to account for fat tails , skewness and excessive kurtosis of empirical probability distributions of returns in real financial markets , it has become increasingly popular to model the dynamics of market factors as a lvy process .lvy models are more realistic than gaussian ones but the latter are much more tractable .indeed , in the gaussian framework , explicit pricing formulas are known for a wide range of options and other contingent claims both without and with early exercise features , whereas in the lvy models , most of the pricing formulas have been obtained for contingent claims of the european type , with the deterministic life - span .there are some explicit analytic results for options with early exercise features : see boyarchenko and levendorski ( 2000 , 2001 , 2002a , b ) , mordecki ( 2002 ) and the bibliography therein for pricing of perpetual american options , and boyarchenko and levendorski ( 2002b , c ) for pricing of barrier options and first touch digitals . however , the pricing formulas are complicated and difficult for numerical implementation except for a rather special case of pricing of perpetual american options under exponential jump - diffusions or spectrally one - sided processes .another obstacle for non - gaussian modelling arises when one considers more general markov processes .the explicit pricing formulas in affine term structure models and certain lvy - driven ornstein - uhlenbeck models are known in the case of contingent claims with the deterministic life span only see duffie et al .( 2000 , 2002 ) , chacko and das ( 2002 ) , and barndorff - nielsen and shephard ( 2001b ) , barndorff - nielsen et al ( 2002 ) , respectively ; for non - gaussian variants of the hjm - model , see eberlein and raible ( 1999 ) . in the general case , the dependance on the state variable does not allow one to obtain explicit analytical answers .the following observation helps to obtain efficient approximate solutions . as barndorff - nielsen and levendorski ( 2001 ) notice ,typically , a good fit to the data can be achieved with lvy processes whose lvy densities decay as or faster , where ( the steepness parameter of the exponential lvy process ) is large .they used this property to derive an asymptotic pricing formula for european options under certain class of feller processes .the same observation was used in boyarchenko and levendorski ( 2002a , b , d ) and kudryavzev and levendorski ( 2002 ) to derive efficient approximate formulas for perpetual american and bermudan options , and first - touch - digitals , respectively .it was shown in boyarchenko and levendorski ( 2002a , b ) that the simple approximate formula is of the same form as the corresponding formula in a gaussian model even when the underlying lvy process has no gaussian component .it can be shown that the leading term of the approximate pricing formula in barndorff - nielsen and levendorski ( 2001 ) can also be written as the pricing formula in a gaussian model .these observations can serve as an analytical explanation of relatively good performance of gaussian models in apparently non - gaussian situations .thus , as far as pricing formulas are concerned , lvy processes with large steepness parameters behave almost as the brownian motion , and feller processes with large steepness parameters considered in barndorff - nielsen and levendorski ( 2001 ) behave almost as gaussian diffusions .it seems reasonable to use the nomer _ pseudo - diffusions _ for lvy processes and more general lvy - like feller processes with large steepness parameters .the modelling with pseudo - diffusions allows one to obtain an efficient approximation to the price ; in some situations , the asymptotic expansion of the price can be obtained , of the form where the leading term , , is the price in the gaussian model with the same instantaneous drift and variance .the first correction term takes into account the moments of order three as well ( skewness ) , the second correction term accounts for moments of order four , etc .notice that though the leading term looks as the pricing formula in the gaussian model , the drift " and variance - covariance matrix " used in the formula for the leading term are not the same as the ones of the gaussian component of the process unless it is purely gaussian .indeed , a lvy process may have no diffusion component at all .the aim of the paper is to apply the approximate pricing approach to quadratic term structure models ( qtsm ) when the stochastic factor follows a mean - reverting pseudo - diffusion process of the simplest form ( it is unlikely that in the qtsm model , an explicit pricing formula can be obtained unless the process process is gaussian ) , and derive a pricing formula of the form ( [ asprice ] ) . for the discussion about advantages of the gaussian qtsm model , see ahn et al ( 2002 , 2003 ) and chen and poor ( 2002 ) .cheng and scaillet ( 2002 ) consider an affine - quadratic model , and allow for jumps but only in the dynamics of affine variables of the model .notice that the use of jumps in qtsm models adds additional flexibility in joint modelling under the historic and a risk - neutral measures , and one may hope that the performance of qtsm models can be improved by introducing jumps .another improvement ( and quite sizable one ) is expected in pricing of out - of - the - money options near expiry , where the main contribution to the price comes from the jump part of the process .near expiry , however , a different approximate formulas are needed , which use more detailed information about the jump part of the processes than the skewness and kurtosis .these formulas are similar to approximate formulas for out - of - the - money options on stocks developed in levendorski ( 2003 ) , and can be derived by the same reasoning .   in section [ sectlevy ] ,we list families of exponential lvy processes used in empirical and theoretical studies of financial markets . in section [ sectgen ] , we formulate the pricing problem for an interest rate derivative of the european type , and by using the feynman - kac theorem , reduce the pricing problem to the boundary problem for an integro - differential equation .we also explain the scheme of the asymptotic pricing . in section [ sectbondg ] ,we recall the solution of the bond pricing problem in the one - factor gaussian case , and indicate the properties of the solution which are crucial for our asymptotic method . in section [ sectbond1 ] , we demonstrate our method in the simplest case of the one - factor lvy model for the bond price , and present numerical examples . in section [ exten ] , we consider possible specifications of the market price of risk , the generalization for the multi - factor case , derive approximate formulas for interest rate derivatives near expiry , and suggest a procedure of parameter fitting based on the asymptotic expansions . in section [ concl ] , we summarize our results , and compare the lvy qtsm with multi - factor gaussian qtsm . in the appendix , technical results are proven .as early as in 1963 , mandelbrot suggested to use stable lvy processes .the modelling with stable lvy processes is not quite realistic since the tails of lvy stable distributions are too fat ( polynomially decaying ) , whereas the tails of distributions of returns observed in real financial markets exhibit exponential decay .moreover , the second moment of a lvy stable distribution is infinite ( unless it is a gaussian one ) .this contradicts the observed convergence to the gaussian distribution over a longer time scale , and even worse , the underlying stock itself should have the infinite price under the stable lvy process , which makes the model inconsistent for pricing purposes .starting with the beginning of the 90th , several families of lvy processes with probability distributions having exponentially decaying tails have been used to describe the behavior of stock prices in real financial markets :  * variance gamma processes ( vgp ) constructed and used by madan and co - authors in a series of papers during 90th ( see madan et al .( 1998 ) and the bibliography therein ) ; * hyperbolic processes ( hp ) were constructed and used by eberlein and co - authors ( see eberlein et al .( 1998 ) , eberlein and prause ( 1999 ) ) ; hyperbolic distributions were constructed by barndorff - nielsen ( 1977 ) ) ; * normal inverse gaussian processes ( nig ) were introduced by barndorff - nielsen ( 1998 ) and used to model german stocks by barndorff - nielsen and jiang ( 1998 ) ; * truncated lvy processes ( tlp ) constructed by koponen ( 1995 ) were used for modeling in real financial markets by bouchaud and potters ( 1997 ) , cont et al ( 1997 ) and matacz ( 2001 ) ; the extended koponen family was constructed in boyarchenko and levendorski ( 2000 ) ( the generalization was needed since probability distribution of koponen s family have tails of the same rate of exponential decay whereas in real financial markets , the left tail is usually much fatter ; in carr et al ( 2002 ) and boyarchenko and levendorski ( 2002a , b ) , the extended koponen family is called cgmy - model and kobol family , respectively ) . *normal tempered stable lvy processes were constructed in barndorff - nielsen and levendorski ( 2001 ) and barndorff - nielsen and shephard ( 2001a ) ; they contain nig as a subclass .  in boyarchenko and levendorski ( 2000 ) ,a general class of lvy processes , which contained all the classes listed above modulo certain reservation about vgp was introduced , under the name generalized truncated lvy processes .later , in barndorff - nielsen and levendorski ( 2001 ) , the name : regular lvy processes of exponential type " ( rlpe ) was suggested . for a more detailed exposition ,see boyarchenko and levendorski ( 2002a , 2002b ) . in order to present examples , recall that a lvy process can be completely specified by its characteristic exponent , , definable from the equality =e^{-t\psi(\xi)} ] and such that as in the tube domain , ( see boyarchenko and levendorski ( 2002b ) ) .the and are called the order and type of the process .  to simplify the justification of the use of the feynman - kac formula, we add unnecessary condition : for any multi - index , there exists a constant such that for all in the tube domain , .notice that this condition holds for all model classes of rppe s .  in the appendix , by making use of the feynman - kac formula , we will prove the following theorem .[ thmfk1 ] let the stochastic factor satisfy ( [ sde ] ) , ( [ cka ] ) , ( [ req ] ) , ( [ cexpect1 ] ) , ( [ rlpe1 ] ) , ( [ rlpe2 ] ) , and ( [ rlpe3 ] ) , let be given by ( [ req ] ) , and let be a continuous function , which admits a bound ( [ boundg ] ).  then a ) the stochastic expression ( [ defcc ] ) defines a continuous function , which admits an estimate latexmath:[\[\label{boundf }  independent of and ;  \b ) is a unique solution to the following problem : where is the infinitesimal generator of .recall that the infinitesimal generator of the lvy process , , can be represented in the form of a pseudo - differential operator ( pdo ) with the symbol : .a pdo with the symbol acts on sufficiently regular functions as follows : where is the fourier transform of : in particular , the partial derivative is the pdo with the symbol .the asymptotic pricing formulas will be derived under the following conditions .assume that the characteristic exponent of the driving lvy process depends on a small parameter : and satisfies the following three conditions .first , we require that the in the definition of the tube domain satisfies .the next two conditions are formulated for in the tube domain :  \1 ) in the region , admits an estimate where  ] , decays as , as , and , the fourier transform of w.r.t .the first argument , decays as , as . to be more specific , .\end{aligned}\ ] ]we conclude that for any , in the region  ] . by making the inverse fourier transform ,we obtain is independent of . by applying the feynman - kac theorem to ( [ fk2])-([bc2 ] ) , the representation of in the form of the stochastic integral results : ,\ ] ] and from ( [ bounddf0 ] ), we derive an estimate latexmath:[\[\label{boundf1 }  of , and  ] .  unlike ( [ boundf1 ] ), we have a polynomially growing factor in the rhs of ( [ boundf2 ] ) .notice , however , that for practical purposes , one needs to know the bond price for small values of , hence for small values of , and therefore the polynomially growing factor does not matter much .   to simplify the calculation of the next terms of the asymptotics ,it is advantageous to change the variables in equations similar to ( [ fk5 ] ) : where is given by ( [ defka2 ] ) , and is the solution to the cauchy problem that is , the same change of the variables simplifies the calculation of .introduce an operator by . under the change of variables( [ changex ] ) , and , therefore and we can rewrite ( [ fk5 ] ) in the form where , .clearly , is a polynomial in of the same order as : and the coefficients can easily be calculated by using formulas for the coefficients of or , better , independently . under the change of variables ( [ changex ] ) , becomes where therefore where and .the solution of ( [ fk55 ] ) subject to is a polynomial in of the same order as : whose coefficients can easily be found by integration : after that we make the inverse change of variables , and calculate .suppose that the approximation of order has been found : where , and , , are polynomials in with coefficients depending on on : we look for the next term of the asymptotics in the form , where is the solution to the problem where is the collection of terms of order in the expression that is , set and multiply ( [ fk6 ] ) by .we obtain equation ( [ fkg1 ] ) allows us to simplify ( [ fk66 ] ) and obtain a problem in the half - space with the unknown : from ( [ deftg1 ] ) , is a polynomial in as well : where , and any of the coefficients may be zero , that is , the order of may be less than .denote by the order of .  by making the change of variables ( [ changex ] ) , we simplify ( [ fk7 ] ) : where .since is a polynomial in of order : ( the coefficients will be calculated in subsection [ subcalcg ] ) , also is : and the coefficients are easily found by integration : for    we can rewrite ( [ deftg1 ] ) as by using the initial data , we find and from ( [ defg ] ) and ( [ solf1])-([solf2 ] ) step by step. in particular , is given by ( [ defg0 ] ) , and where are polynomials of degree 6 and 4 , respectively .hence , where and are polynomials of degree 6 and 4 , respectively , which solve ( [ fk77 ] ) with and in the rhs ( and satisfy the initial condition ) .the reader can use ( [ solf1])-([solf2 ] ) to obtain explicit formulas for coefficients of in terms of coefficients of . notice that though the latter can be written explicitly , in practical implementation of the method , it is simpler to write a program which calculates the coefficients of a polynomial , given coefficients of a polynomial , and use this program to calculate ( and for one wish it , though in applications , it seems not a reasonable thing to do ) .we see that the second order approximation can be written in the form where , , and are polynomials in with coefficients depending on - and parameters , , , . for practical applications,([second1 ] ) can be rewritten in the form where can be inferred from the data .denote by is the price of the bond , and by the price in the gaussian model ; then ( [ second2 ] ) becomes where by using the formulas for the yield and forward rate we obtain approximate formulas where and where    ( the author thanks nina boyarchenko for writing the programs for numerical examples and checking the algebra in the previous two sections . )we take the simplest model for : , and constant , , and .we also fix , and study shapes of correction terms to the bond price , yield and forward rates in ( [ second3 ] ) , ( [ yield ] ) and ( [ frate ] ) ( see figures 13 ) .  from figure 3, we clearly see that it is the first correction term , , that can account for the hump of the forward rate curve - provided the skewness is negative and not too small in modulus . in the next three figures 46 , we plot the bond price , yield and forward rate ; first , the leading term ( dots ) , then the formula with the first correction term taken into account ( solid line ) , and finally , the formula with the two correction terms ( dotted line ) .we take the same parameters as above , and , .we see that fairly large skewness does produce a hump - shaped forward rate curve , when the gaussian curve has no hump ; the asymptotic formulas are applicable since is small w.r.t . , and is small w.r.t . .  in the last series of figures ( figures 79 ) , we fix small , and show how the shapes of the curves vary with the skewness .notice that if one fits the gaussian qtsm and the non - gaussian one to the same data set , then the leading gaussian - like leading term in the lvy model will be determined by slightly different drift and volatility parameters and the spot value of the factor , hence the pictures above do not describe quite accurately the difference between the gaussian and lvy models .however , the shapes of the curves in the gaussian model and the leading gaussian approximation in the non - gaussian models are the same ( and not differ much ) .thus , one can use the pictures to get a feeling what a difference between the gaussian and non - gaussian qtsms can be .all the constructions in the previous two sections admit modifications for the multi - factor case .the only two differences are  * in the gaussian approximation , the , and are matrix functions .the first two can be found by solving a system of linear ode s ( for detailed realization in the gaussian qtsm - model , see kim ( 2003 ) ) , and then is found by the integration ; * the correction terms are polynomials not in one factor but in several factors .this leads to systems of linear ode s whose unknowns are coefficients at factors of order , where depends on the step of the method .  certainly , the systems of linear ode to be solved become significantly larger but they remain linear systems ; therefore , the numerical solution is fairly stable .assume that under the historic measure , the dynamics of is given by where is an -dimensional lvy process with the characteristic exponent .we assume that admits the analytic continuation into the tube domain , where is an open set containing 0 . to specify the interest rate dynamics under a risk - neutral measure , , we consider first the state price deflator in the form , where obeys the sde the vector represent the market prices of risk of the factors .notice that for processes with jumps , one can not use an arbitrary .for the bond to be priced , it is necessary that , and the condition suffices for the bond and options on the yield to be priced ( to be more specific , any payoff which admits a polynomial bound with respect to factors is admissible ; if a payoff grows exponentially , then additional restrictions on must be imposed ) .set .it is easy to check that under the condition , is the characteristic exponent of a lvy process , call it .the new measure is the esscher transform of popular in the literature on the pricing of options on stocks ( see e.g. eberlein et al . (1998 ) and boyarchenko and levendorski ( 2002b ) ) .notice that is analytic in the tube domain , and choose .the straightforward calculations show that the pricing formula \ ] ] can be written as the pricing formula under the risk - neutral measure : .\ ] ] it is possible that in some situations , the specification of risk ( [ sdeq ] ) is not sufficiently flexible , and in the gaussian qtsm , ahn et al .( 2002 ) consider a more general model for prices of risk .this more general specification is not applicable in lvy models with exponentially decaying lvy densities ; however , there is an additional flexibility in model with jumps , which may provide any number of additional degrees of freedom : one can use the pricing formula ( [ prq ] ) with any lvy process provided the difference of lvy densities of and is of finite variation ( this can be shown as in the case of pricing of derivatives on a stock ; see e.g. carr et al .( 2002 ) ) .an additional restriction should be taken into account if we want to use the approximate formulas of sections 45 : the lvy densities under and should decay exponentially , and sufficiently fast .notice that contrary to the change of measure in the gaussian model , the change of measure in the non - gaussian model may lead to the changes ( albeit small ) in the instantaneous moments of order two .let be the price of an european style derivative contract with the pay - off , where .a typical example is a call option on the yield with the pay - off . for a fixed ,the formulas in the preceding section allows one to find an approximation to as a function of the spot values of factors ( and of ) . in the gaussian model, this is a quadratic function , and in the non - gaussian model , it is a fourth - order polynomial with small coefficients of order 3 and 4 .hence , we can calculate the two roots of the equation , which are close to the two roots in the gaussian approximation , by using simple perturbation technique .denote these roots by , and represent the pay - off function in the form where }(x)(r(x)-k).\ ] ] by using the perturbation technique once again , we can calculate the fourier transforms of functions , and reduce the problem of the calculation of to the family of problems considered in sections 45 , with the pay - off instead of 1 .the modification of the asymptotic calculations is straightforward albeit lengthy . in the end, we make the inverse fourier transform , and obtain an asymptotic formula for the price , . for a numerical realization of this formula, one needs to choose an appropriate grid in the frequency domain , and for each from the grid , solve a number of systems of ode s .the inverse fft finishes the job .notice that this is a variant of the standard transform method ( see e.g. duffie et al .( 2000 ) , chacko and das ( 2002 ) and the bibliography therein ) .when using this scheme , one should remember that due to the non - smoothness of the pay - off at the money , the approximate formulas will not work well near expiry , especially near expiry and strike .fortunately , near the strike , a different approximation and much simpler one can be derived .fix , and introduce denote by the solution to the affine model with the short rate modelled by ( [ rx0 ] ) , and the same dynamics of the factors under the risk - neutral measure and pay - off as in the initial qtsm .[ nearexpfirst ] as and .the proof of theorem [ nearexpfirst ] is relatively straightforward , and it relies on the exponential decay of the density of jumps . since the solution in the affine model with jumps is well - known and fairly simple , we get an efficient approximation to the option price in the non - gaussian qtsm. should one wish it , the correction term to the formula can be obtained as the price of a derivative security ( in the same affine term structure model ) , which pays the stream of dividends at rate . for in - the - money options , one can use a constant function instead of .  finally , for out - of - the - money options ,a simpler approximation can be derived , in the form where depends only on the lvy density of the process but not on the gaussian part and drift . for the case of the call option on the yield considered above ,essentially the same calculations as in levendorski ( 2003 ) give for  and the explicit analytic expression in terms of parameters of model classes of rlpe processes can be derived ( the modification of the calculations in levendorski ( 2003 ) made for and is straightforward ) .far from expiry , the leading term of the yields depends on the instantaneous moments of order one and two , and for out - of - the - money options on yields , near expiry , the leading term depends only on the jump part of the process .this allows us to suggest the following scheme of the fitting the model to the data , under a risk - neutral measure .  1 .infer parameters of the gaussian model ( including the spot values of the factors ) from the data on yields . hereone can use the efficient method of moments as in ahn et al .( 2002 ) or an estimation method integrated with the extraction of the state variables ( the extended - kalman - filter - based quasi - maximum - likelihood estimate ) as in kim ( 2003 ) , say .we regard these parameters values as a zero - order approximation to the spot values of factors , drift , mean reverting and variance - covariance parameters of the non - gaussian model .2 .  given the spot values of the factors, one can infer the conditional characteristic function of the process from empirical data as in singleton ( 1999 ) and calculate the moments up to order 4 , or infer these moments as in collin - dufresne et al .however , in order to identify the contribution of jumps more accurately , the following steps seems to be reasonable .3 .  choose a parametrized model for the lvy density ( or one of the standard lvy models described in section 2 ) , and by using the prices on interest rate derivatives near expiry and approximate formulas near expiry , fit the parameters of the lvy density , and calculate the instantaneous moments of order 3 and 4 .calculate the correction terms to the yields in the gaussian approximation by using the asymptotic formulas of section 5 and the zero - order approximations for the parameters in the gaussian approximation , and moments of order 3 and 4 .subtract the correction terms from the data , and use the new data set in the gaussian procedure to infer the corrected values of the spot factors and the first two moments .step 3 can be repeated in order to get a corrected specification of the lvy density and moments of order 3 and 4 .notice that in order that this procedure be consistent , the resulting moments of order 3 and 4 must be small relative to the smallest eigenvalue of the matrix of the instantaneous second moments .we constructed a class of qtsm models with a regular lvy process of exponential type in place of the gaussian one in standard qtsm . by using the feynman - kac formula ,we have reduced the pricing problem for an interest rate derivative to a boundary problem for a pseudo - differential operator . in the case of the bond , we have found an approximate solution to the boundary problem assuming that the tails of probability densities of a process decay sufficiently fast .the leading term of the approximate solution looks as in the gaussian model ( even when the underlying process has no gaussian component ) , and the correction terms depend on skewness and kurtosis .numerical examples are produced to show that by changing skewness and kurtosis , various shapes of the forward rate curve can be obtained .in particular , negative skewness can produce a hump - shaped forward rate curve even when the gaussian curve has no hump : the very non - gaussianity of the process is ( one of ) causes of the hump of the forward rate curve .bond prices and the yield curve also change but the types of the shape of the curves remain essentially the same .  we discussed possible choices of a risk - neutral measure , and indicated additional flexibility which the usage of jumps provides . a brief outline of asymptotic pricing in the multi - factor case , and the pricing of interest rate derivatives is given .we derived simple asymptotic formulas for option prices near expiry , and suggested a procedure of the fitting of the model to the data .notice that the use of pseudo - diffusions for option pricing instead of gaussian models with approximately the same number of parameters is advantageous not only near expiry where the latter are expected to produce serious errors .if the number of parameters is approximately the same , then the number of factors in the former is smaller than in the latter , therefore the use of the inverse fft simplifies significantly in the non - gaussian model , and the numerical procedure becomes much more stable .by using the decomposition , where and are non - negative , we see that it suffices to prove theorem [ thmfk1 ] for continuous non - negative .fix such that for all , , , and for any , set .then is a continuous function with the compact support .define by ( [ defcc ] ) with in the rhs . for any , as , and by the monotone convergence theorem , .notice that in the sense of generalized functions : for any non - negative , below we will show that  a.  let be continuous and satisfy ( [ boundg ] ) ; then problem ( [ fk1])-([bc1 ] ) has a unique continuous solution , which satisfies ( [ boundf ] ) ; b.  in the half - space , is of the class w.r.t . ; c.  as , in the sense of generalized functions ; d.  if is a continuous function of the compact support , then is given by ( [ defcc ] ) .  by ( iv ) , , by ( iii ) and ( ii ) , the limit in ( iii )is a continuous function , and since we already know that pointwise , we conclude that solves the problem ( [ fk1])-([bc1 ] ) , and finish the proof of theorem [ thmfk1 ] .it remains to prove ( i)-(iv ) .we start with the proof of ( i ) .assume that is diagonalizable : there exists a matrix such that is a diagonal matrix with the diagonal entries ( if is not diagonalizable , an additional step is to be made - see the end of the proof of ( i ) ) . by making the change of variables , we reduce to the case of the diagonal matrix ; the becomes . to simplify the notation below ( and without loss of generality ) , we assume that itself is diagonal . in ( [ fk1])-([bc1 ] ) , change the variables : where is the solution to ode subject to , and set we obtain notice that ) ] , and satisfies estimate where and depend on but not on and $ ].  estimates ( [ rlpe1 ] ) , ( [ rlpe2 ] ) , ( [ rlpe3 ] ) , and ( [ estr2der])-([estr2 ] ) allows one to apply the standard technique of construction of the inverse to the operator of a boundary problem for pdo to problem ( [ afk3])-([abc3 ] ) .this technique is based on the construction of an appropriate partition of unity , localization and patching of an approximate inverse from local inverses ; for the realization of this general scheme for many classes of pdo see levendorski ( 1993 ) . in op ., boundary value problems in -based spaces were considered whereas here we need corresponding results for -based spaces .this modification is straightforward : see e.g. the modification in barndorff - nielsen and levendorski ( 2001 ) , for a different class of pdo .  in the result, we obtain that , the continuous solution to problem ( [ afk3])-([abc3 ] ) , which admits estimate ( [ boundf ] ) , exists and it is unique . moreover , it is of the class in the half - plane , and satisfies estimate }|e^{-{\omega}|y|}v(y , \tau)|\le c \sup_{{{{\bf r}}^n } }   if is not diagonalizable , then prior to the change of variables , an additional change of variables , where , is needed .the will be replaced by , which generically has pairwise distinct eigenvalues and hence , diagonalizable ; will change as well , and becomes after that we make the same changes of variables ( using the new and ) .  to prove ( iii ) , we take , and apply the argument above starting with instead of and instead of . since estimate ( [ estv ] ) implies that }|e^{-{\omega}_1|x|}(f(x , t)-f^m(x , t))|\to 0\quad { \rm as}\ m\to \infty,\ ] ] which proves ( iii ) .it remains to prove ( iv ) .we have seen that for a continuous with compact support , is continuous in the half - plane , and of the class in the open half - plane .moreover , decays faster than as , for any ( notice that a continuous of the compact support satisfies ( [ boundg ] ) for any , and in order that the proof of estimate ( [ boundf ] ) remain valid , we may use any , negative ones in particular ) .further , is continuous and semi - bounded from below .these conditions are more than sufficient for the feynman - kac theorem to be applicable ( for instance , at this stage , we can repeat the proof on p.274 in rogers and williams ( 1994 ) ) , which gives ( iv ) .theorem [ thmfk1 ] has been proved .

Abstract: the non - gaussianity of processes observed in financial markets and relatively good performance of gaussian models can be reconciled by replacing the brownian motion with lvy processes whose lvy densities decay as or faster , where is large . this leads to asymptotic pricing models . the leading term , , is the price in the gaussian model with the same instantaneous drift and variance . the first correction term depends on the instantaneous moments of order up to three , that is , the skewness is taken into account , the next term depends on moments of order four ( kurtosis ) as well , etc . in empirical studies , the asymptotic formula can be applied without explicit specification of the underlying process : it suffices to assume that the instantaneous moments of order greater than two are small w.r.t . moments of order one and two , and use empirical data on moments of order up to three or four . as an application , the bond pricing problem in the non - gaussian quadratic term structure model is solved .  for pricing of options near expiry , a different set of asymptotic formulas is developed ; they require more detailed specification of the process , especially of its jump part . the leading terms of these formulas depends on the jump part of the process only , so that they can be used in empirical studies to identify the jump characteristics of the process .     department of economics , the university of texas at austin  key words : quadratic term structure models , lvy processes , asymptotic solutions


high - dimensional data are increasingly collected in statistical applications , which include biological experiments , climate and environmental studies , financial observations and others . the high dimensionality calls for new statistical methodologies which are adaptive to this new feature of the modern statistical data .the covariance matrix for a -dimensional random vector is an important measure on the dependence among components of .the sample covariance , constructed based on independent copies of , is a key ingredient in many statistical procedures in the conventional multivariate analysis [ and ] where the data dimension is regarded as fixed .the widespread use of in the conventional multivariate procedures is largely due to being a consistent estimator of when is fixed or small relative to the sample size .however , for high - dimensional data such that  ] , and that for the tapering estimator is , where is an index value for a `` bandable '' class of covariances \\[-8pt ] & & \hspace*{12pt}\mbox{and  } 0<\varepsilon_{0}\leq\lambda_{\min}(\sigma)\leq \lambda _ { \max}(\sigma)\leq\varepsilon_{0}^{-1 }  \biggr\}.\nonumber\end{aligned}\ ] ] the range of bandwidths in the hypothesis ( [ eq : hypo ] ) should cover the above optimal rates when .we note that is valid if and only if , and the latter implies that .a strategy is to construct an unbiased estimator of and use it to develop the test statistic .let be the sum of squares of the sub - diagonal of .then , . it can be checked that an unbiased estimator of is where denotes summation over mutually different subscripts shown and .the reason to sum over different indices is for easier manipulations with the mean and variance of the final test statistic and to establish the asymptotic normality .the latter leads to a test procedure for the bandedness .we consider the following statistic : as each is invariant under the location shift , is also location shift invariant .hence , without loss of generality , we assume .  to facilitate our analysis , as and , we assume a multivariate model for the high - dimensional data .[ ass1 ] ( i ) are independent and identically distributed ( i.i.d . )-dimensional random vectors such that where is a constant matrix with , , and are i.i.d .-dimensional random vectors such that and .\(ii ) write .each has uniformly bounded moment , and there exist finite constants and such that for , , and for any integers with  whenever are distinct subscripts .the requirement of common third and fourth moments of is not essential and is purely for the sake of simpler notation .our theory allows different third and fourth moments as long as they are uniformly bounded , which are actually assured by having uniformly bounded moment . the asymptotic framework that regulates the sample size , the dimensionality and the covariance is the following .  [ ass2 ] as , , and .we note that includes , the `` large , small '' paradigm , but may not imply .different from the usual approach of specifying an explicit growth rate of with respect to , assumption [ ass2 ] requires ratio of to shrinks at the rate of or smaller .the latter is stronger than .it is needed due to possible diverging bandwidths .let be the class of covariances satisfying the last part of assumption [ ass2 ] .the class includes the `` bandable '' class of given in ( [ eq : bandable ] ) for the banding estimation . to appreciate this ,let be the eigenvalues of .if the smallest and largest eigenvalues are bounded away from 0 and respectively , then therefore , the `` bandable '' covariances are contained in .now suppose that has exactly zero eigenvalues and being the smallest nonzero eigenvalue .then thus , is in as long as is bounded and for some as .the latter means that the class is likely to contain the class considered in , which allows the smallest eigenvalue to diminish to zero .it can be also checked that the following two covariances , are members of if are uniformly bounded from infinity and zero respectively .we first describe the basic properties of the statistic  defined in ( [ eq : wnk0 ] ) .let \\[-8pt ] & & { } + \frac{4}{n}\delta\operatorname{tr}\bigl\{\gamma'\bigl(\sigma -b_{k}(\sigma)\bigr)\gamma \circ \gamma'\bigl(\sigma - b_{k}(\sigma)\bigr)\gamma\bigr\},\nonumber % % \end{aligned}\ ] ] where for two matrices and .[ pro1 ] under assumptions [ ass1 ] and [ ass2 ] , \quad\mbox{and}\quad % %  { \operatorname{var}}(w_{nk})=\nu_{nk}^{2}+o(\nu_{nk}^{2 } ) .% % \ ] ]  the proposition indicates that under , /{n},\ ] ] and is the leading order variance of .it can be shown that . since  and , we have for a constant and .we note that as since .in particular , if is fixed , .the following theorem establishes the asymptotic normality of .[ teo1 ] under assumptions [ ass1 ] and [ ass2 ] , and if , }{\nu _ { nk}}\stackrel{d}{\rightarrow } n(0,1).\ ] ]  in order to formulate a test procedure based on the asymptotic normality , we need to estimate  ] under  .let be the estimator , whose consistency to  ] and , proposition [ pro2 ] means that , under , }\stackrel{p}{\rightarrow}1 ] is the square of frobenius norm of the difference between and its -banded version , and measures the level of noise in the statistic  .then , the power of the test under is }{\nu _ { nk}}\geq\frac{2 z_{\alpha}v_{nk}}{n\nu_{nk}}-\delta_{nk } \biggr ) .% % \end{aligned}\ ] ] since , then for large .hence asymptotically , }{\nu _ { nk}}\geq z_{\alpha}\frac{v_{nk}}{\operatorname{tr}(\sigma ^{2})}-\delta_{nk } \biggr).\ ] ]  to gain more insight on the power , let /\operatorname{tr}(\sigma^2) ] . from propositions [ pro1 ] and [ pro2 ] , since is nonnegative definite ,  ] if is diverging but satisfies ( [ eq : k0 ] ) , and are uniformly bounded away from and .   and their first order differences of the simulated data .the dashed line in the right panel is .the true bandwidth is 5 . ]we would like to remark that the multiplier in s formation leads to  being `` free ranged '' as long as .if such multiplication is not administrated , namely by setting , the range of needs to be restricted properly to ensure convergence . the requirement of is to avoid situations where has segments of zero sub - diagonals followed by nonzero sub - diagonals when one moves away from the main diagonal .our estimator can be modified to suit such situations .however , we would not elaborate here for the sake of simplicity in the presentation . attaining the consistency of with diverging a smaller value .to better understand the theorem and the bandwidth estimator , we conducted a simulation study for , and with generated from model ( [ eq:5.1 ] ) with a multivariate normal distribution .the detailed simulation setting will be provided in section [ sec5 ] .figure [ fig1 ] presents box - plots of the modified statistics ( left panel ) and its first - order difference ( right panel ) , with .we see from the right panel that the first five boxes are relatively large , and is close to 0 while for .this indicates that five would be the bandwidth estimate .  in practical implementations with finite samples ,the bandwidth estimator may be sensitive to the tuning parameters and .note that , as revealed a few paragraphs earlier , should be significantly larger than for and close to for .such a pattern , as displayed in figure [ fig1 ] , indicates that is a change point for .this motivates us to consider a regression change - point detection algorithm for bandwidth estimation .consider , the difference between successive statistics , for , for a sufficiently large that covers the true bandwidth .the idea is to fit , at each candidate , a regression function to such that for all .we may fit a nonparametric , locally weighted linear regression [ ; fan and gijbels ( ) ] on to the left of with the smoothing window - width , where is a smoothing parameter , and fit a flat line at the level for to the right of .if is too small for the above nonparametric regression , a parametric polynomial regression may be conducted .let be the regression estimate , nonparametric or parametric , obtained over the set , and let be the absolute deviation of the fitted errors .then a bandwidth estimator , as we call the change - point estimator , is our empirical studies reported in section [ sec5 ] show this estimator worked quite well .bickel and levina ( , ) proposed a method to select the bandwidth based on a repeated random splitting of the original sample to two sub - samples of sizes and .let and be the sample covariances based the sub - samples of sizes and respectively , where denotes the split , for , where is the total numbers of sample splitting .the risk for each candidate is defined to be , where for a matrix , .an empirical version of the risk is and the bandwidth estimator is  recommended to be , and the number of random splits , , while suggested and using the frobenius norm instead of the norm . considered a similar method to select the bandwidth in their estimator .we note that these approaches can be adversely impacted by high dimensionality , due to the fact that may be a poor estimator of if is much larger than , as found in early works [ johnstone ( ) ; ] .in this section , we report results from simulation studies to verify the proposed test for the bandedness and the bandwidth estimator .we evaluate the performance of the proposed test under several different structures of covariance matrix for normal and gamma random vectors .we generate -dimensional independent and identical multivariate random vectors according to a model where is the bandwidth of the covariance , in all settings and the other coefficients will be specified shortly .two distributions are assigned to the i.i.d . : ( i ) the normal distribution ; ( ii ) the standardized distribution so that it has zero mean and unit variance . to mimic the `` large , small  '' paradigm , we choose and respectively .we first evaluate the size of the proposed test under the null hypothesis for ( diagonal ) , and .the coefficients for are : and 0.5 , respectively , for ; , and and , respectively , for ; and for . to assess the power , we generate data according to ( [ eq:5.1 ] ) so that and test for for and , respectively , with the values being the same with those in the corresponding in the simulation for the size reported above .we note that this design , having the bandwidth of the null hypothesis adjacent to the true bandwidth , is the hardest for the test , as the null and the alternative is the closest , given the setting of the parameters  .all the simulation results are based on 1000 simulations .we also evaluate the test proposed in cai and jiang ( ) , based on the asymptotic distribution of the coherence statistic under the same simulation settings used for the proposed test .the test encountered a very severe size distortion in that the real sizes are much less than the nominal level of , which also caused the power of the test to be unfavorably low .for these reasons , we will not report the simulation results of the test .the coherence statistic is the largest pearson correlation coefficients among all pairs of different components in , and is an extreme value - type statistic .extreme value statistics are known to be slowly converging , and a computing intensive method is needed to speed up its convergence .the asymptotic distribution established in cai and jiang ( ) may be the foundation to justify such a method .@ & & + & & + & & + & & + & * 50 * & * 100 * & * 300 * & * 600 * & * 50 * & * 100 * & * 300 * & * 600 * +  + 20 & 0.069 & 0.065 & 0.061 & 0.066 & 0.055 & 0.056 & 0.065 & 0.075 + 40 & 0.067 & 0.049 & 0.047 & 0.060 & 0.056 & 0.054 & 0.055 & 0.059 + 60 & 0.066 & 0.064 & 0.045 & 0.051 & 0.068 & 0.039 & 0.065 & 0.049 + [ 6pt ] +  + 20 & 0.069 & 0.061 & 0.056 & 0.060 & 0.062 & 0.058 & 0.069 & 0.069 + 40 & 0.061 & 0.048 & 0.048 & 0.069 & 0.059 & 0.049 & 0.069 & 0.075 + 60 & 0.045 & 0.053 & 0.056 & 0.067 & 0.048 & 0.061 & 0.068 & 0.059 + [ 3pt ] + 20 & 0.065 & 0.069 & 0.058 & 0.067 & 0.063 & 0.061 & 0.057 & 0.061 + 40 & 0.063 & 0.052 & 0.047 & 0.068 & 0.059 & 0.055 & 0.066 & 0.071 + 60 & 0.050 & 0.056 & 0.057 & 0.061 & 0.050 & 0.070 & 0.068 & 0.060 + [ 6pt ] +  + 20 & 0.058 & 0.050 & 0.055 & 0.058 & 0.056 & 0.046 & 0.062 & 0.062 + 40 & 0.049 & 0.042 & 0.051 & 0.058 & 0.059 & 0.048 & 0.076 & 0.071 + 60 & 0.050 & 0.043 & 0.065 & 0.064 & 0.040 & 0.063 & 0.065 & 0.052 + [ 3pt ] + 20 & 0.060 & 0.055 & 0.056 & 0.061 & 0.059 & 0.054 & 0.062 & 0.062 + 40 & 0.055 & 0.047 & 0.055 & 0.059 & 0.058 & 0.046 & 0.071 & 0.064 + 60 & 0.044 & 0.043 & 0.058 & 0.060 & 0.042 & 0.060 & 0.067 & 0.061 + [ 6pt ] + 20 & 0.045 & 0.058 & 0.067 & 0.059 & 0.050 & 0.061 & 0.054 & 0.064 + 40 & 0.043 & 0.054 & 0.049 & 0.061 & 0.041 & 0.052 & 0.065 & 0.064 + 60 & 0.031 & 0.046 & 0.065 & 0.069 & 0.034 & 0.040 & 0.053 & 0.048 +  table [ tab1 ] reports the empirical sizes of the proposed test at the 5% nominal significance for with and , respectively , under both the normal and gamma distributions .table [ tab2 ] summarizes the empirical power of the tests whose sizes are reported in table [ tab1 ] . to understand the power results ,table [ tab2 ] also contains the values of for each simulation setting .we observe from table [ tab1 ] that the test has reasonably empirical sizes , around 5% , and that the test is not sensitive to the dimensionality indicated by its robust performance .there is some size inflation , which is due to a number of factors , mainly to the dimensionality , the sample size and the approximation error of the finite sample distribution of the test statistic by the limiting normal distribution .we recall that the test statistic is a linear combination of -statistics , whose convergence to the limiting normal distribution can be slow . in the simulations for power evaluation ( reported in table [ tab2 ] ) , we designed the simulation so that a constant was maintained for a set of different , while was held fixed .the empirical powers reported in table [ tab2 ] show that the power is quite reflective to the sample size  and , namely larger or large leads to higher power .this is because as decreases , the signal of the test increases .so it becomes easier to distinguish the null hypothesis from the alternative .and after we controlled and , the power was not sensitive to at all , confirming a remark made at the end of section [ sec3 ] .@ & & + & & + & & + & & + & * 50 * & * 100 * & * 300 * & * 600 * & * 50 * & * 100 * & * 300 * & * 600 * +  +  + 20 & 0.300 & 0.313 & 0.330 & 0.336 & 0.315 & 0.312 & 0.340 & 0.312 + 40 & 0.683 & 0.722 & 0.711 & 0.702 & 0.710 & 0.721 & 0.752 & 0.741 + 60 & 0.962 & 0.964 & 0.952 & 0.954 & 0.958 & 0.955 & 0.950 & 0.949 + [ 3pt ] + 20 & 0.146 & 0.144 & 0.139 & 0.152 & 0.148 & 0.140 & 0.147 & 0.143 + 40 & 0.269 & 0.253 & 0.258 & 0.279 & 0.256 & 0.281 & 0.311 & 0.311 + 60 & 0.406 & 0.443 & 0.455 & 0.451 & 0.438 & 0.449 & 0.458 & 0.441 + [ 6pt ] +  + 20 & 0.090 & 0.112 & 0.119 & 0.123 & 0.096 & 0.112 & 0.108 & 0.118 + 40 & 0.149 & 0.181 & 0.178 & 0.200 & 0.161 & 0.169 & 0.218 & 0.196 + 60 & 0.261 & 0.284 & 0.328 & 0.314 & 0.246 & 0.297 & 0.290 & 0.284 +  for bandwidth estimation , we generate according to ( [ eq:5.1 ] ) . while we keep , the other coefficients for are :   , for ;   for ;   for and for ;   for and for .  the covariances have bandwidth 3 , 5 , 10 and 15 respectively .we evaluate two bandwidth estimators .one is given in ( [ eq : fixed ] ) with and , namely , and the other is the change - point estimator given in ( [ eq : empirical ] ) , applied on candidate whose -values for are larger than .we employ the loess algorithm in r to carry our the nonparametric regression estimation to the left of a  , with a default smoothing parameter ..9d2.9d2.9d2.9@ & & & + & & & + & & * method * & & & & + 20&40 & fixed & 0.58 ( 1.465 ) & 0.07 ( 0.946)&-0.5 ( 1.114 ) & -1.63 ( 1.931 ) + & & change - point & 0.60 ( 0.569)&-0.21 ( 0.518)&-1.48 ( 2.134 ) & 0.06 ( 1.734 ) + & & bla & -0.66 ( 0.855)&-0.86 ( 1.287)&-4.72 ( 2.202)&-9.19 ( 2.246 ) + & & blb & 0.59 ( 1.036)&-0.53 ( 1.460)&-3.97 ( 2.932)&-6.63 ( 4.403 ) + & & rlz & 0.11 ( 1.363)&-0.18 ( 1.855)&-2.55 ( 2.732)&-8.02 ( 2.760 ) + [ 3pt ] & 100 & fixed & 0.14 ( 0.636 ) & 0.1 ( 0.659 ) & -0.22 ( 0.440)&-0.96 ( 0.875 ) + & & change - point & 0.56 ( 0.499)&-0.07 ( 0.293)&-0.52 ( 0.882 ) & 0.18 ( 0.968 ) + & & bla & -0.09 ( 1.272 ) & 0.45 ( 1.617)&-2.33 ( 2.010)&-6.14 ( 2.686 ) + & & blb & 0.7 ( 1.219)&-0.26 ( 1.561)&-3.88 ( 2.772)&-7.29 ( 3.506 ) + & & rlz & 0.45 ( 1.861)&-0.59 ( 1.799)&-3.79 ( 2.203)&-8.8 ( 2.103 ) + [ 3pt ] & 200 & fixed & 0.01 ( 0.1 ) & & -0.12 ( 0.327)&-0.66 ( 0.728 ) + & & change - point & 0.67 ( 0.473 ) & & -0.18 ( 0.435 ) & 0.09 ( 0.379 ) + & & bla & 0.78 ( 2.077 ) & 1.14 ( 2.327)&-0.58 ( 2.637)&-2.55 ( 3.560 ) + & & blb & 1.18 ( 1.935)&-0.1 ( 2.302 ) & -2.91 ( 2.878)&-6.14 ( 3.579 ) + & & rlz & 0.55 ( 1.641)&-0.29 ( 1.719)&-4.6 ( 1.928 ) & -9.51 ( 1.823 ) + [ 6pt ] 40 & 80 & fixed & 0.14 ( 0.551 ) & 0.08 ( 0.464)&-0.01 ( 0.1 ) & -0.10 ( 0.302 ) + & & change - point & 0.47 ( 0.502)&-0.01 ( 0.100)&-0.12 ( 0.383 ) & 0.08 ( 0.273 ) + & & bla & -0.24 ( 0.780 ) & 0.23 ( 1.014)&-1.32 ( 1.663)&-3.55 ( 2.907 ) + & & blb & 1.5 ( 1.514 ) & 0.94 ( 1.427 ) & 0.06 ( 2.210)&-0.17 ( 3.260 ) + & & rlz & 1.05 ( 1.629 ) & 0.71 ( 2.222 ) & 0.72 ( 2.374 ) & 1.28 ( 3.229 ) + [ 3pt ] & 200 & fixed & & & & -0.04 ( 0.197 ) + & & change - point & 0.55 ( 0.500 ) & & -0.04 ( 0.281 ) & 0.02 ( 0.141 ) + & & bla & 0.29 ( 1.200 ) & 1.03 ( 1.322 ) & 0.28 ( 1.633)&-1.30 ( 2.285 ) + & & blb & 1.64 ( 1.605 ) & 1.24 ( 1.837 ) & 0.58 ( 2.833)&-0.1 ( 2.976 ) + & & rlz & 1.36 ( 2.435 ) & 1.16 ( 2.465 ) & 2.07 ( 3.647 ) & 1.07 ( 2.861 ) + [ 3pt ] & 400 & fixed & & & & + & & change - point & 0.56 ( 0.499 ) & & & + & & bla & 0.88 ( 1.754 ) & 1.5 ( 1.962 ) & 1.25 ( 2.240 ) & 0.22 ( 2.642 ) + & & blb & 2.61 ( 2.457 ) & 1.74 ( 2.493 ) & 0.68 ( 3.396 ) & 0.09 ( 3.715 ) + & & rlz & 2.19 ( 2.943 ) & 1.98 ( 3.369 ) & 1.17 ( 3.420)&-0.39 ( 2.821 ) +  .9d1.9d1.9d2.9@ & & & + & & & + & & * method * & & & & + 60&120 & fixed & 0.02 ( 0.141 ) & 0.08 ( 0.706 ) & 0.02 ( 0.2 ) & -0.01 ( 0.1 ) + & & change - point & 0.52 ( 0.502 ) & & & -0.01 ( 0.1 ) + & & bla & 0.22 ( 0.938 ) & 0.85 ( 0.989 ) & 0.14 ( 1.363)&-0.88 ( 1.659 ) + & & blb & 1.71 ( 1.458 ) & 1.52 ( 1.541 ) & 1.67 ( 2.108 ) & 1.49 ( 2.615 ) + & & rlz & 1.24 ( 1.753 ) & 0.71 ( 1.431 ) & 2.03 ( 2.683 ) & 2.13 ( 2.845 ) + [ 3pt ] & 300 & fixed & & & & + & & change - point & 0.58 ( 0.496 ) & & & + & & bla & 0.47 ( 1.439 ) & 1.56 ( 1.683 ) & 1.06 ( 2.136 ) & 0.70 ( 2.452 ) + & & blb & 2.15 ( 2.017 ) & 2.04 ( 2.474 ) & 1.73 ( 2.877 ) & 1.74 ( 2.922 ) + & & rlz & 1.68 ( 2.188 ) & 1.02 ( 2.383 ) & 2.45 ( 3.686 ) & 2.75 ( 3.331 ) + [ 3pt ] & 600 & fixed & & & & + & & change - point & 0.54 ( 0.501 ) & & & + & & bla & 1.05 ( 1.702 ) & 1.92 ( 2.102 ) & 2.01 ( 2.393 ) & 1.06 ( 2.490 ) + & & blb & 3.16 ( 2.631 ) & 2.87 ( 2.699 ) & 2.97 ( 3.532 ) & 1.33 ( 3.254 ) + & & rlz & 3.3 ( 3.721 ) & 3.23 ( 3.787 ) & 3.82 ( 4.029 ) & 2.7 ( 3.506 ) +  for each , we compare the proposed bandwidth estimators with the estimators advocated in bickel and levina ( , ) and .we choose to be 20 , 40 and 60 . for each , is chosen 2 times , 5 times and 10 times of , respectively .following the settings of bickel and levina ( , ) , is chosen to be and , respectively , and the number of random splits in ( [ eq : bl ] ) is .table [ tab3 ] reports the average empirical bias and standard deviation of the five bandwidth estimators based on 100 replications .we observe from table [ tab3 ] that the overall performance of the proposed estimators is better than those of bickel and levina ( , ) and , with smaller standard deviation and bias .moreover , as is increased , both the bias and standard deviation of the proposed estimators decreased , and are quite robust to , which is a nice property to have . for the estimators of bickel and levina ( , ) and , the bias and the standard deviation could increase along with the increase of , and are much larger than those of the proposed estimators .these are likely caused by the problems associated with the sample covariance matrix when the data dimension is high .in this section , we report an empirical study on a prostate cancer data set [ adam et al .( ) ] from protein mass spectroscopy , which was aimed to distinguish the healthy people from the ones with the cancer by analyzing the constituents of the proteins in the blood .adam et al . ( ) recorded for each blood serum sample , the intensity for a large number of time - of - flight values  .the time of flight is related to the mass over charge ratio of the constituent proteins .they collected the intensity in the total of 48,538 -sites and the full data set consisted of 157 healthy patients and 167 with cancer .-values and the first order differences for the healthy and the cancer groups for bandwidths larger than 50 .the -values of the test for for are too small to be considered for bandwidth estimation . ]tibshirani et al .( ) analyzed the data by the fused lasso .they ignored -ratios below 2000 to avoid chemical artifacts , and averaged the intensity recordings in consecutive blocks of 20 .these gave rise to a total of 2181 dimensions per observation . estimated the inverse of the covariance matrix of the intensities by an adaptive banding approach with a nested lasso penalty .they carried out additional averaging of the data of tibshirani et al .( ) in consecutive blocks of 10 , resulting in a total of 218 dimensions .we considered the standardized data of , and tested for the banded structure of the covariance matrix of the intensities .the test statistics , -values and the first order differences for the healthy and cancer groups are displayed in figure [ fig2 ] for bandwidths .we do not display in the figure for bandwidths less than 50 since the values of the test statistics are too large , and the associated -values for are too small for .these bandwidth estimates together with the shapes of the curves for the test statistics and the -values in figure [ fig2 ] suggest that the covariance matrix of the healthy group is likely to be banded , while the covariance of the cancer group may not be banded at all , given the very large bandwidth and the shape of the curve . for the cancer group ,as shown in figure [ fig2 ] , the test statistics are relatively flat for , and then fall sharply afterward , which indicates relatively small values in the covariance matrix from sub - diagonal 120 to 140 .however , there is a substantial contribution from sub - diagonals for .these are echoed in the -values displayed in panel ( b ) with almost stationary -values within the above mentioned range , followed by a sharp increase .panel ( d ) of figure [ fig2 ] displays a rather unsettled curve for , the difference between successive statistics  .these are all in sharp contrasts to those of the healthy group , indicating rather different covariance structures between the two groups .  at , we reject a when the statistic is larger than 3.29 . for the healthy group ,the smallest such that is not rejected is , while for the cancer group is 191 .we apply the bandwidth estimator ( [ eq : fixed ] ) with and .the estimated bandwidth for the health group is 121 and for the cancer group is 212 . at the same time , the bandwidth estimates , by employing bickel and levina s ( ) approach , are 144 for the healthy group and 193 for the cancer group .the one for the healthy group is much larger than the 121 we obtained earlier , using the estimator ( [ eq : fixed ] ) .we then apply the proposed regression change - point bandwidth estimator over a range of bandwidths whose associated -values for testing  are larger than . for the healthy group , the bandwidth range is ; for the cancer groupthe range is .we set the smoothing parameter in the loess procedure in r. the regression bandwidth estimator is for the healthy group , which is slightly larger than the 121 obtained from the estimator ( [ eq : fixed ] ) . for the cancer group ,the estimated bandwidth is 215 .this rather large estimated bandwidth suggests that , compared to the healthy group , there is substantially more dependence among the protein mass spectroscopy measurements among the cancer patients , and , in particular , the covariance may not be banded at all for this group of patients .we first introduce some notation . for , define and then , , and .let and for .we first establish some lemmas for later use .[ lem1 ] under assumptions [ ass1 ] and [ ass2 ] , .  since , by the independence between different observations, we have   note that let and =\sigma _ { l_1l_2}\sigma_{l_3l_4}+\sigma_{l_1l_3}\sigma_{l_2l_4}+\sigma _ { l_1l_4}\sigma_{l_2l_3} ] and for and any , the first statement is readily obtained .the second statement follows by applying lemmas [ lem1 ] , [ lem4 ] and the fact that .  proof of proposition [ pro2 ] it can be carried out following the same routes as those in lemmas [ lem1 ] and [ lem2 ] .specifically , it can be shown that . hence , .it is clear from the proof of proposition [ pro1 ] that .therefore , in order to derive the asymptotical distribution of the statistic , we only need to consider the asymptotical normality of .let , and for , be a sequence of -field generated by the data sequence .let denote the conditional expectation with respect to .write , where . then for every  , , is a martingale difference sequence with respect to the -fields .[ lem5 ] let . under assumptions [ ass1 ] and [ ass2 ] , as ,   first establish the first part of ( [ ea1 ] ) .noting that , we need only to show .note that \\ & & { } + \frac{2}{n } \biggl(\sum_{|l_1-l_2|>k}x_{tl_1}x_{tl_2}\sigma _ { l_1l_2}-\sum _ { |l_1-l_2|>k}\sigma^2_{l_1l_2 } \biggr ) . \ ] ] denote .let be the matrix with the entry being and ; then where is a constant and and   to prove , we intend to prove the variance of each is of small order of .  for , denote for any , then .note that for any and for any , except and .thus for any , we only need to verify that and are of small orders of .note that where is a constant . since , which indicates that .similarly , we can also show that .thus   following the same procedure , we can prove that for all the other , . since , we have .thus we have , and hence the first part of ( [ ea1 ] ) .  for the second part of ( [ ea1 ] ) , by simple algebra , we can rewrite as , where ,\\ s_{t3}&=&\frac{2}{n } \{x'_{t}\sigma x_{t}-\operatorname{tr}(\sigma ^2 ) \}\end{aligned}\ ] ] and .\ ] ] since , we have for a positive constant , in the following , we will prove the four terms on the right are of small orders of , respectively . to this end , note that where is a positive constant . since , and , then we have .thus ,   similarly , we can show that for and , . combining all the four parts together , we have , which leads to the second part of ( [ ea1 ] ) .denote and . then . for diverging , but satisfying ( [ eq : k0 ] ) ,we intend to prove diverging to uniformly on for any .and uniformly converges to in probability for any and , where and .[ lem6 ] under assumptions [ ass1 ] , [ ass2 ] and ( [ eq : k0 ] ) , if and are uniformly bounded away from and , for any , as :   , for any for any ;   , for any for any ;   , for any for any , where and .\(a ) if is bounded away from , similarly to the proof of lemmas [ lem1 ] and [ lem2 ] , it can be checked that .therefore , by chebyshev s inequality , for any , where the last inequality comes from the fact that .hence , which converge to since satisfies ( [ eq : k0 ] ) .consider , and denote by the above argument , as . for any , we have /v_{nk}\leq1/(1-\varepsilon r_{k})\leq1 + 2\varepsilon r_{k}\ ] ] for any .hence , for any , which implies that diverge uniformly on , by choosing  small enough .therefore , for any , by choosing small enough , there exists a such that for any , the conclusion follows by noting that as .the other two parts of the conclusion can be obtained similarly .for simplicity in the presentation , we omit them here .  proof of theorem [ teo1 ] by lemmas [ lem1 ] , [ lem5 ] and the martingale central limit theorem [ ] , it is readily shown that as , substituting for , theorem [ teo1 ] follows by noting .proof of theorem [ teo2 ] note that , and .it can be shown that for any , where .this means that for any , there exists a positive integer , such that for all , .then from ( [ eq : beta ] ) , }{\nu_{nk}}\geq z_{\alpha}\frac{v_{nk}}{{\operatorname { tr}}(\sigma ^{2})}-\delta_{nk } , b_{n,\eta } \biggr)\\[-2pt ] & \geq & p \biggl(\frac{w_{nk}-\operatorname{tr}(\sigma^2)+{\operatorname { tr}}[\{b_{k}(\sigma)\ } ^{2}]}{\nu_{nk}}\geq z_{\alpha}(1+\eta)-\delta_{nk } , b_{n,\eta } \biggr)\\[-2pt ] & \geq & p \biggl(\frac{w_{nk}-\operatorname{tr}(\sigma^2)+{\operatorname { tr}}[\{b_{k}(\sigma)\ } ^{2}]}{\nu_{nk}}\geq z_{\alpha}(1+\eta)-\delta_{nk } \biggr)-p(b_{n,\eta}^{c } ) . \ ] ] therefore , from theorem [ teo1 ] , }{\nu _ { nk}}\geq z_{\alpha}(1+\eta)-\delta_{nk } \biggr\}\\[-2pt ] & & { } -\limsup_{n\to\infty}p(b_{n,\eta}^{c})\\[-2pt ] & \geq & 1-\phi\bigl\{z_{\alpha}(1+\eta)-\liminf_{n\to\infty}\delta _{ nk}\bigr\ } -\varepsilon . \ ] ] the first part of the theorem follows by taking and .\(ii ) the condition implies that as .hence , .proof of theorem [ teo3 ] first consider the case where is bounded .consider to be a fixed sufficiently large integer . recall that , where by ( [ eq : wnk ] ) , since , we have , for any . note that thus , from ( [ eq : wnk ] ) , for , the condition implies that in probability , where .therefore , for and for .hence , for any , as ,   therefore , for any and any , for each , there exists a positive integer such that for all , and note that both and are finite , we can set an , which is larger than all  such that the above are satisfied .define , for , and for .then , for any , . hence , for any and , .  forthe case of diverging , consider and .for any and , let and .denote and then for any , we have for any and for any , which lead to for any .therefore , and from ( [ eq : fixed ] ) , we have .it follows that . since as by lemma [ lem6 ], we have .we thank an associate editor and two referees for constructive comments and suggestions which have lead to improvements in the presentation of the paper . we also thank professor ji zhu for sharing the data .

Abstract: motivated by the latest effort to employ banded matrices to estimate a high - dimensional covariance , we propose a test for being banded with possible diverging bandwidth . the test is adaptive to the `` large , small '' situations without assuming a specific parametric distribution for the data . we also formulate a consistent estimator for the bandwidth of a banded high - dimensional covariance matrix . the properties of the test and the bandwidth estimator are investigated by theoretical evaluations and simulation studies , as well as an empirical analysis on a protein mass spectroscopy data .  .  .


semantic information can be helpful in almost all aspects of natural language understanding , including word sense disambiguation , selectional restrictions , attachment decisions , and discourse processing .semantic knowledge can add a great deal of power and accuracy to natural language processing systems .but semantic information is difficult to obtain . in most cases ,semantic knowledge is encoded manually for each application .there have been a few large - scale efforts to create broad semantic knowledge bases , such as wordnet  and cyc  .while these efforts may be useful for some applications , we believe that they will never fully satisfy the need for semantic knowledge .many domains are characterized by their own sublanguage containing terms and jargon specific to the field .representing all sublanguages in a single knowledge base would be nearly impossible .furthermore , domain - specific semantic lexicons are useful for minimizing ambiguity problems . within the context of a restricted domain ,many polysemous words have a strong preference for one word sense , so knowing the most probable word sense in a domain can strongly constrain the ambiguity .we have been experimenting with a corpus - based method for building semantic lexicons semi - automatically .our system uses a text corpus and a small set of seed words for a category to identify other words that also belong to the category .the algorithm uses simple statistics and a bootstrapping mechanism to generate a ranked list of potential category words .a human then reviews the top words and selects the best ones for the dictionary .our approach is geared toward fast semantic lexicon construction : given a handful of seed words for a category and a representative text corpus , one can build a semantic lexicon for a category in just a few minutes .  in the first section , we describe the statistical bootstrapping algorithm for identifying candidate category words and ranking them .next , we describe experimental results for five categories . finally , we discuss our experiences with additional categories and seed word lists , and summarize our results .our work is based on the observation that category members are often surrounded by other category members in text , for example in conjunctions ( _ lions and tigers and bears _ ) , lists ( _ lions , tigers , bears ... _ ) , appositives ( _ the stallion , a white arabian _ ) , and nominal compounds ( _ arabian stallion _ ; _ tuna fish _ ) .given a few category members , we wondered whether it would be possible to collect surrounding contexts and use statistics to identify other words that also belong to the category .our approach was motivated by yarowsky s word sense disambiguation algorithm  and the notion of statistical salience , although our system uses somewhat different statistical measures and techniques .we begin with a small set of seed words for a category .we experimented with different numbers of seed words , but were surprised to find that only 5 seed words per category worked quite well .as an example , the seed word lists used in our experiments are shown below .the input to our system is a text corpus and an initial set of seed words for each category .ideally , the text corpus should contain many references to the category .our approach is designed for domain - specific text processing , so the text corpus should be a representative sample of texts for the domain and the categories should be semantic classes associated with the domain . given a text corpus and an initial seed word list for a category c , the algorithm for building a semantic lexicon is as follows :  1 .we identify all sentences in the text corpus that contain one of the seed words .each sentence is given to our parser , which segments the sentence into simple noun phrases , verb phrases , and prepositional phrases .for our purposes , we do not need any higher level parse structures .we collect small context windows surrounding each occurrence of a seed word as a head noun in the corpus . restricting the seed words to be head nounsensures that the seed word is the main concept of the noun phrase .also , this reduces the chance of finding different word senses of the seed word ( though multiple noun word senses may still be a problem ) .we use a very narrow context window consisting of only two words , the first noun to the word s right and the first noun to its left .we collected only nouns under the assumption that most , if not all , true category members would be nouns .the context windows do not cut across sentence boundaries .note that our context window is much narrower than those used by other researchers  .we experimented with larger window sizes and found that the narrow windows more consistently included words related to the target category .3 .  given the context windows for a category , we compute a category score for each word , which is essentially the conditional probability that the word appears in a category context .the category score of a word w for category c is defined as : + _ score(w , c ) = _ + note that this is not exactly a conditional probability because a single word occurrence can belong to more than one context window .for example , consider the sentence : _ i bought an ak-47 gun and an m-16 rifle . _ the word _m-16 _ would be in the context windows for both _ gun _ and _ rifle _ even though there was just one occurrence of it in the sentence .consequently , the category score for a word can be greater than 1 . 4 .next , we remove stopwords , numbers , and any words with a corpus frequency 5 .we used a stopword list containing about 30 general nouns , mostly pronouns ( e.g. , _i , he , she , they _ ) and determiners ( e.g. , _ this , that , those _ ) . the stopwords and numbers are not specific to any category and are common across many domains , so we felt it was safe to remove them .the remaining nouns are sorted by category score and ranked so that the nouns most strongly associated with the category appear at the top . 5 .the top five nouns that are not already seed words are added to the seed word list dynamically .we then go back to step 1 and repeat the process .this bootstrapping mechanism dynamically grows the seed word list so that each iteration produces a larger category context . in our experiments ,the top five nouns were added automatically without any human intervention , but this sometimes allows non - category words to dilute the growing seed word list .a few inappropriate words are not likely to have much impact , but many inappropriate words or a few highly frequent words can weaken the feedback process .one could have a person verify that each word belongs to the target category before adding it to the seed word list , but this would require human interaction at each iteration of the feedback cycle .we decided to see how well the technique could work without this additional human interaction , but the potential benefits of human feedback still need to be investigated .after several iterations , the seed word list typically contains many relevant category words .but more importantly , the ranked list contains many additional category words , especially near the top .the number of iterations can make a big difference in the quality of the ranked list .since new seed words are generated dynamically without manual review , the quality of the ranked list can deteriorate rapidly when too many non - category words become seed words . in our experiments , we found that about eight iterations usually worked well .the output of the system is the ranked list of nouns after the final iteration .the seed word list is thrown away .note that the original seed words were already known to be category members , and the new seed words are already in the ranked list because that is how they were selected .finally , a user must review the ranked list and identify the words that are true category members . how one defines a `` true '' category member is subjective andmay depend on the specific application , so we leave this exercise to a person .typically , the words near the top of the ranked list are highly associated with the category but the density of category words decreases as one proceeds down the list .the user may scan down the list until a sufficient number of category words is found , or as long as time permits .the words selected by the user are added to a permanent semantic lexicon with the appropriate category label .our goal is to allow a user to build a semantic lexicon for one or more categories using only a small set of known category members as seed words and a text corpus .the output is a ranked list of potential category words that a user can review to create a semantic lexicon quickly .the success of this approach depends on the quality of the ranked list , especially the density of category members near the top . in the next section ,we describe experiments to evaluate our system .we performed experiments with five categories to evaluate the effectiveness and generality of our approach : _ energy _ , _ financial _ , _ military _ , _ vehicles _ , and _ weapons_.the muc-4 development corpus ( 1700 texts ) was used as the text corpus  .we chose these five categories because they represented relatively different semantic classes , they were prevalent in the muc-4 corpus , and they seemed to be useful categories .  for each category , we began with the seed word lists shown in figure [ seed - words ] .we ran the bootstrapping algorithm for eight iterations , adding five new words to the seed word list after each cycle .after the final iteration , we had ranked lists of potential category words for each of the five categories .the top 45 words from each ranked list are shown in figure [ top - ranked - words ] .while the ranked lists are far from perfect , one can see that there are many category members near the top of each list .it is also apparent that a few additional heuristics could be used to remove many of the extraneous words .for example , our number processor failed to remove numbers with commas ( e.g. , _2,000 _ ) . andthe military category contains several ordinal numbers ( e.g. , _10th 3rd 1st _ ) that could be easily identified and removed .but the key question is whether the ranked list contains many true category members . since this is a subjective question , we set up an experiment involving human judges .  for each category , we selected the top 200 words from its ranked list and presented them to a user .we presented the words in random order so that the user had no idea how our system had ranked the words .this was done to minimize contextual effects ( e.g. , seeing five category members in a row might make someone more inclined to judge the next word as relevant ) .each category was judged by two people independently .the judges were asked to rate each word on a scale from 1 to 5 indicating how strongly it was associated with the category . since category judgements can be highly subjective , we gave them guidelines to help establish uniform criteria .the instructions that were given to the judges are shown in figure [ instructions ] .we asked the judges to rate the words on a scale from 1 to 5 because different degrees of category membership might be acceptable for different applications .some applications might require strict category membership , for example only words like _ gun _ , _ rifle _ , and _ bomb _ should be labeled as weapons .but from a practical perspective , subparts of category members might also be acceptable .for example , if a _ cartridge _ or _ trigger _ is mentioned in the context of an event , then one can infer that a gun was used . and for some applications , any word that is strongly associated with a category might be useful to include in the semantic lexicon .for example , words like _ ammunition _ or _ bullets _ are highly suggestive of a weapon . in the umass/ muc-4 information extraction system  , the words _ ammunition _ and _ bullets _ were defined as weapons , mainly for the purpose of selectional restrictions .the human judges estimated that it took them approximately 10 - 15 minutes , on average , to judge the 200 words for each category . since the instructions allowed the users to assign a zero to a word if they did not know what it meant , we manually removed the zeros and assigned ratings that we thought were appropriate .we considered ignoring the zeros , but some of the categories would have been severely impacted .for example , many of the legitimate weapons ( e.g. , m-16 and ar-15 ) were not known to the judges .fortunately , most of the unknown words were proper nouns with relatively unambiguous semantics , so we do not believe that this process compromised the integrity of the experiment .  finally , we graphed the results from the human judges .we counted the number of words judged as 5 s by either judge , the number of words judged as 5 s or 4 s by either judge , the number of words judged as 5 s , 4 s , or 3 s by either judge , and the number of words judged as either 5 s , 4 s , 3 s , or 2 s .we plotted the results after each 20 words , stepping down the ranked list , to see whether the words near the top of the list were more highly associated with the category than words farther down .we also wanted to see whether the number of category words leveled off or whether it continued to grow .the results from this experiment are shown in figures [ energy - graph]-[weapon - graph ] .  with the exception of the energy category, we were able to find 25 - 45 words that were judged as 4 s or 5 s for each category .this was our strictest test because only true category members ( or subparts of true category members ) earned this rating .although this might not seem like a lot of category words , 25 - 45 words is enough to produce a reasonable core semantic lexicon .for example , the words judged as 5 s for each category are shown in figure [ 5-words ] .  figure [ 5-words ] illustrates an important benefit of the corpus - based approach . by sifting through a large text corpus, the algorithm can find many relevant category words that a user would probably not enter in a semantic lexicon on their own .for example , suppose a user wanted to build a dictionary of vehicle words .most people would probably define words such as _ car _ , _ truck _ , _ plane _ , and _but it is doubtful that most people would think of words like _ gunships _ , _ fighter _ , _ carrier _ , and _ambulances_. the corpus - based algorithm is especially good at identifying words that are common in the text corpus even though they might not be commonly used in general . as another example , specific types of weapons ( e.g. , _m-16 _ , _ ar-15 _ , _ m-60 _ , or _m-79 _ ) might not even be known to most users , but they are abundant in the muc-4 corpus .  if we consider all the words rated as 3 s , 4 s , or 5 s , then we were able to find about 50 - 65 words for every category except energy .many of these words would be useful in a semantic dictionary for the category .for example , some of the words rated as 3 s for the vehicle category include : _ flight _ , _ flights _ , _ aviation _ , _ pilot _ , _ airport _ , and _ highways_.most of the words rated as 2 s are not specific to the target category , but some of them might be useful for certain tasks .for example , some words judged as 2 s for the energy category are : _ spill _ , _ pole _ , _ tower _ , and_ fields_. these words may appear in many different contexts , but in texts about energy topics these words are likely to be relevant and probably should be defined in the dictionary .therefore we expect that a user would likely keep some of these words in the semantic lexicon but would probably be very selective .  finally , the graphs show that most of the acquisition curves displayed positive slopes even at the end of the 200 words .this implies that more category words would likely have been found if the users had reviewed more than 200 words .the one exception , again , was the energy category , which we will discuss in the next section .the size of the ranked lists ranged from 442 for the financial category to 919 for the military category , so it would be interesting to know how many category members would have been found if we had given the entire lists to our judges .when we first began this work , we were unsure about what types of categories would be amenable to this approach .so we experimented with a number of different categories .fortunately , most of them worked fairly well , but some of them did not .we do not claim to understand exactly what types of categories will work well and which ones will not , but our early experiences did shed some light on the strengths and weaknesses of this approach .in addition to the previous five categories , we also experimented with categories for location , commercial , and person .the location category performed very well using seed words such as _ city _ , _ town _ , and _province_. we did nt formally evaluate this category because most of the category words were proper nouns and we did not expect that our judges would know what they were .but it is worth noting that this category achieved good results , presumably because location names often cluster together in appositives , conjunctions , and nominal compounds .for the commercial category , we chose seed words such as _ store _ , _ shop _ , and _market_. only a few new commercial words were identified , such as _ hotel _ and _ restaurant_. in retrospect , we realized that there were probably few words in the muc-4 corpus that referred to commercial establishments .( the muc-4 corpus mainly contains reports of terrorist and military events . )the relatively poor performance of the energy category was probably due to the same problem .if a category is not well - represented in the corpus then it is doomed because inappropriate words become seed words in the early iterations and quickly derail the feedback loop .the person category produced mixed results .some good category words were found , such as _ rebel _ , _ advisers _ , _ criminal _ , and _citizen_. but many of the words referred to organizations ( e.g. , _ fmln _ ) , groups ( e.g. , _ forces _ ) , and actions ( e.g. , _ attacks _ ) . some of these words seemed reasonable , but it was hard to draw a line between specific references to people and concepts like organizations and groups that may or may not consist entirely of people .the large proportion of action words also diluted the list .more experiments are needed to better understand whether this category is inherently difficult or whether a more carefully chosen set of seed words would improve performance .more experiments are also needed to evaluate different seed word lists .the algorithm is clearly sensitive to the initial seed words , but the degree of sensitivity is unknown . for the five categories reported in this paper , we arbitrarily chose a few words that were central members of the category .our initial seed words worked well enough that we did not experiment with them very much .but we did perform a few experiments varying the number of seed words . in general, we found that additional seed words tend to improve performance , but the results were not substantially different using five seed words or using ten .of course , there is also a law of diminishing returns : using a seed word list containing 60 category words is almost like creating a semantic lexicon for the category by hand !building semantic lexicons will always be a subjective process , and the quality of a semantic lexicon is highly dependent on the task for which it will be used .but there is no question that semantic knowledge is essential for many problems in natural language processing .most of the time semantic knowledge is defined manually for the target application , but several techniques have been developed for generating semantic knowledge automatically .some systems learn the meanings of unknown words using expectations derived from other word definitions in the surrounding context ( e.g. ,  ) .other approaches use example or case - based methods to match unknown word contexts against previously seen word contexts ( e.g. ,  ) .our task orientation is a bit different because we are trying to construct a semantic lexicon for a target category , instead of classifying unknown or polysemous words in context .to our knowledge , our system is the first one aimed at building semantic lexicons from raw text without using any additional semantic knowledge .the only lexical knowledge used by our parser is a part - of - speech dictionary for syntactic processing .although we used a hand - crafted part - of - speech dictionary for these experiments , statistical and corpus - based taggers are readily available ( e.g. ,  ) .our corpus - based approach is designed to support fast semantic lexicon construction .a user only needs to supply a representative text corpus and a small set of seed words for each target category .our experiments suggest that a core semantic lexicon can be built for each category with only 10 - 15 minutes of human interaction .while more work needs to be done to refine this procedure and characterize the types of categories it can handle , we believe that this is a promising approach for corpus - based semantic knowledge acquisition .this research was funded by nsf grant iri-9509820 and the university of utah research committee .we would like to thank david bean , jeff lorenzen , and kiri wagstaff for their help in judging our category lists .lehnert , w. , c. cardie , d. fisher , j. mccarthy , e. riloff , and s. soderland .1992 . . in _ proceedings of the fourth message understanding conference ( muc-4 ) _ , pages 282288 , san mateo , camorgan kaufmann .

Abstract: semantic knowledge can be a great asset to natural language processing systems , but it is usually hand - coded for each application . although some semantic information is available in general - purpose knowledge bases such as wordnet and cyc , many applications require domain - specific lexicons that represent words and categories for a particular topic . in this paper , we present a corpus - based method that can be used to build semantic lexicons for specific categories . the input to the system is a small set of seed words for a category and a representative text corpus . the output is a ranked list of words that are associated with the category . a user then reviews the top - ranked words and decides which ones should be entered in the semantic lexicon . in experiments with five categories , users typically found about 60 words per category in 10 - 15 minutes to build a core semantic lexicon .  #