On the density of generalised quadratic forms with applications to asymptotic expansions for test statistics

In this note we derive a general formula useful to express the density of generalised noncentral quadratic forms (i.e. of a scalar random variable obtained by contracting non zero mean multivariate normal vectors over multidimensional arrays) in terms of linear combinations of noncentral chi square random variables. The formula can be used to obtain explicit expressions for the terms appearing in the asymptotic expansions for test statistics under a local alternative.Edgeworth expansions; Generalised noncentral quadratic forms; Local alternatives.

Edgeworth expansions for errors-in-variables models

AbstractEdgeworth expansions for sums of independent but not identically distributed multivariate random vectors are established. The results are applied to get valid Edgeworth expansions for estimates of regression parameters in linear errors-in-variable models. The expansions for studentized versions are also developed. Further, Edgeworth expansions for the corresponding bootstrapped statistics are obtained. Using these expansions, the bootstrap distribution is shown to approximate the sampling distribution of the studentized estimators, better than the classical normal approximation

Exponential Operators, Dobinski Relations and Summability

We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods of multivariate Bell polynomials which enable us to understand the meaning of perturbation-like expansions of exponential operators. Such expansions, obtained as formal power series, are everywhere divergent but the Pade summation method is shown to give results which very well agree with exact solutions got for simplified quantum models of the one mode bosonic systems.Comment: Presented at XIIth Central European Workshop on Quantum Optics, Bilkent University, Ankara, Turkey, 6-10 June 2005. 4 figures, 6 pages, 10 reference

Multivariate Gram-Charlier Densities

This paper introduces a new family of multivariate distributions based on Gram-Charlier and Edgeworth expansions. This family encompasses many of the univariate seminonparametric densities proposed in the financial econometrics as marginal distributions of the different formulations. Within this family, we focus on the specifications that guarantee positivity so obtaining a well-defined multivariate density. We compare different "positive" multivariate distributions of the family with the multivariate Edgeworth-Sargan, Normal and Student’s t in an in- and out-sample framework for financial returns data. Our results show that the proposed specifications provide a quite reasonably good performance being so of interest for applications involving the modelling and forecasting of heavy-tailed distributions.Multivariate distributions; Gram-Charlier and Edgeworth-Sargan densities; MGARCH models; financial data

Asymptotic comparison of negative multinomial and multivariate normal experiments

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.Comment: 12 pages, 0 figure

An Edgeworth-type expansion for the distribution of a likelihood-based discriminant function

The exact distribution of a classification function is often complicated to allow for easy numerical calculations of misclassification errors. The use of expansions is one way of dealing with this difficulty. In this paper, approximate probabilities of misclassification of the maximum likelihood-based discriminant function are established via an Edgeworth-type expansion based on the standard normal distribution for discriminating between two multivariate normal populations

Transformations for multivariate statistics

This paper derives transformations for multivariate statistics that eliminate asymptotic skewness, extending the results of Niki and Konishi (1986, Annals of the Institute of Statistical Mathematics 38, 371-383). Within the context of valid Edgeworth expansions for such statistics we first derive the set of equations that such a transformation must satisfy and second propose a local solution that is sufficient up to the desired order. Application of these results yields two useful corollaries. First, it is possible to eliminate the first correction term in an Edgeworth expansion, thereby accelerating convergence to the leading term normal approximation. Second, bootstrapping the transformed statistic can yield the same rate of convergence of the double, or prepivoted, bootstrap of Beran (1988, Journal of the American Statistical Association 83, 687-697), applied to the original statistic, implying a significant computational saving. The analytic results are illustrated by application to the family of exponential models, in which the transformation is seen to depend only upon the properties of the likelihood. The numerical properties are examined within a class of nonlinear regression models (logit, probit, Poisson, and exponential regressions), where the adequacy of the limiting normal and of the bootstrap (utilizing the k-step procedure of Andrews, 2002, Econometrica 70, 119-162) as distributional approximations is assessed

Maximum-likelihood estimation for diffusion processes via closed-form density expansions

This paper proposes a widely applicable method of approximate maximum-likelihood estimation for multivariate diffusion process from discretely sampled data. A closed-form asymptotic expansion for transition density is proposed and accompanied by an algorithm containing only basic and explicit calculations for delivering any arbitrary order of the expansion. The likelihood function is thus approximated explicitly and employed in statistical estimation. The performance of our method is demonstrated by Monte Carlo simulations from implementing several examples, which represent a wide range of commonly used diffusion models. The convergence related to the expansion and the estimation method are theoretically justified using the theory of Watanabe [Ann. Probab. 15 (1987) 1-39] and Yoshida [J. Japan Statist. Soc. 22 (1992) 139-159] on analysis of the generalized random variables under some standard sufficient conditions.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1118 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org