On the UMVU estimator for the generalized variance of the multinomial family

Abstract

The generalized variance is an important statistical indicator which appears in a number of statistical topics. It is a successful measure for multivariate data concentration. In this article, we established, in a closed form, the bias of the generalized variance maximum likelihood estimator of the Multinomial family. We also derived, with a complete proof, the uniformly minimum variance unbiased estimator (UMVU) for the generalized variance of this family. These results rely on explicit calculations, the completeness of the exponential family and the Lehmann–Scheffé theorem.     

Parameter estimation for Ornstein–Uhlenbeck processes of the second kind driven by α-stable Lévy motions

In this article, we study the problem of parameter estimation for Ornstein–Uhlenbeck processes of the second kind driven by α-stable Lévy motions, based on continuous and discrete observations, respectively. Using the trajectory fitting method combined with the weighted least-squares technique, we discuss the consistency and the asymptotic distributions of the estimators for general weights in both the ergodic and the non ergodic cases.     

Multi-modal tempered stable distributions and prosses with applications to finance

Abstract

The assumption of underlying return distribution plays an important role in asset pricing models. While the return distribution used in the traditional theories of asset pricing is the unimodal distribution, numerous studies which have investigated the empirical behavior of asset returns in financial markets use multi-modal distribution. We introduce a new parsimonious multi-modal distribution, referred to as the multi-modal tempered stable (MMTS) distribution. In this article we also generate the exponential Lévy market models and derive the value-at-risk (VaR) induced from them. To demonstrate the advantages, we will present the results of the parameter estimation and the VaRs for financial data.     

On the optimality of the sample variance

The present paper explores the structure of linear exponential families for which the sample variance is a uniformly minimum variance unbiased estimator.     

On efficient inferences in familial-longitudinal binary models with two variance components

When a generalized linear mixed model (GLMM) with multiple (two or more) sources of random effects is considered, the inferences may vary depending on the nature of the random effects. For example, the inference in GLMMs with two independent random effects with two distinct components of dispersion will be different from the inference in GLMMs with two random effects in a two factor factorial design set-up. In this paper, we consider a familial-longitudinal model for repeated binary data where the binary response of an individual member of a family at a given time point is assumed to be influenced by the past responses of the member as well as two but independent sources of random family effects. For the estimation of the parameters of the proposed model, we discuss the well-known maximum-likelihood (ML) method as well as a generalized quasi-likelihood (GQL) approach. The main objective of the paper is to examine the relative asymptotic efficiency performance of the ML and GQL estimators for the regression effects, dynamic (longitudinal) dependence and variance parameters of the random family effects from two sources.     

On quasi‐likelihood inference in generalized linear mixed models with two components of dispersion

The authors propose a quasi‐likelihood approach analogous to two‐way analysis of variance for the estimation of the parameters of generalized linear mixed models with two components of dispersion. They discuss both the asymptotic and small‐sample behaviour of their estimators, and illustrate their use with salamander mating data.     

On Infinitely Divisible Exponential Dispersion Model Related to Poisson-Exponential Distribution

We construct a univariate exponential dispersion model comprised of discrete infinitely divisible distributions. This model emerges in the theory of branching processes. We obtain a representation for the Lévy measure of relevant distributions and characterize their laws as Poisson mixtures and/or compound Poisson distributions. The regularity of the unit variance function of this model is employed for the derivation of approximations by the Poisson-exponential model. We emphasize the role of the latter class. We construct local approximations relating them to properties of special functions and branching diffusions.     

Stein type confidence interval of the disturbance variance in a linear regression model with multivariate student-t distributed errors

This paper considers the interval estimation of the disturbance variance in a linear regression model with multivariate Student-t errors. The distribution function of the Stein type estimator under multivariate Student-t errors is derived, and the coverage probability of the Stein type confidence interval which is constructed under the normality assumption is numerically calculated under the multivariate Student-t distribution. It is shown that the coverage probability of the Stein type confidence interval is sometimes much smaller than the nominal level, and that it is larger than that of the usual confidence interval in almost all cases. For the case when it is known that errors have a multivariate Student-t distribution, sufficient conditions under which the Stein type confidence interval improves on the usual confidence interval are given, and the coverage probability of the stein type confidence interval is numerically evaluated.     

Data cloning estimation of GARCH and COGARCH models

GARCH models include most of the stylized facts of financial time series and they have been largely used to analyse discrete financial time series. In the last years, continuous-time models based on discrete GARCH models have been also proposed to deal with non-equally spaced observations, as COGARCH model based on Lévy processes. In this paper, we propose to use the data cloning methodology in order to obtain estimators of GARCH and COGARCH model parameters. Data cloning methodology uses a Bayesian approach to obtain approximate maximum likelihood estimators avoiding numerically maximization of the pseudo-likelihood function. After a simulation study for both GARCH and COGARCH models using data cloning, we apply this technique to model the behaviour of some NASDAQ time series.     

On a class of distributions on the simplex

In the present paper we define and investigate a novel class of distributions on the simplex, termed normalized infinitely divisible distributions, which includes the Dirichlet distribution. Distributional properties and general moment formulae are derived. Particular attention is devoted to special cases of normalized infinitely divisible distributions which lead to explicit expressions. As a by-product new distributions over the unit interval and a generalization of the Bessel function distribution are obtained.     

Estimation of variance components in the mixed effects models: A comparison between analysis of variance and spectral decomposition

The mixed effects models with two variance components are often used to analyze longitudinal data. For these models, we compare two approaches to estimating the variance components, the analysis of variance approach and the spectral decomposition approach. We establish a necessary and sufficient condition for the two approaches to yield identical estimates, and some sufficient conditions for the superiority of one approach over the other, under the mean squared error criterion. Applications of the methods to circular models and longitudinal data are discussed. Furthermore, simulation results indicate that better estimates of variance components do not necessarily imply higher power of the tests or shorter confidence intervals.     

An efficient procedure for computing minque of variance components and generalized least squares estimates of fixed effects

An efficient method for computing minimum norm quadratic unbiased estimates (MINQUE) of variance components and generalized least squares estimates of the fixed effects in the mixed model is developed. The computing algorithm uses a modification of the W transformation.     

Modelling and Prediction of Financial Time Series

We consider statistical aspects of the modelling and prediction theory of time series in one and many dimensions. We discuss Lévy-based and general models, and the stationary and non-stationary cases. Our starting point is the recent pair of surveys, Szeg'ó's theorem and its probabilistic descendants and Multivariate prediction and matrix Szeg'ó theory, by this author.     

Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values

Let X _{U (1)} < X _{U (2)} < … < X _{U ( n )} < … be the sequence of the upper record values from a population with common distribution function F. In this paper, we first give a theorem to characterize the generalized mixtures of geometric distribution by the relation between E[(X _{U ( n +1)}–X _{U ( n )})²|X _{U ( n )} = x] and the function of the failure rate of the distribution, for any positive integer n. Secondly, we also use the same relation to characterize the generalized mixtures of exponential distribution. The characterizing relations were motivated by the work of Balakrishnan and Balasubramanian (1995). Received: March 31, 1999; revised version: November 22, 1999     

On the characterization of generalized extreme value, power function, generalized Pareto and classical Pareto distributions by conditional expectation of record values

In the present paper, we give some theorems to characterize the generalized extreme value, power function, generalized Pareto (such as Pareto type II and exponential, etc.) and classical Pareto (Pareto type I) distributions based on conditional expectation of record values. Received: June 23, 1998; revised version: September 20, 1999     

The effect of number of clusters and cluster size on statistical power and Type I error rates when testing random effects variance components in multilevel linear and logistic regression models

When using multilevel regression models that incorporate cluster-specific random effects, the Wald and the likelihood ratio (LR) tests are used for testing the null hypothesis that the variance of the random effects distribution is equal to zero. We conducted a series of Monte Carlo simulations to examine the effect of the number of clusters and the number of subjects per cluster on the statistical power to detect a non-null random effects variance and to compare the empirical type I error rates of the Wald and LR tests. Statistical power increased with increasing number of clusters and number of subjects per cluster. Statistical power was greater for the LR test than for the Wald test. These results applied to both the linear and logistic regressions, but were more pronounced for the latter. The use of the LR test is preferable to the use of the Wald test.     

Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏_γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏_γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏_γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏_γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏_γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏_γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.