Efficient estimators for the good family

We consider the problem of estimating the two parameters of the discrete Good distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. The maximum likelihood estimators (MLE's) of the parameters are obtained by solving numerically a system of equations involving the Lerch zeta function and the sufficient statistics. We find an expression for the asymptotic variance-covariance matrix of the MLE's, which can be evaluated numerically. We show that the probability mass function satisfies a simple recurrence equation linear in the two parameters, and propose the quadratic distance estimator (QDE) which can be computed with an ineratively reweighted least-squares algorithm. the QDE is easy to calculate and admits a simple expression for its asymptotic variance-covariance matrix. We compute this matrix for the MLE's and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. Finally, we present a numerical example.     

Inference for the positive stable laws based on a special quadratic distance

Inference methods for the positive stable laws, which have no closed form expression for the density functions are developed based on a special quadratic distance using negative moments. Asymptotic properties of the quadratic distance estimator (QDE) are established. The QDE is shown to have asymptotic relative efficiency close to 1 for almost all the values of the parameter space.Goodness-of-fit tests are also developed for testing the parametric families and practical numerical techniques are considered for implementing the methods. With simple and efficient methods to estimate the parameters, positive stable laws could find new applications in actuarial science for modelling insurance claims and lifetime data.     

A multivariate two-factor skew model

It is well known that many data, such as the financial or demographic data, exhibit asymmetric distributions. In recent years, researchers have concentrated their efforts to model this asymmetry. Skew normal model is one of such models that are skew and yet possess many properties of the normal model. In this paper, a new multivariate skew model is proposed, along with its statistical properties. It includes the multivariate normal distribution and multivariate skew normal distribution as special cases. The quadratic form of this random vector follows a χ² distribution. The roles of the parameters in the model are investigated using contour plots of bivariate densities.     

On approximating the distribution of indefinite quadratic forms

This paper provides a simple methodology for approximating the distribution of indefinite quadratic forms in normal random variables. It is shown that the density function of a positive definite quadratic form can be approximated in terms of the product of a gamma density function and a polynomial. An extension which makes use of a generalized gamma density function is also considered. Such representations are based on the moments of a quadratic form, which can be determined from its cumulants by means of a recursive formula. After expressing an indefinite quadratic form as the difference of two positive definite quadratic forms, one can obtain an approximation to its density function by means of the transformation of variable technique. An explicit representation of the resulting density approximant is given in terms of a degenerate hypergeometric function. An easily implementable algorithm is provided. The proposed approximants produce very accurate percentiles over the entire range of the distribution. Several numerical examples illustrate the results. In particular, the methodology is applied to the Durbin–Watson statistic which is expressible as the ratio of two quadratic forms in normal random variables. Quadratic forms being ubiquitous in statistics, the approximating technique introduced herewith has numerous potential applications. Some relevant computational considerations are also discussed.     

On Quadratic Forms of Multivariate t Distribution with Applications

This note mainly aims to illustrate that some quadratic problems are robust in a sense with respect to the probabilistic distributions involved. The secondary moments of the quadratic forms of a multivariate t distribution are calculated. Then, the resulting formulae are applied to the quadratic problems of quadratic sufficiency and quadratic prediction. It is shown by revisiting the two problems that the same conclusions hold when the multivariate normal distribution is replaced with a multivariate t distribution.     

A History of the Development of Craig's Theorem

Craig's theorem on the independence of quadratic forms in normal variates is traced from its first form, for iid standard normal variates, to the form for variates following an arbitrary nonsingular joint normal distribution. This article gives the main thrust of the development and makes recommendations on coverage of the theorem in courses and textbooks. The history of Craig's theorem is not a happy one. The authors of the earlier articles in the literature tended to make errors of a linear-algebraic nature. Authors of more recently published textbooks have given incorrect or misleadingly incomplete coverage of Craig's theorem and its proof.     

Characteristic function-based inference for GARCH models with heavy-tailed innovations

We consider estimation and goodness-of-fit tests in GARCH models with innovations following a heavy-tailed and possibly asymmetric distribution. Although the method is fairly general and applies to GARCH models with arbitrary innovation distribution, we consider as special instances the stable Paretian, the variance gamma, and the normal inverse Gaussian distribution. Exploiting the simple structure of the characteristic function of these distributions, we propose minimum distance estimation based on the empirical characteristic function of properly standardized GARCH-residuals. The finite-sample results presented facilitate comparison with existing methods, while the new procedures are also applied to real data from the financial market.     

Moments of generalized quadratic forms and distributions of certain multivariate test statistics

Summary Moments and distributions of quadratic forms or quadratic expressions in normal variables are available in literature. Such quadratic expressions are shown to be equivalent to a linear function of independent central or noncentral chi-square variables. Some results on linear functions of generalized quadratic forms are also available in literature. Here we consider an arbitrary linear function of matrix-variate gamma variables. Moments of the determinant of such a linear function are evaluated when the matrix-variate gammas are independently distributed. By using these results, arbitrary non-null moments as well as the non-null distribution of the likelihood ratio criterion for testing the hypothesis of equality of covariance matrices in independent multivariate normal populations are derived. As a related result, the distribution of a linear function of independent matrix-variate gamma random variables, which includes linear functions of independent Wishart matrices, is also obtained. Some properties of generalized special functions of several matrix arguments are used in deriving these results.     

Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix

In this article we consider a set of t repeated measurements on p variables (or characteristics) on each of the n individuals. Thus, data on each individual is a p ×t matrix. The n individuals themselves may be divided and randomly assigned to g groups. Analysis of these data using a MANOVA model, assuming that the data on an individual has a covariance matrix which is a Kronecker product of two positive definite matrices, is considered. The well-known Satterthwaite type approximation to the distribution of a quadratic form in normal variables is extended to the distribution of a multivariate quadratic form in multivariate normal variables. The multivariate tests using this approximation are developed for testing the usual hypotheses. Results are illustrated on a data set. A method for analysing unbalanced data is also discussed.     

On the Admissibility of Linear Estimators in a Multivariate Normal Distribution Under LINEX Loss Function

Consider an estimation problem of a linear combination of population means in a multivariate normal distribution under LINEX loss function. Necessary and sufficient conditions for linear estimators to be admissible are given. Further, it is shown that the result is an extension of the quadratic loss case as well as the univariate normal case.     

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.     

Estimation of the mean of a spherically symmetric distribution with constraints on the norm

The authors consider the problem of estimating, under quadratic loss, the mean of a spherically symmetric distribution when its norm is supposed to be known and when a residual vector is available. They give a necessary and sufficient condition for the optimal James‐Stein estimator to dominate the usual estimator. Various examples are given that are not necessarily variance mixtures of normal distributions. Consideration is also given to an alternative class of robust James‐Stein type estimators that take into account the residual vector. A more general domination condition is given for this class.     

Form-invariance under weighted sampling

In this paper, form-invariant weighted distributions are considered in an exponential family. The class of bivariate distribution with invariant property is identified under exponential weight function. The class includes some of the custom bivariate models. The form-invariant multivariate normal distributions are obtained under a quadratic weight function.     

A note on the joint distribution of correlated quadratic forms

In the present paper, the authors obtained exact and asymptotic expressions for the joint distribution of correlated quadratic forms when the underlying distribution is a multivariate normal.     

An approximation to the distribution of the ratio of two general quadratic forms with application

Based on mixed cumulants up to order six, this paper provides a four moment approximation to the distribution of a ratio of two general quadratic forms in normal variables. The approximation is applied to calculate the percentile points of modified F-test statistics for testing treatment effects when standard F-ratio test is misleading because of dependence among observations. For the special case, when data is generated by an AR(1) process, the approximation is evaluated by a simulation study. For the general SARMA (p,q)(P,Q)_s process, a modified F-test statistic Is given, and its distribution for the (0,1)(0,l)₁₂ process, is approximated by the moment approximation technique.     

Bayesian multivariate normal analysis with a wishart prior

This paper considers the Bayesian analysis of the multivariate normal distribution when its covariance matrix has a Wishart prior density under the assumption of a multivariate quadratic loss function. New flexible marginal posterior distributions of the mean μ and of the covariance matrix Σ are developed and univariate cases with graphical representations are given.     

CONVERGENT SERIES EXPRESSIONS FOR INVERSE MOMENTS OF QUADRATIC FORMS IN NORMAL VARIABLES

Using relatively recent results from multivariate distribution theory, a direct approach to evaluating the inverse moments of a quadratic form in normal variables is proposed. Convergent infinite series expressions involving the invariant polynomials of matrix argument are obtained. The solution also depends upon a positive scalar which is arbitrarily chosen. For the solution to converge an upper bound upon this scalar is derived.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

An upper bound for the distribution function of a positive definite quadratic form

In this paper the researchers are presenting an upper bound for the distribution function of quadratic forms in normal vector with mean zero and positive definite covariance matrix. They also will show that the new upper bound is more precise than the one introduced by Okamoto [4] and the one introduced by Siddiqui [5]. Theoretical Error bounds for both, the new and Okamoto upper bounds are derived in this paper. For larger number of terms in any given positive definite quadratic form, a rough and easier upper bound is suggested.     

Prbposterior expected loss as a scoring rule for prior distributions

A scoring rule for evaluating the usefulness of an assessed prior distribution should reflect the purpose for which the distribution is to be used. In this paper we suppose that sample data is to become available and that the posterior distribution will be used to estimate some quantity under a quadratic loss function. The utility of a prior distribution is consequently determined by its preposterior expected quadratic loss. It is shown that this loss function has properties desirable in a scoring rule and formulae are derived for calculating the scores it gives in some common problems. Many scoring rules give a very poor score to any improper prior distribution but, in contrast, the scoring rule proposed here provides a meaningful measure for comparing the usefulness of assessed prior distributions and non-informative (improper) prior distributions. Results for making this comparison in various situations are also given.