spectral gmm estimation of continuous time processes abstract this paper derives a methodology for the estimation of continuous time stochastic models based on the characteristic function the estimation method does not require discretization of the stochastic process and it is simple to apply in practice the method is essentially generalized method of moments on the complex plane hence it shares the efficiency and distribution properties of gmm estimators we illustrate the method with some applications to relevant estimation problems in continuous time finance we estimate a model of stochastic volatility a jump diffusion model with constant volatility and a model that nests both the stochastic volatility model and the jump diffusion model we find that negative jumps are important to explain skewness and asymmetry in excess kurtosis of the stock return distribution while stochastic volatility is important to capture the overall level of this kurtosis positive jumps are not statistically significant once we allow for stochastic volatility in the model we also estimate a non affine model of stochastic volatility and find that the power of the diffusion coefficient appears to be between one and two rather than the value of one half that leads to the standard affine stochastic volatility model however we find that including jumps into this non affine stochastic volatility model reduces the power of the diffusion coefficient to one half finally we offer an explanation for the observation that the estimate of persistence in stochastic volatility increases dramatically as the frequency of the observed data falls based on a multiple factor stochastic volatility model

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

bounds and expansions for fisher information when the moments are known summary an expansion of the score statistic as a polynomial in the random variable can be used to provide a nondecreasing sequence of lower bounds for the fisher information when the moments are known as functions of the parameter if the random variable is asymptotically normal the first three bounds provide the first three terms in the asymptotic expansion for the fisher information this leads to a definition of efficiency which may be used to advantage with small sample sizes providing lower bounds for the efficiency of any statistic whose moments are known but whose exact distribution is intractable fisher 1925 suggested that the efficiency of a statistic be measured by its squared correlation with the maximum likelihood estimate this idea was extended by rao 1962 who defined first and second order efficiency by considering how well a linear or quadratic function of the statistic correlated with the derivative of the log likelihood in 2 of the present paper we develop this further by looking at polynomial expansions of the score statistic to obtain a nondecreasing sequence of lower bounds for the fisher information 4y 0 of a statistic y y x which is some function of the data x these bounds may be expressed in terms of the moments of the statistic and their first derivatives with respect to the unknown parameter 0 if we return to fisher s 1925 definition of efficiency as the ratio of fy 0 ox 0 whre ox 0 is the fisher information in the whole sample we can use this to obtain an efficiency for y which is invariant under one one transformations and which does not need to appeal to asymptotic arguments the results are particularly useful for random variables whose moments are known but whose exact sampling distribution is either unknown or intractable in 3 we consider other implications such as the possibility of using different sequences of functions and the question of convergence a recent application of these results by brockwell brown 1981 is given as an example rao 1962 pointed out that the limit of the ratio fy 0 fx o in large samples was the asymptotic squared correlation between the statistic y and the derivative of the log likelihood function lx o alx 0 ao for the full data set when this ratio tends to one both fisher and rao suggested using lim fx 0 fy 0 as n oo as a means of discriminating between alternative statistics this asymptotic concept of second order

nonparametric estimation for a class of levy processes we consider estimation for a class of levy processes modelled as a sum of a drift a symmetric stable process and a compound poisson process we propose a nonparametric approach to estimating unknown parameters of our model including the drift the scale and index parameters in the stable law the mean of the poisson process and the underlying jump size distribution we show that regression and nonparametric deconvolution methods based on the empirical characteristic function can be used for inference interesting connections are shown to exist between properties of our estimators and of those found in conventional deconvolution

estimation of stable distributions by indirect inference this article deals with the estimation of the parameters of an stable distribution by the indirect inference method with the skewed t distribution as an auxiliary model the latter distribution appears as a good candidate for an auxiliary model since it has the same number of parameters as the stable distribution with each parameter playing a similar role to improve the properties of the estimator in finite sample we use a variant of the method called constrained indirect inference in a monte carlo study we show that this method delivers estimators with good properties in finite sample in particular they are much more efficient than two other prevalent methods based on the characteristic function and the empirical quantiles we provide an empirical application to hedge fund returns

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

estimating stable latent factor models by indirect inference abstract cross sections of financial returns are characterized by common underlying factors and exhibit fat tails that may be captured by stable distributions this paper focuses on estimating factor models with independent latent factors and idiosyncratic noises featuring a multivariate stable distribution constant over time static factor models or a time varying conditional multivariate stable distribution garch factor models although the simulation of such a distribution is straightforward the estimation of its parameters encounters difficulties these difficulties are overcome in this paper by implementing the indirect inference estimation method with the multivariate student s t as the auxiliary distribution

fisher s information for discretely sampled levy processes this paper studies the asymptotic behavior of the fisher information for a levy process discretely sampled at an increasing frequency we show that it is possible to distinguish not only the continuous part of the process from its jumps part but also different types of jumps and derive the rates of convergence of efficient estimators

efficient estimation of general dynamic models with a continuum of moment conditions there are two difficulties with the implementation of the characteristic function based estimators first the optimal instrument yielding the ml efficiency depends on the unknown probability density function second the need to use a large set of moment conditions leads to the singularity of the covariance matrix we resolve the two problems in the framework of gmm with a continuum of moment conditions a new optimal instrument relies on the double indexing and as a result has a simple exponential form the singularity problem is addressed via a penalization term we introduce hac type estimators for non markov models a simulated method of moments is proposed for non analytical cases

diverging moments and parameter estimation heavy tailed distributions are enjoying increased popularity and are becoming more readily applicable as the arsenal of analytical and numerical tools grows they play key roles in modeling approaches in networking finance and hydrology to name but a few areas the tail parameter is of central importance because it governs both the existence of moments of positive order and the thickness of the tails of the distribution some of the best known tail estimators such as those of koutrouvelis and hill are either parametric or show a lack of robustness or accuracy this article develops a shift and scale invariant nonparametric estimator for both upper and lower bounds for orders with finite moments the estimator builds on the equivalence between tail behavior and the regularity of the characteristic function at the origin and achieves its goal by deriving a simplified wavelet analysis that is particularly suited to characteristic functions

a new approach to function based hypothesis testing in location scale families motivated by two applications in the mining industry we introduce a new approach to testing the hypothesis that two sampled distributions are simply location and scale changes of one another the test applicable to both paired data and two sample data is based on the empirical characteristic function more conventional techniques founded on the empirical distribution function suffer from serious drawbacks when used to test for location scale families in the motivating applications knowing that the distributions differ only in location and scale has significant operational and economic advantages enabling protocols for one type of data to be applied directly to another supplementary material in the form of matlab code is available online

on some fourier methods for inference abstract common statistical procedures such as maximum likelihood and m estimation admit generalized representations in the fourier domain the fourier domain provides fertile ground for approaching a number of difficult problems in inference in particular the empirical characteristic function and its extension for stationary time series are shown to be fundamental tools which support numerically simple inference procedures having arbitrarily high asymptotic efficiency and certain robustness features as well a numerical illustration involving the symmetric stable laws is given

estimation of affine asset pricing models using the empirical characteristic function abstract the known functional form of the conditional characteristic function ccf of discretely sampled observations from an affine diffusion is used to develop computationally tractable and asymptotically efficient estimators of the parameters of affine diffusions and of asset pricing models in which the state vectors follow affine diffusions both time domain estimators based on fourier inversion of the ccf and frequency domain estimators based directly on the ccf are constructed a method of moments estimator based on the ccf is shown to approximate the efficiency of maximum likelihood for affine diffusion and asset pricing models

estimation for multivariate stable distributions with generalized empirical likelihood abstract this paper considers the generalized empirical likelihood gel method for estimating the parameters of the multivariate stable distribution the gel method is considered to be an extension of the generalized method of moments gmm the multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails we treat the spectral measure which summarizes scale and asymmetry by discretization in order to estimate all the model parameters simultaneously we apply the estimating function constructed by equating empirical and theoretical characteristic functions the efficacy of the proposed gel method is demonstrated in monte carlo studies an illustrative example involving daily returns of market indexes is also included

barycentric formulae for cardinal sinc interpolants we present a barycentric representation of cardinal interpolants as well as a weighted barycentric formula for their efficient evaluation we also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and in other cases is even more accurate in numerical examples we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work

barycentric formulae for cardinal sinc interpolants by jean paul berrut a formula for the efficient evaluation of the truncated cardinal series is known to be numerically unstable near the interpolation abscissae here it is shown how the series can be evaluated in an entirely stable manner

inferences on the parameters of the birnbaum saunders fatigue life distribution based on maximum likelihood estimation this paper provides tests of hypotheses and confidence intervals for the scale and shape parameters singly of the birnbaum saunders fatigue life model each with the other an unknown nuisance parameter the procedure for each parameter is based on its maximum likelihood estimate although in the case of the scale parameter it is necessary to consider conditional procedures

the method of simulated quantiles we introduce an inference method based on quantiles matching which is useful for situations where the density function does not have a closed form but it is simple to simulate and or moments do not exist functions of theoretical quantiles which depend on the parameters of the assumed probability law are matched with sample quantiles which depend on observations since the theoretical quantiles may not be available analytically the optimization is based on simulations we illustrate the method with the estimation of stable distributions a thorough monte carlo study and an illustration to 22 nancial indexes show

bivariate sub gaussian model for stock index returns abstract financial time series are commonly modeled with methods assuming data normality however the real distribution can be nontrivial also not having an explicitly formulated probability density function in this work we introduce novel parameter estimation and high powered distribution testing methods which do not rely on closed form densities but use the characteristic functions for comparison the approach applied to a pair of stock index returns demonstrates that such a bivariate vector can be a sample coming from a bivariate sub gaussian distribution the methods presented here can be applied to any nontrivially distributed financial data among others

estimation of location and scale in cauchy distributions using the empirical characteristic function summary the iterative method for the parameter estimation of stable laws proposed by koutrouvelis 1981 is modified to handle the case where the distribution is assumed to belong to a cauchy family with location and or scale parameter unknown it is shown that the determination of optimum values at which the empirical characteristic function needs to be evaluated reduces in this case to the determination of the asymptotically optimum quantiles for the parameter estimation of an exponential distribution by linear functions of order statistics the estimators of the scale and location parameters are asymptotically independent and normally distributed and achieve asymptotic efficiencies close to unity the proposed procedure is compared to a method of estimation using linear functions of order statistics

maximum likelihood estimation of stochastic frontier models by the fourier transform the paper is concerned with several kinds of stochastic frontier models whose likelihood function is not available in closed form first with output oriented stochastic frontier models whose one sided errors have a distribution other than the standard ones exponential or half normal the gamma and beta distributions are leading examples second with input oriented stochastic frontier models which are common in theoretical discussions but not in econometric applications third with two tiered stochastic frontier models when the one sided error components follow gamma distributions fourth with latent class models with gamma distributed one sided error terms fifth with models whose two sided error component is distributed as stable paretian and the one sided error is gamma the principal aim is to propose approximations to the density of the composed error based on the inversion of the characteristic function which turns out to be manageable using the fourier transform procedures that are based on the asymptotic normal form of the log likelihood function and have arbitrary degrees of asymptotic efficiency are also proposed implemented and evaluated in connection with output oriented stochastic frontiers the new methods are illustrated using data for us commercial banks electric utilities and a sample from the national youth longitudinal survey

variable selection in a log linear birnbaum saunders regression model for high dimensional survival data via the elastic net and stochastic em the birnbaum saunders bs distribution is broadly used to model failure times in reliability and survival analysis in this article we propose a simultaneous parameter estimation and variable selection procedure in a log linear bs regression model for high dimensional survival data to deal with censored survival data we iteratively run a combination of the stochastic em algorithm sem and variable selection procedure to generate pseudo complete data and select variables until convergence treating pseudo complete data as uncensored data via sem makes it possible to incorporate iterative penalized least squares and simplify computation we demonstrate the efficacy of our method using simulated and real datasets

an information theoretic alternative to generalized method of moments estimation while optimally weighted gmm estimation has desirable large sample properties its small sample performance is poor in some applications we propose a computationally simple alternative for weakly dependent data generating mechanisms based on minimization of the kullback leibler information criterion conditions are derived under which the large sample properties of this estimator are similar to gmm i e the estimator will be consistent and asymptotically normal with the same asymptotic covariance matrix as gmm in addition we propose overidentifying and parametric restrictions tests as alternatives to analogous gmm procedures

detection of dependent heavy tailed signals this paper examines the problem of detection of dependent alpha stable signals measurements of several phenomena exhibit non gaussian heavy tailed behavior in their probability density functions p d f we use the class of alpha stable distributions to characterize these signals when two sensors make simultaneous measurements of such phenomena these heavy tailed realizations are dependent across sensors the intersensor dependence is modeled using copulas we consider a two sided test in the neyman pearson framework and present an asymptotic analysis of the generalized likelihood test glrt both nested and non nested models are considered in the analysis the performance of the proposed scheme is evaluated numerically on simulated data as well as indoor seismic data with appropriately selected models our results demonstrate that a high probability of detection can be achieved for false alarm probabilities of the order of 10 4

spectral estimation of the fractional order of a levy process we consider the problem of estimating the fractional order of a levy process from low frequency historical and options data an estimation methodology is developed which allows us to treat both estimation and calibration problems in a unified way the corresponding procedure consists of two steps the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain while the second step is identical for both calibration and estimation the first one depends on the problem at hand minimax rates of convergence for the fractional order estimate are derived the asymptotic normality is proved and a data driven algorithm based on aggregation is proposed the performance of the estimator in both estimation and calibration setups is illustrated by a simulation study

the modified yule walker method for α stable time series models abstract this paper discusses the problem of parameters estimation for stable periodic autoregressive par time series considered models generalize popular and widely accepted autoregressive ar time series by examining measures of dependence for stable processes first we introduce new empirical estimator of autocovariation for stable sequences based on this approach we generalize yule walker method for estimation of parameter for par time series thus we fill a gap in estimation methods for non gaussian models we test proposed procedure and show its consistency moreover we use our approach to model real empirical data thus showing usefulness of heavy tailed models in statistical modelling

a log linear model for the birnbaum saunders distribution the birnbaum saunders distribution was derived to model times to failure for metals subject to fatigue in this article we formulate and develop a log linear model for the birnbaum saunders distribution the model may be used for accelerated life testing or to compare the median lives of several populations methods of analyzing data for this log linear model are discussed with maximum likelihood and least squares methods being compared it is found that for commonly occurring conditions the notorious intractability of the birnbaum saunders distribution is not a serious problem because least squares and normal theory procedures provide a reasonable alternative

first applications of a formula for the error of finite sinc interpolation in former articles we have given a formula for the error committed when interpolating a several times differentiable function by the sinc interpolant on a fixed finite interval in the present work we demonstrate the relevance of the formula through several applications correction of the interpolant through the insertion of derivatives to increase its order of convergence improvement of the barycentric formula rational sinc interpolants with and without replacement of the usually unknown derivatives with finite differences convergence acceleration through extrapolation and improvement of one sided interpolants

chebyshev polynomial approximations and characteristic function estimation on developpe une nouvelle procedure pour estimer les fonctions caracteristiques basee sur un developpement en polynomes de chebyshev pour la densite du cosinus des resultats de simulation confirment l utilite de la methode

on the efficiency of empirical characteristic function procedures summary the asymptotic normality and arbitrarily high efficiency of some new statistical procedures based on the empirical characteristic function is established under general conditions

estimation of regression coefficients with the help of characteristic functions abstract in this paper some easily applicable estimators of the regression coefficient in a simple regression model with errors in variables are derived usually one uses sample moments to estimate the regression coefficient however we use empirical characteristic functions in several points besides theoretical derivations a small study with generated data is discussed this study compares our method with estimating methods which use higher order moments