n ( t n ) = k u 0 -t/n
(i) We assume that K is of bounded variation with compact support on [-

1 2 , 1 2 ] and b n → 0, b n n → ∞. ψ(λ) is also of bounded variation. Since V 1 b n K( u 0 -• b n ) = O( 1 b n ) it is obvious that φ n ∞,V , φ n V,∞ , φ n V,V and φ n ∞,∞ are of order O( 1 bn ). Furthermore, ρ (h n ) 2 (φ n ) = ρ 2 (φ n ) = 1 b n K 2 (x) dx |ψ(λ)| 2 dλ 1/2 (27) that is (24) is fulfilled if nb (log n) 2 . For f (•, λ) continuous at u 0 we have c (hn) E (φ nj , φ nk ) = 2π 1 0 1 b 2 n K u 0 -u b n 2 π -π ψ j (λ) ψ k (λ) + ψ k (-λ) f 2 (u, λ) dλ du + κ 4 1 0 1 b 2 n K u 0 -u b n 2 π -π ψ j (λ 1 )f (u, λ 1 )dλ 1 π -π ψ k (λ 2 )f (u, λ 2 )dλ 2 du ∼ 1 b n K 2 (x) dx 2π π -π ψ j (λ) ψ k (λ) + ψ k (-λ) f 2 (u 0 , λ) dλ + κ 4 π -π ψ j (λ 1 )f (u 0 , λ 1 ) dλ 1 π -π ψ k (λ 2 )f (u 0 , λ 2 ) dλ 2 =: 1 b n Γ jk
that is ( 25) is also fulfilled and we obtain from ( 26) the central limit theorem

b n n F n (φ nj ) -F (h n ) (φ nj ) j=1,...,d D → N (0, Γ). ( 28
)
One may also allow for different u 0 = u j and u 0 = u k . Since

1 b n K u j -u b n 1 b n K u k -u b n du = o( 1 b n )
for u j = u k we obtain by straightforward calculations that the corresponding covariance structure tends to 0.

(ii) The additional taper h n (u) = I [u 0 -bn/2,u 0 +bn/2] (u) implies that we use only data from the interval [u 0b n /2, u 0 + b n /2]. We obtain in this case

ρ (h n ) 2 (φ n ) = 1 0 h 4 n (u) 1 b 2 n K u 0 -u b n 2 du π -π |ψ(λ)| 2 dλ 1/2 = 1 0 1 b 2 n K u 0 -u b n 2 du π -π |ψ(λ)| 2 dλ 1/2
, i.e. we have the same ρ

(h n ) 2 (φ n ) as above. Furthermore, c (h n ) E (φ n , φ n )
is the same. Thus we obtain the same asymptotic distribution and the same rate of convergence.

(iii) If K(•) = k(•) 2 we obtain in this case 1 b n ρ (h n ) 2 (φ n ) = 1 0 1 b 2 n K u 0 -u b n 2 du π -π |ψ(λ)| 2 dλ 1/2
i.e. we obtain again the same expression. Furthermore,

1 b 2 n c (h n ) E (φ nj , φ nk ) is the same as c (h n )
E (φ nj , φ nk ) above. Thus we have again the same asymptotic distribution and the same rate of convergence.

Example 3.5 (Local covariance estimator)

If we set ψ(λ) = cos(λk) in the last remark then estimate (i) becomes the kernel covariance estimator of c(u 0 , k)

c n (u 0 , k) := F n (φ n,u 0 ,k ) = 1 n n t=1 1 b n K u 0 -t/n b n π -π cos(λk) J n t n , λ dλ = 1 n t 1 b n K u 0 -t/n b n X [t+1/2+k/2],n X [t+1/2-k/2],n . (29)
Apart from the bias (which is treated below) (28) gives

b n n c n (u k , k) -c(u k , k) k=1,...,d D → N (0, Γ) (30) with Γ k, = δ u k ,u K 2 (x)dx 2π π -π cos(λk) cos(λ )f (u k , λ) 2 dλ + κ 4 c(u k , k) c(u k , ) .
By using Parseval's formula and cos(λk

) = 1 2 [exp(iλk) + exp(-iλk)] the term in the [...] - brackets is the same as ∞ j=-∞ c(u k , j) c(u k , j -k -) + c(u k , j) c(u k , j -k + ) + κ 4 c(u k , k) c(u k ,
) which is the usual asymptotic variance of the covariance-estimator (cf. [START_REF] Fuller | Introduction to Statistical Time Series[END_REF], Theorem 6.4.1).

We finally have to treat the bias. We have

F (φ n,u 0 ,k ) -c(u 0 , k) = 1 0 π -π 1 b n K u 0 -u b n f (u, λ) -f (u 0 , λ) cos(λk) dλ du = O(b 2 n ) if f (u, λ
) is differentiable in u with Lipschitz continuous derivative in some neighborhood of u 0 and the Lipschitz constant is uniform in λ. We therefore obtain (30

) if √ b n n b 2 n → 0 (i.e. for b n n -1/5 ).
The same follows for the estimates (ii) and (iii). We mention that the estimate (iii) with ψ(λ) = cos(λk) is just the Fourier transform of the classical tapered periodogram.

Example 3.6 (Curve estimation by local Whittle-estimates) Whittle estimates were introduced by [START_REF] Whittle | Estimation and information in stationary time series[END_REF] for stationary processes and were generalized by [START_REF] Dahlhaus | A likelihood approximation for locally stationary processes[END_REF] to locally stationary processes. Parametric Whittle estimates for locally stationary processes have been discussed in [START_REF] Dahlhaus | A likelihood approximation for locally stationary processes[END_REF] and in combination with the empirical spectral measure in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF], Example 3.1.

Here we consider semiparametric curve estimation where the time varying spectral density is of the form

f (u, λ) = f θ(u) (λ) with θ(u) ∈ Θ ⊆ R d for all u ∈ [0, 1]. An example are tvAR-processes. Let θ n (u) := argmin θ∈Θ L n (u, θ) with L n (u, θ) := 1 4π 1 n n t=1 1 b n K u -t/n b n π -π log 4π 2 f θ (λ) + J n ( t n , λ) f θ (λ) dλ. (31)
In case of a tvAR(p)-process θ n (u) is the solution of the local Yule-Walker equations: Let

C n (u) = ( c n (u, 1), . . . , c n (u, p)) and Σ n (u) = { c n (u, i -j)} i,j=1,...,p with the covariance estimator c n (u, k) as in (29). If θ n (u) = ( α 1 (u), . . . , α p (u), σ 2 (u)) then it is not difficult to show that α 1 (u), . . . , α p (u) = -Σ n (u) -1 C n (u)
and

σ 2 (u) = c n (u, 0) + p k=1 α k (u) c n (u, k).
The basic steps for proving asymptotic normality of θ n (u) are well known. We mention that the empirical spectral measure plays a key role in the proof since the difference of the likelihood and its first and second order derivatives can be represented with the spectral measure. Large parts of the proof (in particular consistency of θ n (u)) follow in the same way as in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF], Theorem 4.1 where a uniform rate of convergence for θ n (u)θ 0 (u) 2 has been derived. The final step of the proof is to show asymptotic normality of the score function. We have with

∇ i := ∂ ∂θ i b n n ∇ i L n u, θ 0 (u) = b n n F n (φ n,u,i ) -F (φ n,u,i ) + O 1 √ b n n (32) + b n n 1 4π 1 0 π -π 1 b n K u -v b n f θ 0 (v) (λ) -f θ 0 (u) (λ) ∇ i f -1 θ (λ) |θ=θ 0 (u) dλ dv where φ n,u,i (v, λ) := 1 b n K( u-v b n ) 1 4π ∇ i f -1 θ (λ) |θ=θ 0 (u)
(for tvAR(p)-processes this score function is simpler -cf. ( 50)). If θ 0 (u) is differentiable with Lipschitz continuous derivative in some neighborhood of u and f θ (λ) is differentiable in θ with uniform Lipschitz continuous derivative in some neighborhood of θ 0 (u) then the last term is of lower order provided b n n -1/5 .

As in Remark 3.4 we therefore obtain from Theorem 3.2 (cf.( 27))

b n n ∇L n u, θ 0 (u) D → N 0, K 2 (x)dx Γ(u) + Λ(u) (33) with Γ(u) k, := 1 4π π -π ∇ k log f θ (λ) |θ=θ 0 (u) ∇ log f θ (λ) |θ=θ 0 (u) dλ and Λ(u) k, := κ 4 1 4π π -π ∇ k log f θ (λ) |θ=θ 0 (u) dλ 1 4π π -π ∇ log f θ (λ) |θ=θ 0 (u) dλ .
Furthermore, under appropriate smoothness assumptions on f and θ, it can be shown that

∇ 2 L n u, θn P → Γ(u)
for every θn with θn

P → θ 0 (u) (34)
which implies with a Taylor expansion and consistency of θ n (u)

b n n θ n (u) -θ 0 (u) D → N 0, K 2 (x)dx Γ(u) -1 Γ(u) + Λ(u) Γ(u) -1 . (35)
Similarly, it can be shown that √ b n n θ n (u)θ 0 (u) and √ b n n θ n (v)θ 0 (v) are jointly asymptotically normal with zero limit covariance.

f θ (λ) dλ.
where h u,n ( t n ) := k u-t/n bn . This estimate has the same asymptotic properties provided k(•) 2 = K(•). We omit details.

Related estimators

Many estimators are "approximately" of the form discussed in Sections 2 and 3. In this section we derive the asymptotic properties of such estimators rigorously and give examples beyond the examples already discussed in the last sections.

Sum estimators

We first consider the empirical spectral process with the integral statistic F n (φ) being replaced by the sum statistic

F Σ n (φ) := 2π n 2 n t=1 n j=1 φ( t n , λ j ) J (h n ) n ( t n , λ j ) ( 36
)
where λ j := 2πj n . This estimator has already been used in Example 2.7 (testing for stationarity). The empirical spectral process in this case is defined by

E Σ n (φ) := √ n F Σ n (φ) -F (φ) . ( 37
)
Theorem 4.1 (i) Suppose Assumptions 2.1 and 3.1 hold. Furthermore, let φ n be a function possibly depending on n. Then

n ρ (hn) 2 (φ n ) 2 E F Σ n (φ n ) -F n (φ n ) 2 ≤ K (log n) φ n 2 ∞,V n ρ (hn) 2 (φ n ) 2 . ( 38
f n (u, λ) := 2π n 2 n t=1 n j=1 1 b t K t u -t/n b t 1 b f K f λ -λ j b f J (hn) n t n , λ j (39)
with different bandwidths and kernels in frequency and time direction. We have

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT f n (u, λ) = F Σ n (φ n,u,λ ) with φ n,u,λ (v, µ) := 1 b t K t ( u-v b t ) 1 b f K f ( λ-µ b f
). Below we deduce from Theorem 4.1 that

nb t b f f n (u i , λ i ) -f (u i , λ i ) i=1,...,d D → N (0, Γ) (40) with Γ i,j = 2π f (u 1 , λ 1 ) 2 δ u i ,u j δ λ i ,λ j K 2 t (x) dx K 2 f (x)
dx (and an additional factor 2 if λ i = 0). The situation is the same as in Remark 3.4 but in addition with a kernel in frequency direction, i.e.

ψ(µ) = 1 b f K f ( λ-µ b f ).
In particular we are having again 3 different choices for constructing estimates (by using kernels and data-tapers). The 3 estimates are numerically different but with the same asymptotic distribution. Besides (39) we mention the ordinary smoothed periodogram

f n (u, λ) := 2π n n j=1 1 b f K f λ -λ j b f I (h u,n ) n (λ) (41) with data-taper h n ( t n ) = k u-t/n b t , k 2 (•) = K t (•) and H 2,n = n t=1 h 2 n ( t n ) ≈ b t n.
We now derive (40) by checking the assumptions of Theorem 4.1. We assume that the kernels

K t , K f have compact support on [-1 2 , 1 2
] and are of bounded variation with

1/2 -1/2 x K(x) dx = 0 and 1/2 -1/2 K(x) dx = 1. Furthermore let b t , b f → 0 and nb t b f (log n) 2 as n → ∞. The kernel K f is periodically extended (i.e. K f ( x b f ) = K f ( x±2π b f
) due to periodicity of f ) to include estimates for frequencies around ±π.

It is obvious that

φ n,u,λ ∞,V , φ n,u,λ V,∞ , φ n,u,λ V,V and φ n,u,λ ∞,∞ are of order O( 1 b t b f ). Furthermore, ρ 2 (φ n,u,λ ) 2 = 1 b t b f K 2 t (x) dx K 2 f (x) dx that is (24) is fulfilled if nb t b f (log n) 2
. We now check (25). By using

1 b t K t u 1 -v b t 1 b t K t u 2 -v b t dv = o 1 b t
for u 1 = u 2 (and the same for K f ) we obtain for continuous f (u, λ) by straightforward calculations in the case λ 1 = 0

c (I (0,1] ) E (φ n,u 1 ,λ 1 , φ n,u 2 ,λ 2 ) = 2π f (u 1 , λ 1 ) 2 δ u 1 ,u 2 δ λ 1 ,λ 2 1 b t b f K 2 t (x) dx K 2 f (x) dx + o( 1 b t b f )
(the term with κ 4 is of lower order) and therefore

lim n→∞ c (I (0,1] ) E (φ n,u 1 ,λ 1 , φ n,u 2 ,λ 2 ) ρ 2 (φ n,u 1 ,λ 1 ) ρ 2 (φ n,u 2 ,λ 2 ) = 2π f (u 1 , λ 1 ) 2 δ u 1 ,u 2 δ λ 1 ,λ 2 .
If λ 1 = 0 the same holds with a factor 2 on the right hand side. Thus we obtain

nb t b f f n (u i , λ i ) -F (φ n,u i ,λ i ) i=1,...,d D → N (0, Γ)
with Γ as above. We now treat the bias. We have

F (φ n,u,λ ) -f (u, λ) = 1 0 π -π 1 b t K t u -v b t 1 b f K f λ -µ b f f (v, µ) -f (u, λ) dµ du.
Suppose f (u, λ) is twice differentiable in u and λ with uniformly bounded derivatives

∂ 2 ∂u 2 f (u, λ), ∂ 2 ∂u ∂λ f (u, λ) and ∂ 2 ∂λ 2 f (u, λ). Then a Taylor expansion yields F (φ n,u,λ )-f (u, λ) = O p (b 2 t + b 2 f ) and therefore the CLT in (40) holds if nb t b f (b 2 t + b 2 f ) → 0 (e.g. for b t = b f n -1/6 ).

Shifted estimates

By using Parseval's equality F n (φ) can be written in the form

F n (φ) = 1 2πn n t=1 k φ( t n , -k) X (h n ) [t+1/2+k/2],n X (h n ) [t+1/2-k/2],n (42)
with φ(u, k) := π -π φ(u, λ) exp(iλk) dλ. (43)
There exist several quadratic forms which are stochastically equivalent to F n (φ), for example the statistic 1 2πn

n t=1 k : 1≤t-k≤n φ( t n , -k) X (hn) t,n X (hn) t-k,n . (44)
This is equal to

F * n (φ) = 1 2πn n t=1 k φt,-k X (hn) [t+1/2+k/2],n X (hn) [t+1/2-k/2],n with φt,-k = φ( [t+1/2+k/2] n , -k) which implies t φt,k -φ( t n , k) ≤ K min{|k| + 1, n} V ( φ(•, k)) and sup t | φt,k | ≤ K sup u | φ(u, k)|.
By using Lemma 5.1 we therefore have

t φt,k -φ( t n , k) ≤ K min{|k|, n} |k| φ V,V k = 0 , t φt,0 -φ( t n , 0) ≤ K φ V,∞ , (45) and sup
t | φt,k | ≤ K |k| φ ∞,V k = 0 , sup t | φt,0 | ≤ K φ ∞,∞ . (46)
We will use these technical assumptions since they cover several other cases as well.

Let

E * n (φ) := √ n F * n (φ) -F (φ) (47)
be the corresponding empirical spectral process.

Theorem 4.3 (i) Suppose Assumptions 2.1 and 3.1 hold. Furthermore, let φ n be a function possibly depending on n with Fourier-transform φn (u, k), and φn,t,k be complex numbers such that ( 45) and ( 46) hold (with φ replaced by φ n and K independent of n) and (φ n ) .

(φ n ) 2 E F * n (φ n ) -F n (φ n ) 2 ≤ K 1 + φ n ∞,∞ φ n V,∞ + 1 + log n φ n ∞,V φ n V,V n ρ
Furthermore, we have with ρ ∞ (φ) := ∞ k=-∞ φ(k)

ρ ∞ φ * n ρ (h n ) 2 (φ n ) ≤ 2π φ n ∞,∞ ρ (h n ) 2 (φ n ) + K φ n ∞,V ρ (h n ) 2 (φ n ) ∞ k=1 1 |k| exp(-k 2 b 2 /2) ≤ K φ n ∞,∞ + log(b -1 ) φ n ∞,V ρ (hn) 2 (φ n ) .
For b = 1/n (24) implies that the above terms are all of order o( √ n). In addition

ρ 2,n (φ) 2 ≤ ρ 2 (φ) 2 + 1 n ρ ∞ (φ) sup k V φ(•, k) leading to ρ (hn) 2,n φ * n ρ (h n ) 2 (φ n ) 2 ≤ ρ (h n ) 2 (φ * n ) 2 ρ (h n ) 2 (φ n ) 2 + K φ n ∞,∞ + log(b -1 ) φ n ∞,V √ n ρ (h n ) 2 (φ n ) φ n V,V √ n ρ (h n ) 2 (φ n ) ≤ K.
Therefore the remainder term R n in Lemma 5.6(i) and the higher cumulants in Lemma 5.7(i) in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] (φ nj ) j=1...,d .

From Lemma 5.5(i) in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] we obtain with (k) as in (2) and Lemma 5.1

√ n EF n φ n ρ (h n ) 2 (φ n ) -F φ n ρ (h n ) 2 (φ n ) ≤ K φ n ∞,∞ + (log n) φ n ∞,V + φ n V,∞ + φ n V,V √ n ρ (hn) 2 (φ n ) = o(1)
which finally proves Theorem 3.2.

Furthermore, we obtain from (57), ( 58) and ( 46)

n EF * n (φ) -EF n (φ) 2 ≤ K n k 1 (k) t φt,k -φ( t n , k) 2 ≤ K φ V,∞ + φ V,V n (63)
that is the result.(ii) and (iii) are immediate consequences.

Acknowledgement: The author is very grateful to two anonymous referees whose comments have led to substantial improvements. This work has been supported by the Deutsche Forschungsgemeinschaft (DA 187/15-1).

  and k(•) are kernel functions and b n is the bandwidth. Dependent on the function ψ(λ) this leads to different applications (see the examples below). If K(•) = k(•) 2 then the resulting estimates all have the same asymptotic properties -see below. However, they are numerically different and their finite sample behaviour needs to be investigated. For example the last estimate is a stationary estimate on a segment which uses the classical tapered periodogram or the nontapered periodogram for k(•) = I [-1/2,1/2] (•) .

  The above estimator corresponds to case (i) in Remark 3.4. Case (iii) in Remark 3.4 leads instead to the classical tapered Whittle likelihood on the segment, that is θn

  )(ii) (Fixed index functions) Thus under the assumptions of Theorem 2.4√ n F Σ n (φ) -F n (φ) P → 0 and as a consequenceE Σ n (φ j ) j=1,...,d D → E(φ j ) j=1,...,dwhere E(φ j ) j=1,...,d has the same Gaussian distribution as in Theorem 2.4.(iii) (Varying index functions) Thus under the assumptions of Theorem 3.2E Σ n (φ j /ρ (hn) 2 (φ nj )) j=1,...,d has the same asymptotic distribution as given in Theorem 3.2. Example 4.2 (Spectral density estimates) Consider a kernel estimate of the time-varying spectral density

  (Fixed index functions) Thus under the assumptions of Theorem 2.4√ n F * n (φ) -F n (φ) P → 0 and as a consequence E * n (φ j ) j=1,...,d D → E(φ j ) j=1,...,dwhere E(φ j ) j=1,...,d has the same Gaussian distribution as in Theorem 2.4.(iii) (Varying index functions) Thus under the assumptions of Theorem 3.2E * n (φ j /ρ (h n ) 2(φ nj )) j=1,...,d has the same asymptotic distribution as given in Theorem 3.2.

  converge to zero.With (25) this implies the convergence of all cumulants and therefore

	√ n F n	φ * nj (hn) ρ 2 (φ nj )	-EF n	φ * nj (hn) ρ 2 (φ nj ) j=1...,d	D → N (0, Σ)
	and with (53) the same CLT for	√ n F n φ nj /ρ	(hn) 2	(hn) 2

(φ nj ) -EF n φ nj /ρ

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

Example 4.4 (Local least squares estimates) Suppose we want to fit a tvAR(p)-model

by the local least squares estimate ( α * 1 (u), . . . , α * p (u)) := argmin

α j X t-j,n 2 and σ * 2 (u) := 1

Both equations can be summarized with θ(u) = α 1 (u), . . . , α p (u), σ 2 (u) to

As in Example 3.6 the asymptotic distribution is determined by the score function ∇L * n u, θ 0 (u) . Below we show that this is of the form F * n (φ n ) with ( 45) and ( 46) while the score function of the local Whittle-estimator from Example 3.6 is the corresponding F n (φ n ) from (42). We have in the special situation of tvAR(p)-processes due to Kolmogorov's equation (cf. (32))

. The first term is the same as in ∇ i L * n u, θ 0 (u) . We now calculate F n (φ n,u,i ). The time varying spectral density is given by f

ACCEPTED MANUSCRIPT

Therefore we have with α 0 = 1, α = α 0, (u) ( = 1, . . . , p), α = 0 ( / ∈ {0, . . . , p}) and

For the local least squares estimate we have for i = 1, . . . , p

It is now straightforward to check ( 45) and ( 46). Thus Theorem 4.3 implies the equivalence of F n (φ n,u,i ) and F * n (φ n,u,i ). In a similar way this follows for F n (φ n,u,0 ) and F * n (φ n,u,0 ) (the score with respect to σ 2 ). As a consequence the convergence result (33) also holds for ∇L * n u, θ 0 (u) . In addition convergence of the second derivative as in (34) needs to be proved for L * n which again follows from the convergence of the empirical spectral measure. This implies that the convergence result (35) also holds for θ * n (u).

ACCEPTED MANUSCRIPT

Example 4.5 (Symmetrized estimator)

We have

Of course it is more natural to use a 'symmetrized pre-periodogram' with the second summand being

instead. It is easy to see that the resulting estimator is of the form

) , k odd, and it is easy to check assumptions ( 45) and ( 46) for this φt,k . Theorem 4.3 now implies that all results also hold with the symmetrized pre-periodogram.

Proofs

where φ R (u, dλ) is the signed measure corresponding to φ R (u, λ) := lim µ↓λ φ(u, µ) (since φ is of bounded variation in λ the limit exists; for the same reason φ R (u, dλ) is a signed measure). This implies sup

The other results follow similarly.

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

ACCEPTED MANUSCRIPT

(Proof of Theorem 3.2)

The result is established via the method of cumulants. In particular we want to adapt cumulant calculations in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] where the cumulants have been calculated under stronger smoothness assumptions. The trick is to smooth φ n (u, λ) in λ -direction and to prove asymptotic normality instead for the resulting sequence of approximations:

with φ n (u, k) being the Fourier transform as in ( 43) and k b (k) = exp(-k 2 b 2 /2). Thus we obtain from Lemma 5.7(i) in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] and Lemma 5.1

We now prove a CLT for

(φ nj ) j=1...,d by applying Lemma 5.6(i) and Lemma 5.7(i) of [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF]. We obtain from (51) and Lemma 5.1 with φ(k

ACCEPTED MANUSCRIPT

We now prove the properties of the sum estimator F Σ n (φ) from Section 4.1. It is easy to show that

with φΣ (

Recall that

We need the following lemma.

Lemma 5.3 We have for |k| < n

Proof. The proof can e.g. be found in [START_REF] Dahlhaus | On the asymptotic distribution of Bartlett's U p -statistic[END_REF], Lemma 3.1.

5.4 (Proof of Theorem 4.1) (i) As in the proof of Lemma 5.7(i) in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] we obtain with φ( t n , -k) replaced by φΣ ( t n , -k) -φ( t n , -k)

.

With Lemma 5.3 and the relation

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

ACCEPTED MANUSCRIPT

we now obtain as an upper bound for the first term in the . . . -brackets in (55)

The substitution s = t 1t 2 and the fact that there are at most n -|k 2 | values of t 2 (due to the restriction 1

The second term in the . . . -brackets in (55) can be estimated in the same way and the third term obviously is smaller than the first one.

Furthermore, we have

which implies assertion (i). (ii) and (iii) are immediate consequences.

5.5 (Proof of Theorem 4.3) (i) We set φ = φ n for short. We obtain from Lemma 5.7(i) in [START_REF] Dahlhaus | Empirical spectral processes for locally stationary time series[END_REF] with φt,kφ( t n , k) instead of φ( t n , k) , ( 45), (46) and Lemma 5.1


The Psychosocial Effects of Adult Strabismus -A Review

Correction of adult strabismus is not a cosmetic procedure but one that restores normality to an individual's appearance that has been altered by a disease process.

Two fundamental principles underpinning facial attractiveness are symmetry and averageness -manifest strabismus effects both of these giving an unconscious signal of poor genetic history.

The presence of manifest strabismus adversely affects many aspects of patients' lives including finding a partner, job prospects, interaction with peers and may manifest more seriously as psychiatric disorders. Surgical correction has been shown to be safe and effective for the functional problems of strabismus in adults but the hugely positive effects on the psychosocial aspects are only now becoming apparent. The advent of a new adult strabismus specific quality-of-life questionnaire and its subsequent validation will make this quantification of improvement easier. The wider medical community and the public at large should be made aware of the benefits of corrective strabismus surgery in adults.

Introduction

Adult strabismus is a common problem, present in approximately 4% of the population. [START_REF] Coats | Reasons for delay of surgical intervention in adult strabismus[END_REF] Visual maturity occurs at approximately 8 years of age. If strabismus occurs after this age, then functional problems such as diplopia give a clear indication for treatment. Diplopia is not usually an issue for those adult patients in whom strabismus developed before visual maturity. Strabismus has been shown to have a more detrimental effect to patients' quality of life (QoL) than diabetic retinopathy, with levels similar to that seen with macula degeneration or following a mild cerebrovascular accident. [START_REF] Beauchamp | The cost utility of strabismus surgery in adults[END_REF] The treatment of adult strabismus in those patients without binocular potential or who are not suffering from diplopia is regarded, by some, as "cosmetic" -this terminology is incorrect. Cosmetic surgery enhances or beautifies. Strabismus is the result of a disease process, which leads to a deviation from normality. Strabismus surgery is performed to restore normality.

There is significant evidence that shows that adult strabismus is associated not only with functional effects but also negative psychosocial changes that have wide ranging effects on all aspects of the patients' lives. This review brings together the current published literature with regard to human perceptions of facial attractiveness and the deep-rooted prejudices associated with deviations from normality, along with the psychosocial effects of strabismus and the current work in developing tools to quantify these difficulties and finally document the effects of surgery on these patients.

Perceptions of Body Image & Attractiveness

It takes just 150ms to judge a stranger's facial attractiveness [START_REF] Goldstein | Judgments of facial attractiveness in the absence of eye movements[END_REF] and when doing this, the center of the eyes, mouth and nose are the targets for gaze. [START_REF] Mertens | Gaze motor asymmetries in the perception of faces during a memory task[END_REF] Facial attractiveness preferences are present from an early age, long before environmental influences have been acquired [START_REF] Slater | Newborn infants prefer attractive faces[END_REF] and are similar across cultures [START_REF] Rhodes | Perceived health contributes to the attractiveness of facial symmetry, averageness, and sexual dimorphism[END_REF][START_REF] Little | Preferences for symmetry in human faces in two cultures: data from the UK and the Hadza, an isolated group of hunter-gatherers[END_REF][START_REF] Little | Symmetry is related to sexual dimorphism in faces: data across culture and species[END_REF] indicating that attractiveness has a large evolutionary component. Evolutionary psychologists believe that facial attractiveness is a 'honest' indicator of health and reproductive value, playing an important part in the sexual selection of species. [START_REF] Bashour | History and current concepts in the analysis of facial attractiveness[END_REF][START_REF] Zaidel | Appearance of symmetry, beauty, and health in human faces[END_REF][START_REF] Rhodes | Attractiveness of facial averageness and symmetry in non-western cultures: in search of biologically based standards of beauty[END_REF][START_REF] Scheib | Facial attractiveness, symmetry and cues of good genes[END_REF][START_REF] Rhodes | Perceived health contributes to the attractiveness of facial symmetry, averageness, and sexual dimorphism[END_REF][START_REF] Rhodes | The evolutionary psychology of facial beauty[END_REF][START_REF] Rhodes | Do facial averageness and symmetry signal health?[END_REF][START_REF] Mealey | Symmetry and perceived facial attractiveness: a monozygotic co-twin comparison[END_REF][START_REF] Rg | The two sides of beauty: Laterality and the duality in facial attractiveness[END_REF][START_REF] Grammer | Human (Homo sapiens) facial attractiveness and sexual selection: the role of symmetry and averageness[END_REF][START_REF] Kowner | Facial asymmetry and attractiveness judgment in developmental perspective[END_REF] Facial attractiveness gives an impression of being nicer, more intelligent and healthier. [START_REF] Bashour | History and current concepts in the analysis of facial attractiveness[END_REF] It has even been shown that more attractive defendants are given lighter sentences in the judicial system. [START_REF] Patry | Attractive but guilty: deliberation and the physical attractiveness bias[END_REF] But what constitutes attractiveness? Is it really in the eye of the beholder?

There are four major cues underpinning facial attractiveness:

• Averageness

• Symmetry

• Sexual dimorphism (males having masculine features and females, feminine)

• Youthfulness.

Average and symmetric faces may reflect resistance to developmental stressors, both environmental and genetic. [START_REF] Rhodes | The evolutionary psychology of facial beauty[END_REF] The neoclassical canons describing average facial proportions date back to the Renaissance artists and earlier. [START_REF] Bashour | History and current concepts in the analysis of facial attractiveness[END_REF] Out of the original ten canons, two include references to the ratio of the interocular distance to other aspects of the facial features; 1) the distance between the eyes is equal to the width of the nose and; 2) the distance between the eyes is equal to the width of each eye. These 'golden ratios' are substantiated in the modern literature, with the ideal female facial features having an interpupillary distance of 46% of face width. [START_REF] Pallett | New "golden" ratios for facial beauty[END_REF] Ocular facial features are thought of as being incorporated into both a 'length ratio' -the vertical distance between the eyes and the mouth and the 'width ratio' -the interpupillary distance.

The eye-mouth-eye angle has also been reported to be a good indicator of both masculinity and symmetry. [START_REF] Danel | Eye-mouth-eye angle as a good indicator of face masculinization, asymmetry, and attractiveness (Homo sapiens)[END_REF] Obvious asymmetry of facial features is immediately unattractive, but as a species we have evolved into finding facial asymmetry a more basic, genetic warning sign. In nature, many animal species depend on their ability to perceive symmetry in potential mates and it may be a marker of both developmental stability and phenotypic quality. [START_REF] Rhodes | Perceived health contributes to the attractiveness of facial symmetry, averageness, and sexual dimorphism[END_REF] Asymmetry implies poor health and bad genes with symmetry of physical traits being a sign of 'high quality' development. [START_REF] Rhodes | Perceived health contributes to the attractiveness of facial symmetry, averageness, and sexual dimorphism[END_REF] Facial symmetry may be an indication of good health rather than attractiveness. [START_REF] Zaidel | Appearance of symmetry, beauty, and health in human faces[END_REF] This can be supported by studies finding males with less asymmetric features having more sexual partners, better sperm counts and more offspring. [START_REF] Moller | A meta-analysis of the heritability of developmental stability[END_REF][START_REF] Moller | Bilateral symmetry and sexual selection: A meta-analysis[END_REF][START_REF] Manning | Developmental stability, ejaculate size, and sperm quality in men[END_REF] More symmetric females have increased fertility. [START_REF] Manning | Breast asymmetry and phenotypic quality in women[END_REF] Psychometric testing has shown that in male faces, both increasing symmetry and averageness had a positive effect on attractiveness whereas the effect of symmetry was less important for female faces. [START_REF] Komori | Averageness or symmetry: which is more important for facial attractiveness?[END_REF][START_REF] Komori | Effect of averageness and sexual dimorphism on the judgment of facial attractiveness[END_REF] Separate studies show that females show more minor elements of asymmetry than males [START_REF] Ercan | Facial asymmetry in young healthy subjects evaluated by statistical shape analysis[END_REF] but prefer symmetric features in a chosen mate. [START_REF] Oinonen | An inverse relationship between typical alcohol consumption and facial symmetry detection ability in young women[END_REF] This visual preference increases during the more fertile days of the menstrual cycle. [START_REF] Little | Preferences for symmetry in faces change across the menstrual cycle[END_REF][START_REF] Oinonen | Facial symmetry detection ability changes across the menstrual cycle[END_REF] Several studies have indicated that acute alcohol intoxication reduces the ability to detect asymmetry and there is a concurrent increase in the attractiveness ratings of photographs. [START_REF] Souto | Alcohol intoxication reduces detection of asymmetry: an explanation for increased perceptions of facial attractiveness after alcohol consumption?[END_REF][START_REF] Halsey | An explanation for enhanced perceptions of attractiveness after alcohol consumption[END_REF] Chronic alcohol use may cause this reduction in asymmetry detection to become a permanent trait. [START_REF] Oinonen | An inverse relationship between typical alcohol consumption and facial symmetry detection ability in young women[END_REF] Manifest strabismus is not a feature of the average human face. Its presence affects both the golden ratio and the width ratio and increases the amount of visible asymmetry greatly. Although perceptions of 'beauty' may be a higher cognitive function, [START_REF] Honekopp | Once more: is beauty in the eye of the beholder? Relative contributions of private and shared taste to judgments of facial attractiveness[END_REF] unconscious sexual selection is a far more deep-rooted instinct and the presence of ocular misalignment sends out messages of poor genetic history. A common complaint in strabismic adults is difficulty in finding a life-partner. This may be a partial explanation.

Objective and Subjective views of Strabismus

Strabismic individuals are likely to encounter social difficulties at all stages of life.

Negative responses of children to a strabismic peer are evident from the age of five, with children as young as four describing 'differences' when presented with an altered, strabismic toy. [START_REF] Paysse | Age of the emergence of negative attitudes toward strabismus[END_REF] Patching regimes have also been shown to have a negative impact on a child's psychosocial wellbeing, increasing bullying and stigmatization, [START_REF] Koklanis | Psychosocial impact of amblyopia and its treatment: a multidisciplinary study[END_REF] which can continue into adulthood. [START_REF] Menon | Study of the psychosocial aspects of strabismus[END_REF] Negative psychosocial consequences have been documented in parents of strabismic children, causing some difficulties in undertaking the motherhood role. [START_REF] Kothari | Comparison of psychosocial and emotional consequences of childhood strabismus on the families from rural and urban India[END_REF] Reports of the negative psychosocial effects of strabismus in adults were first published in 1993 with patients reporting all aspects of their lives being affected by manifest strabismus; self image, job prospects, relationships, education and sports [START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF] and have been confirmed by many since. [START_REF] Menon | Study of the psychosocial aspects of strabismus[END_REF][START_REF] Burke | Psychosocial implications of strabismus surgery in adults[END_REF][START_REF] Hatt | The effects of strabismus on quality of life in adults[END_REF][START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF][START_REF] Jackson | The psychosocial benefits of corrective surgery for adults with strabismus[END_REF][START_REF] Durnian | The psychosocial aspects of strabismus; correlation between the AS-20 and DAS59 quality-of-life questionnaires[END_REF] These difficulties have been reported to worsen both with increasing age [START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF] and with increasing strabismus size [START_REF] Jackson | The psychosocial benefits of corrective surgery for adults with strabismus[END_REF] especially if the magnitude of deviation is greater than 25 Prism Diopters [START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF].

Young strabismic adults report difficulties in making and maintaining relationships, both sexual and platonic, with members of the opposite sex -describing a lack of confidence and low self-esteem as the main contributing factors. [START_REF] Menon | Study of the psychosocial aspects of strabismus[END_REF][START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF][START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF] It has suggested that patients perceive that others would rate themselves less negatively than they would rate themselves in various personality traits, indicative of an inherent lack of confidence. [START_REF] Burke | Psychosocial implications of strabismus surgery in adults[END_REF] More serious manifestations include an increased incidence of psychiatric disorders in young adults who developed strabismus, especially exotropia, as a child. One report found that 41.3% of strabismic patients developed mental health problems compared with 30.7% of controls. [START_REF] Bez | Adult strabismus and social phobia: a casecontrolled study[END_REF][START_REF] Mckenzie | Prevalence and sex differences of psychiatric disorders in young adults who had intermittent exotropia as children[END_REF][START_REF] Mohney | Mental illness in young adults who had strabismus as children[END_REF] There are reports of a genetic linkage between constant exotropia and schizophrenia in the PMX2B gene. [START_REF] Toyota | Association between schizophrenia with ocular misalignment and polyalanine length variation in PMX2B[END_REF] When viewed by others, strabismic individuals are rated more negatively than controls in various attributes including: perceived health, [START_REF] Uretmen | Negative social bias against children with strabismus[END_REF] communication skills, [START_REF] Olitsky | The negative psychosocial impact of strabismus in adults[END_REF] intelligence, [START_REF] Olitsky | The negative psychosocial impact of strabismus in adults[END_REF][START_REF] Mojon-Azzi | Strabismus and employment: the opinion of headhunters[END_REF] attractiveness [START_REF] Mojon-Azzi | Opinions of dating agents about strabismic subjects' ability to find a partner[END_REF] and successfulness. [START_REF] Mojon-Azzi | Opinions of dating agents about strabismic subjects' ability to find a partner[END_REF][START_REF] Goff | Effect of adult strabismus on ratings of official U.S army photographs[END_REF] These perceptions seem to translate into real difficulties gaining employment. One study reported that more than 70% of employment headhunters would consider patients with strabismus to have great difficulty in finding employment, with only severe acne and visible missing teeth being the only facial anomalies rated more negatively. [START_REF] Mojon-Azzi | Strabismus and employment: the opinion of headhunters[END_REF] Large angle horizontal strabismus has also been shown to be detrimental in gaining employment when the patients are judged against orthophoric controls.[55] Those with manifest strabismus have been shown more unlikely to gain promotion or further their career in the U.S. military. [START_REF] Goff | Effect of adult strabismus on ratings of official U.S army photographs[END_REF] When questioned, 92.5% of dating agencies stated that strabismic individuals would struggle to find a partner, as they are perceived as less attractive and less erotic. [START_REF] Mojon-Azzi | Opinions of dating agents about strabismic subjects' ability to find a partner[END_REF] There is over-whelming evidence describing these negative effects on patients. There is debate on the effect of both direction of deviation and sex of the patient. In studies with patients self-reporting experiences, generally there is no difference between the difficulties faced between those with eso-and exo-deviations. [START_REF] Menon | Study of the psychosocial aspects of strabismus[END_REF][START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF][START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF] Esotropic males report that they have their intelligence underestimated more and are discriminated against in the workplace more than exotropic males. [START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF] When being rated by others, esotropes are generally rated worse. [START_REF] Uretmen | Negative social bias against children with strabismus[END_REF][START_REF] Olitsky | The negative psychosocial impact of strabismus in adults[END_REF][START_REF] Goff | Effect of adult strabismus on ratings of official U.S army photographs[END_REF] A 2007 study found that asymmetry of the face is rated as more unattractive, the more medial the asymmetric feature is. With esotropia the pupil is much close to the midline than in exotropia and this may explain this finding. [START_REF] Spinger | Facial attractiveness: visual impact of symmetry increases significantly towards the midline[END_REF] When it was reported, strabismic females were rated more negatively than males by others. [START_REF] Mojon-Azzi | Opinions of dating agents about strabismic subjects' ability to find a partner[END_REF][START_REF] Goff | Effect of adult strabismus on ratings of official U.S army photographs[END_REF][START_REF] Coats | Impact of large angle horizontal strabismus on ability to obtain employment[END_REF] Women, in particular, are detrimentally affected by the idealized image of the female body portrayed by the media [START_REF] Monro | Media-portrayed idealized images, body shame and appearance anxiety[END_REF][START_REF] Strahan | Comparing to perfection: How cultural norms for appearance affect social comparisons and self-image[END_REF] and we suggest that a similar response is seen in strabismic patients.. There is inherent bias in papers dealing with a cohort of patients that have presented themselves for treatment. A large numbers of people with manifest strabismus often do not present for treatment. Does this self-selected group not show any negative psychosocial effects? Or is it that they (and maybe their primary care provider) are still unaware of the treatment options? [START_REF] Coats | Reasons for delay of surgical intervention in adult strabismus[END_REF] Development of robust tools to document the improvement that can be gained from strabismus surgery has been previously recommended. [START_REF] Mills | Strabismus surgery for adults: a report by the American Academy of Ophthalmology[END_REF] The subsequent dissemination of results would help in making the wider medical community aware of the potential health (both physical and mental) benefits.

Development of assessment tools quantifying psychosocial difficulties

Unlike generic methods, condition-specific QoL assessment tools focus on problems associated with the pathology of interest, it is therefore important that these tools are developed. Development of a strabismus-specific questionnaire is important primarily to assess the success or failure of service delivery in a clinical setting and also to identify those patients that may require psychosocial counseling. Previous attempts have been made to quantify the negative effects of strabismus by using various generic questionnaires [START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF][START_REF] Jackson | The psychosocial benefits of corrective surgery for adults with strabismus[END_REF][START_REF] Bez | Adult strabismus and social phobia: a casecontrolled study[END_REF][START_REF] Archer | Social and emotional impact of strabismus surgery on quality of life in children[END_REF]; non-strabismus specific ophthalmic instruments [START_REF] Hatt | The effects of strabismus on quality of life in adults[END_REF]; and locally produced questionnaires. [START_REF] Menon | Study of the psychosocial aspects of strabismus[END_REF][START_REF] Kothari | Comparison of psychosocial and emotional consequences of childhood strabismus on the families from rural and urban India[END_REF][START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF][START_REF] Burke | Psychosocial implications of strabismus surgery in adults[END_REF][START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF][START_REF] Beauchamp | The management of strabismus in adults--III. The effects on disability[END_REF] Until recently there were no strabismus-specific questionnaires dealing with the psychosocial aspects of strabismus. The Amblyopia and Strabismus Questionnaire (ASQE) has been previously used, however only four of the twenty-six questions deal with the psychosocial elements of strabismus which, although perfectly adequate when dealing with the subject as a whole is not sufficient when the psychosocial aspects are to be investigated and documented. [START_REF] Felius | The Amblyopia and Strabismus Questionnaire: English translation, validation, and subscales[END_REF] The 20-item Adult Strabismus questionnaire (AS-20) has been made available. It was developed by distilling down a 181-item questionnaire, gained from patient interviews, to 20 questions, 10 dealing with the psychosocial elements of strabismus and 10 with the functional problems. [START_REF] Hatt | Development of a quality-of-life questionnaire for adults with strabismus[END_REF] The questions used are the best discriminators. The AS-20 is a freely available QoL questionnaire developed specifically for strabismic adults. The overall score is the mean of all the questions answered, with a score from 0 to 100 (0 being worst and 100 being best). The threshold for a normal, non-strabismic, score is 84 [START_REF] Hatt | Comparison of quality-of-life instruments in adults with strabismus[END_REF]. The test-retest reliability of the AS-20 is good, indicating its potential use in assessing changes in strabismus over a long time period. [START_REF] Leske | Test-retest reliability of health-related quality-of-life questionnaires in adults with strabismus[END_REF] The AS-20 has been compared directly to the 25-item National Eye Institute Visual Function Questionnaire (VFQ-25) in strabismic adults and although both detected the reduction in health-related QoL, the AS-20 had a greater sensitivity to the effects of strabismus. The VFQ-25 identified strabismus with diplopia far better than nonfunctional strabismus, a representation of how it was designed [START_REF] Hatt | Comparison of quality-of-life instruments in adults with strabismus[END_REF]. The AS-20 also showed a strong correlation to the widely used Derriford Appearance Scale 59 (DAS59) QoL research tool, again being more specific to strabismus patients and not being influenced by other bodily factors that can impinge on a patient's perception of their own appearance [START_REF] Durnian | The psychosocial aspects of strabismus; correlation between the AS-20 and DAS59 quality-of-life questionnaires[END_REF].

Effects of strabismus surgery

A report from the American Academy of Ophthalmology (2004) concluded that strabismus surgery in adults is safe and effective in: 1) restoring ocular alignment; 2) eliminating diplopia; 3) restoring binocularity when achievable; 4) expanding the visual field; and 5) improving any head position. It did not have any conclusions regarding subjective or psychosocial benefits due to lack of studies at that time [START_REF] Mills | Strabismus surgery for adults: a report by the American Academy of Ophthalmology[END_REF].

The cost effectiveness of strabismus surgery has been estimated at $1632/Quality of life years (QALY), less than cataract surgery or vitrectomy for diabetic vitreous haemorrhage and these figures are further improved, the earlier the surgery is performed. [START_REF] Beauchamp | The cost utility of strabismus surgery in adults[END_REF] All aspects of psychosocial problems have been shown to improve following surgery.

The first report by Satterfield in 1993 documented the negative psychosocial effects of strabismus but did not try to quantify the effects of strabismus surgery. [START_REF] Satterfield | Psychosocial aspects of strabismus study[END_REF] Burke retrospectively reported on 15 psychosocial personality traits in 31 strabismic adults and all traits improved following surgery. The traits most improved with surgery were: confidence; attractiveness; self-esteem; being at ease; sociable; interactions with the opposite sex. Patients perceived that people viewed them more positively after surgery but did feel that others would rate them less highly than they rated themselves and all scores were significantly less than they would have been in an 'ideal world'. Females had a more positive effect from surgery than males, as did esotropes. There was no difference in scores when correlated with age. [START_REF] Burke | Psychosocial implications of strabismus surgery in adults[END_REF] Jackson prospectively reported on the results of 3 different scores of social anxiety and depression in 46 patients. The early signs of mental health issues related to strabismus improved following surgery with social anxiety scores and social avoidance reducing to a normative level and general anxiety levels improving. There was no direct correlation between the magnitude of deviation improvement and any quantitative score. In contrast to Burke, they reported that exotropic patients obtain greater benefits from the surgery. [START_REF] Jackson | The psychosocial benefits of corrective surgery for adults with strabismus[END_REF] Menon reported prospectively on 40 young adults. 97.5% improved their subjective appearance and 95% had a significant improvement in self-confidence and personal relationships following correction of their strabismus. [START_REF] Menon | Study of the psychosocial aspects of strabismus[END_REF] Beauchamp retrospectively reported on 101 patients and significant improvements were found in: health; daily tasks; social interactions; concerns; self image; job prospects. [START_REF] Beauchamp | The management of strabismus in adults--III. The effects on disability[END_REF] Nelson retrospectively reported on 128 strabismic adults, 85% had improved selfesteem following surgery with 65% having an improved ability to meet new people, this was more significant for females rather than male patients. Patients below the age of 35 had a better psychosocial improvement following surgery with respect to selfesteem, meeting new people, and trying new activities. Job opportunities also improved, with females reporting a 15% increase in opportunities and males, 6% . [START_REF] Nelson | The psychosocial aspects of strabismus in teenagers and adults and the impact of surgical correction[END_REF] The published reports of the psychosocial benefits of strabismus surgery show that patients do improve greatly with surgery and this element of strabismus practice should become paramount in the minds of clinicians. The idea of adult strabismus surgery being 'cosmetic' is both misleading and under-appreciates the value of restoring normal appearance to these patients. There is a paucity of prospective studies, however the introduction of the AS-20 tool and its subsequent validation should be useful in increasing this body of evidence. The levels of psychosocial distress experienced by strabismus patients are beyond the clinical environment, affecting all aspects of lives -from the most basic needs of finding a life-partner to their job prospects and subsequent career. Providing patients with additional psychosocial support in the future will have to address this and will clearly require a multi-disciplinary response.

The authors have no proprietary or financial interest in any aspect of the study.

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non-exclusive for government employees) on a worldwide basis to the BMJ Group and co-owners or contracting owning societies (where published by the BMJ Group on their behalf), and its Licensees to permit this article (if accepted) to be published in the British Journal of Ophthalmology and any other BMJ Group products and to exploit all subsidiary rights, as set out in our licence.


MD Josefina P Salgado

Lohmann Cp

Winkler

Corneal Collagen Crosslinking in post-LASIK keratectasia

Keywords: Crosslinking, Keratectasia, LASIK, Cornea

Background/Aims: To evaluate the effect of corneal collagen crosslinking with riboflavin and UV-A as a treatment option for post-laser in situ keratomileusis (LASIK) keratectasia.

Methods: Crosslinking was carried out in 22 eyes of 15 patients with iatrogenic keratectasia. Follow-up, according to a standardized protocol (UCVA, BCVA, slitlamp examination, pachymetry and topography), was performed preoperatively, 1, 3, 6 and 12 months after crosslinking.

Results: The mean BCVA was 0.19 (SD ±0.21) logMAR preoperatively, 0.25 (SD ±0.17) 1 month, 0.20 (SD ±0.20) 3 months, 0.18 (SD ±0.21) 6 months and 0.15 (SD ±0.14) 12 months postoperatively (statistically significant postop1-postop6, p=0.0335). The maximum k-readings were 44.12 (SD ±3.97) preoperatively, 46.23 (SD ±4.14) 1 month, 43.88 (SD ±4.25) 3 months, 45.06 (SD ±5.07) 6 months and 44.43 (SD ±4.06) 12 months postoperatively (statistically significant preop -postop1, p=0.0281).

Conclusion:

Crosslinking in patients with iatrogenic keratectasia stabilized the UCVA and BCVA as well as the maximum k-readings in our cohort. It seems to be a safe and promising procedure to stabilize the refraction and the corneal topography, hence to stop the progression of visual loss, thereby avoiding or delaying disease progression and keratoplasty.

Introduction:

Since the first description of laser in situ keratomileusis (LASIK) [START_REF] Pallikaris | A corneal flap technique for laser in situ keratomileusis. Human studies[END_REF] several million patients have undergone this surgery. The intra and post operative complications were reduced with the improvement of the mechanical microkeratomes and the introduction of new femtosecond laser systems. However refractive surgery cuts and removes corneal tissue by creating a flap and removing tissue from the stromal bed.

Therefore, in all excimer laser procedures, the corneal biomechanics are clearly weakened. [START_REF] Dupps | Biomechanics and wound healing in the cornea[END_REF][START_REF] Krueger | Biomechanical effects of femtosecond and microkeratome-based flap creation: prospective contralateral examination of two patients[END_REF] Iatrogenic keratectasia is a rare but major sight-threatening complication after LASIK [START_REF] Comaish | Progressive post-LASIK keratectasia: biomechanical instability or chronic disease process?[END_REF] and was first described in 1998. [START_REF] Seiler | Iatrogenic keratectasia after laser in situ keratomileusis[END_REF] Since then, many reports of keratectasia have been published. [START_REF] Ou | Keratectasia after laser in situ keratomileusis (LASIK): evaluation of the calculated residual stromal bed thickness[END_REF][START_REF] Kymionis | Unilateral corneal ectasia after laser in situ keratomileusis in a patient with uncomplicated photorefractive keratectomy in the fellow eye[END_REF][START_REF] Argento | Corneal ectasia after laser in situ keratomileusis[END_REF] Keratectasia is characterized by a progressive corneal steepening which can occur centrally or inferiorly combined with severe refractive changes, loss of visual acuity (VA) and a stromal thinning of the cornea. [START_REF] Kohlhaas | A new treatment of keratectasia after LASIK by using collagen with riboflavin/UVA light cross-linking[END_REF] Risk factors for its development are thin corneas, a thin residual stromal bed, deep ablations, enhancement treatments and preoperative abnormalities like forme fruste keratoconus. [START_REF] Binder | Ectasia after laser in situ keratomileusis[END_REF][START_REF] Hafezi | Corneal collagen crosslinking with riboflavin and ultraviolet A to treat induced keratectasia after laser in situ keratomileusis[END_REF] Keratectasia also appears in eyes treated according to the current guidelines without any risk factors. [START_REF] Kohlhaas | A new treatment of keratectasia after LASIK by using collagen with riboflavin/UVA light cross-linking[END_REF] It is important to diagnose this condition at an early stage in order to achieve or maintain good visual acuity and avoid severe procedures such as a keratoplasty.

Various methods have been developed to address keratoconus which have also been used in post-LASIK keratectasia, such as rigid gas permeable contact lenses, intracorneal ring segments (e.g. INTACS), anterior lamellar and penetrating keratoplasty. [START_REF] Argento | Corneal ectasia after laser in situ keratomileusis[END_REF][START_REF] Tan | Current treatment options for corneal ectasia[END_REF][START_REF] Rabinowitz | Intacs for keratoconus[END_REF][START_REF] Bilgihan | Excimer laser-assisted anterior lamellar keratoplasty for keratoconus, corneal problems after laser in situ keratomileusis, and corneal stromal opacities[END_REF][START_REF] Rodriguez | Penetrating keratoplasty versus intrastromal corneal ring segments to correct bilateral corneal ectasia: preliminary study[END_REF] However rigid contact lenses and intracorneal ring segments do not stop the progression of the disease.

Corneal collagen crosslinking was first described in 1997 [START_REF] Sporl | Increased rigidity of the cornea caused by intrastromal cross-linking[END_REF] and soon utilised to treat keratoconus. [START_REF] Wollensak | Riboflavin/ultraviolet-a-induced collagen crosslinking for the treatment of keratoconus[END_REF] Crosslinking represents a photooxidative technique of collagen crosslinking using riboflavin as a photosensitizer for UV-A-light [START_REF] Raiskup-Wolf | Collagen crosslinking with riboflavin and ultraviolet-A light in keratoconus: long-term results[END_REF] and has the ability to increase the corneal biomechanical stability by inducing chemical covalent bonds, bridging amino groups of stromal collagen fibrils, thereby increasing their intra-and inter-fibrillar rigidity. [START_REF] Wollensak | Crosslinking treatment of progressive keratoconus: new hope[END_REF] It has been used to arrest the progression of keratoconus since 2003. [START_REF] Wollensak | Riboflavin/ultraviolet-a-induced collagen crosslinking for the treatment of keratoconus[END_REF] The research and clinical studies have demonstrated that corneal crosslinking stabilizes the weakened cornea and refraction, halting the progression of keratoconus. [START_REF] Raiskup-Wolf | Collagen crosslinking with riboflavin and ultraviolet-A light in keratoconus: long-term results[END_REF][START_REF] Kohlhaas | Collagen crosslinking with riboflavin and UVA-light in keratoconus[END_REF][START_REF] Wittig-Silva | A randomized controlled trial of corneal collagen cross-linking in progressive keratoconus: preliminary results[END_REF] Thus, it reduces the need for a keratoplasty in a majority of cases. Since keratectasia formation appears to be very similar to other ectatic conditions of the cornea, such as keratoconus or pellucid marginal degeneration, similar treatments have been used in order to manage post-LASIK ectasia.

The aim of this study was to evaluate the effect of corneal crosslinking in the treatment of post-LASIK keratectasia.

Materials and Methods:

In this prospective, non randomised clinical study 22 eyes of 15 patients (6 female, 9 male) with a history of LASIK and the development of iatrogenic keratectasia after LASIK were included. Ectasia was diagnosed according to Pallikaris et al. by the slitlamp appearance of corneal thinning in the area of ectasia, unstable topographical steepening (more than 1.0 diopter [D] , figure 1), decreased visual acuity, and/or unstable refraction. [START_REF] Pallikaris | Corneal ectasia induced by laser in situ keratomileusis[END_REF] The patients were referred to our clinic after the onset of symptoms of ectasia between 2 weeks and 10 years after the LASIK procedure (mean: 38.7 months, SD ±35.1). Symptoms included progressive loss of vision, blurry vision, and frequently changing refraction, with increasing myopia and astigmatism.

All patients underwent a full ophthalmic evaluation before corneal crosslinking treatment. This included a detailed history and a complete examination with objective The inclusion criteria for treatment were progressive keratectasia after refractive surgery as well as a pachymetry greater then 400µm. Exclusion criteria included corneal thickness less than 400µm and central corneal scars.

The corneal crosslinking was performed in the outpatient operating room under sterile conditions. The ocular surface was anaesthetized with topical proxymetacaine hydrochloride 0.5% eye drops (Proparacain®, Ursapharm, Saarbrücken, Germany, 2 drops at 30 sec interval) followed by a mechanical corneal debridement of 8mm in diameter.

To photosensitize and saturate the cornea, a topical 0.1% riboflavin isotonic solution (Medio-Haus Medizinprodukte GmbH, Neudorf, Germany) was applied every five minutes for 30 minutes. After this period, a UV-A-irradiation (370nm) with a surface irradiance of 3mW/cm 2 was carried out with 10cm distance for 30 minutes using a commercially available system (UV-X TM , Peschke, Nürnberg, Germany). [START_REF] Wollensak | Crosslinking treatment of progressive keratoconus: new hope[END_REF] This surface irradiance was controlled and calibrated using the UV-A meter (LaserMate-Q, LASER 2000) before the patient was exposed. During the UV-A-exposure period, riboflavin was applied every 5 minutes to guarantee the required concentration and keep the cornea moistened. At the end of the surgery, the corneal surface was washed thoroughly with BSS, and a bandage contact lens (Pure Vision, Bausch & Lomb, Berlin, Germany) soaked in levofloxacin 5mg/ml (Oftaquix® sine, Santen, Germering, Germany) was applied until the epithelial closure was complete.

Until the removal of the bandage contact lens, the postoperative therapy consisted of topical preservative-free antibiotics (levofloxacin 5mg/ml) and lubrication (carbomer 2%, Vidisic EDO® Gel, Dr. Mann GmbH, Berlin, Germany) six times a day.

Diclofenac tablets (Diclofenac-ratiopharm® 50 mg, Ratiopharm, Ulm, Germany) and Pregabalin 75 mg (Lyrica®, Pfizer Pharma, Berlin, Germany) were distributed to the patient to be taken as needed.

Postoperatively, the patients were examined daily until complete reepithelialisation.

Follow-up examinations, according to a standardized protocol, assessed patients' objective and manifest refraction, UCVA, BSCVA, slit-lamp biomicroscopy, ultrasound corneal pachymetry (Tomey SP-3000) and corneal topography (Tomey Topographic Modelling System, TMS-4) and were performed 1, 3, 6 and 12 months after corneal crosslinking. We also analysed following topographic indices: the surface asymmetry index (SAI) which detects alteration of corneal asymmetry by comparing areas of the cornea 180° apart and acts as a disease progression marker; the surface regularity index (SRI) which can be used to predict the optical outcome one might expect from corneal topography values. The third index assessed was the keratoconus prediction index (KPI), a composite calculated index that gives a measure of certainty of detection of the diagnosis of keratoconus. [START_REF] Burns | Keratoconus: an analysis of corneal asymmetry[END_REF] For statistical analysis GraphPad Instat software for Windows was used (GraphPad Software, Inc La Jolla, CA, USA). To test for normal distribution of the parameters we used the method of Kolmogorov and Smirnov and to evaluate the statistical significance we used a two-tailed paired t test. P values >0.05 were interpreted as statistical not significant, values between 0.01 to 0.05 were interpreted as statistical significant (*) and values between 0.001 to 0.01 as statistical very significant (**).

Results:

The mean age of the patients at the time of onset of keratectasia was 38.4 years (SD ±8.13, range 27-51). The surgical procedure was uneventful in all cases.

Postoperatively, a trace haze was present in 10 eyes and disappeared during the first three months.

The mean UCVA and BSCVA are shown in The topographic astigmatic values (Figure 3) were - Preoperatively the mean KPI was 0.32 (SD ±0.12), 1 month after crosslinking it increased to 0.38 (SD ±0.12), 3 months it decreased to 0.30 (SD ±0.11). 6 months after crosslinking the KPI was 0.31 (SD ±0.12) and 12 months postoperatively it was 0.29 (SD ±0.11). The change from the first postoperative month to the third postoperative month was statistically significant (p=0.0291 *). All other changes were not significant.

Discussion:

Iatrogenic keratectasia still lacks an efficacious and predictable treatment option.

Corneal crosslinking is known to work mainly in the anterior two thirds of the cornea. [START_REF] Kohlhaas | A new treatment of keratectasia after LASIK by using collagen with riboflavin/UVA light cross-linking[END_REF][START_REF] Koller | Complication and failure rates after corneal crosslinking[END_REF] This represents a challenge using corneal crosslinking in post-LASIK ectasia, for the flap does not contribute to the mechanical stability of the cornea.

Hence, we would expect it to be less effective in this situation compared to the treatment of keratoconus or PMD.

Up to now only two papers have been published regarding corneal crosslinking exclusively in post-LASIK ectasia. Kohlhaas et al. reported first about a successful treatment of a post-LASIK ectasia using corneal crosslinking back in 2005 in one patient with ectasia on both eyes. [START_REF] Kohlhaas | A new treatment of keratectasia after LASIK by using collagen with riboflavin/UVA light cross-linking[END_REF] In 2007 Hafezi et al. [START_REF] Hafezi | Corneal collagen crosslinking with riboflavin and ultraviolet A to treat induced keratectasia after laser in situ keratomileusis[END_REF] published first results of corneal crosslinking in post-LASIK ectasia showing an arrest and even a partial reversal in the keratectasia.

Our results showed a worsening of both the uncorrected as well as the best spectacle corrected visual acuity in the first postoperative month (a non significant loss of 1.5 lines in UCVA and of 0.5 lines in BSCVA). This change in visual acuity could be related to an increase in myopia and a consequent statistical non significant increase in the spherical equivalent. This myopic shift would explain the decrease in visual acuity and could be, in our opinion, due to a modification in the corneal shape resulting from the compaction of the stromal collagen following the crosslink process.

The increasing myopia goes along with increased topographic values (k-max, k-min, keratometric cylinder, SRI, SAI and KPI) reflecting a steepening of the cornea and an increase of an irregular astigmatism. A trace haze was present in 10 eyes which disappeared within the first three postoperative months and could also explain a slight myopic shift. After this period, the myopia and the spherical equivalent continuously diminished. The spherical equivalent improved significantly from the preoperative value as well as the first postoperative to the sixth postoperative month.

There were some non significant fluctuations in the manifest cylinder along with the corneal topography.

UCVA as well as BSCVA recovered after the third postoperative month and remained stable throughout the follow-up period of 12 months with a statistically significant increase of BSCVA. Parallel to these changes the corneal indices also give us hints about the corneal shape and ectasia existence/evolution. In our series preoperative SRI and SAI were outside of the normal range published by Burns et al. [START_REF] Burns | Keratoconus: an analysis of corneal asymmetry[END_REF],

revealing a corneal asymmetry compatible with post LASIK ectasia. [START_REF] Argento | Corneal ectasia after laser in situ keratomileusis[END_REF] KPI also showed preoperative values consistent with corneal ectasia (according to TMS-4 >0.24 ectasia suspect).

All three analysed indices regressed after the first postoperative month during the follow-up period. According to the index definitions of Burns et al. [START_REF] Burns | Keratoconus: an analysis of corneal asymmetry[END_REF] the statistical significant reduction of KPI and statistically significant regression of the SRI support a regression of the corneal ectasia with a flattening of the cornea.

In our study, corneal crosslinking seems to be an effective option in the treatment of post-LASIK ectasia.

The stabilisation of VA and of the corneal curvature, and in some cases even an improvement in VA and reversal of the corneal ectasia in our study correlates well with the results published by Kohlhaas and Hafezi.

There are, however, a few disadvantages to corneal crosslinking: thus far, researchers have not been able to monitor the long-term experience of patients, nor can they accurately predict the long-term effects of a single procedure.

Nevertheless, the results from the first six years of the crosslinking treatment of keratoconus are very promising. There are few reports of patients requiring a second crosslinking in the same eye. [START_REF] Raiskup-Wolf | Collagen crosslinking with riboflavin and ultraviolet-A light in keratoconus: long-term results[END_REF] Complications are very rare and include sterile infiltrates, stromal scars and haze. Corneal infection is a reported complication which is unlikely directly related to the crosslinking itself, since crosslinking acts as a bactericide and fungicide . [START_REF] Koller | Complication and failure rates after corneal crosslinking[END_REF] Iatrogenic keratectasia remains a rather obscure subject. The underlying problem has not been yet fully understood which makes keratectasia difficult to control. As we see in our cohort, ectasia can present quickly after a refractive procedure, but it can also appear years after surgery. In our opinion, the most important step in the management of iatrogenic keratectasia is to monitor refractive patients regularly, even years after the laser procedure, in order to diagnose and try to treat such a complication at an early stage.

Corneal crosslinking is a promising procedure in the treatment of post-LASIK keratectasia, but we need to await the long-term results of a greater number of patients in randomized controlled trials in order to confirm its efficacy, safety and durability.

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Figure 2 .

 2 The UCVA changed from 0.53 (SD ±0.38) logMAR preoperatively to 0.67 (SD ±0.43) logMAR 1 month after corneal crosslinking. After 3 months, UCVA was 0.54 (SD ±0.35) logMAR. The 6-month postoperative evaluation showed an UCVA of 0.53 (SD ±0.35) logMAR, and after the first year, we had a mean UCVA of 0.40 (SD ±0.27) logMAR. The changes were not statistically significant.The preoperative BSCVA was 0.19 (SD ±0,21) logMAR and changed to 0.25 (SD ±0.17) logMAR after the first postoperative month. After 3 months, BSCVA was 0.20 (SD ±0.20) logMAR and was 0.18 (SD ±0.21) logMAR 6 months after corneal crosslinking. The 12-month results showed a mean BSCVA of 0.15 (SD ±0.14) logMAR. The change from 1 month postoperatively to 6 months postoperatively was statistically significant (p=0.0335 *). All other changes were not statistically significant. The refractive changes are shown in Figure 3. The spherical equivalent changed from -2.39 D (SD ±2.03) preoperatively to -3.31 D (SD ±3.33) 1 month, -2.67 D (SD ±2.38) 3 months, -2.56 D (SD ±2.65) 6 months and -2.07 D (SD ±2.18) 12 months after corneal crosslinking. The changes from preoperatively to 6 months postoperatively (p=0.0404 *) and from 1 month postoperatively to 6 months postoperatively (p=0.0269 *) were statistically significant. All other changes were not statistically significant. The manifest cylinder (Figure 3) changed from -2.59 D (SD±1.86) preoperatively to -2.17 D (SD ±1.38) 1 month, -1.88 D (SD ±1.57) 3 months, -2.15 D (SD±1.59) 6 months and -2.10 D (SD ±1.58) 12 months after corneal crosslinking. The changes were not statistically significant. The k-values are shown in Figure 4. The maximum and minimum k-readings were, respectively, 44.12 D (SD ±3.97) and 41.78 D (SD ±2.69) preoperatively, 46.23 D (SD ±4.14) and 43.25 D (SD ±2.66) 1 month, 43.88 D (SD ±4.25) and 41.20 D (SD ±2.88) 3 months, 45.06 D (SD ±5.07) and 42.20 D (SD ±3.22) 6 months and 44.43 D (SD ±4.06) and 42.04 D (SD ±2.67) 12 months after corneal crosslinking. The change from preoperatively to 1 month postoperatively was statistically significant (p=0.0281 *) in the maximum k-readings. All other changes in the maximum k-readings as well as all changes in the minimum k-readings were not statistically significant.

Figure 1 :Figure 2 :

 12 Figure 1:Example of untreated corneal ectasia evolution over 7 months

Figure 3 :Figure 4 :

 34 Figure 3: Changes of sphere, cylinder and spherical equivalent

Competing Interest: none