Moments for matrix normal variables

In order to obtain moments for matrix normally distributed variables the moment generating function is differentiated by aid of matrix derivatives. Moments of arbitrary order as well as a recursive relation are obtained. Further, some more details are given for the first four moments     

A note on moments of order statistics from exchangeable variates

Joshi (1973) and Balakrishnan and Malik (1985) have derived some some identities for the moments of order statistics from independent and identically distributed random variables. In this paper, we make use of a basic result due to David and Joshi (1968) and show that these identities for the moments also hold when the order statistics arise from exchangeable variables.     

Non–zero inflated modified power series distributions

In this paper, we study the class of inflated modified power series distributions (IMPSD) where inflation occurs at any of the support points. This class include among other the generalized Poisson, the generalized negative binomial, the generalized logarithmic series and the lost games distributions. We give expressions for the moments, factorial moments and central moments of the IMPSD. The maximum likelihood estimation of the parameters of the IMPSD and the variance – covariance matrix of the estimators is obtained. We derive these estimators and their information matrices for mentioned above particular members of IMPSD class. The second part of this paper deals with the distribution of sum of independent and identically distributed random variables taking values s, s+1. s + 2, …, s ≥ 0, with modified power series distributions inflated at the point s.     

Forward irrelevance

We investigate how to combine marginal assessments about the values that random variables assume separately into a model for the values that they assume jointly, when (i) these marginal assessments are modelled by means of coherent lower previsions and (ii) we have the additional assumption that the random variables are forward epistemically irrelevant to each other. We consider and provide arguments for two possible combinations, namely the forward irrelevant natural extension and the forward irrelevant product, and we study the relationships between them. Our treatment also uncovers an interesting connection between the behavioural theory of coherent lower previsions, and Shafer and Vovk's game-theoretic approach to probability theory.     

Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.     

Lower Bounds for the Efficiency of Designs with Respect to the D-Criterion when the Observations are Correlated

In the general linear model consider the experimental design problem for the Gauß-Markov estimator or least squares estimator when the observations are correlated. We prove new formulas for the efficiency of an exact design with respect to the D-criterion. For models with intercept term, for example, these formulas are useful to derive better lower bounds for the efficiency than the bounds recently given for an arbitrary linear model. These bounds are applied in examples to symmetrical regular circulants as covariance matrices. A byproduct of the investigations is some insight as to what kinds of designs might retain their optimality or high efficiency (for the uncorrelated homoscedastic case) under correlated observations.     

Alternative covariance estimates in a replicated measurement error model with correlated,heteroscedastic errors

The article concerns covariance estimates in a replicated measurement error model with correlated, heteroscedastic errors. Freedman has conjectured that using more of the data will improve estimates of covariance matrices and result in a more efficient estimate of the coefficient of the regression model. The paper confirms the conjecture asymptotically for the case that all random variables are normally distributed, but the gain is not substantial.     

Higher-Order Moments Using the Survival Function: The Alternative Expectation Formula

Undergraduate and graduate students in a first-year probability (or a mathematical statistics) course learn the important concept of the moment of a random variable. The moments are related to various aspects of a probability distribution. In this context, the formula for the mean or the first moment of a nonnegative continuous random variable is often shown in terms of its c.d.f. (or the survival function). This has been called the alternative expectation formula. However, higher-order moments are also important, for example, to study the variance or the skewness of a distribution. In this note, we consider the rth moment of a nonnegative random variable and derive formulas in terms of the c.d.f. (or the survival function) paralleling the existing results for the first moment (the mean) using Fubini's theorem. Both nonnegative continuous and discrete integer-valued random variables are considered. These formulas may be advantageous, for example, when dealing with the moments of a transformed random variable, where it may be easier to derive its c.d.f. using the so-called c.d.f. method.     

Robustness of Parameter Estimation Procedures in Multilevel Models When Random Effects are MEP Distributed

In this paper we examine maximum likelihood estimation procedures in multilevel models for two level nesting structures. Usually, for fixed effects and variance components estimation, level-one error terms and random effects are assumed to be normally distributed. Nevertheless, in some circumstances this assumption might not be realistic, especially as concerns random effects. Thus we assume for random effects the family of multivariate exponential power distributions (MEP); subsequently, by means of Monte Carlo simulation procedures, we study robustness of maximum likelihood estimators under normal assumption when, actually, random effects are MEP distributed.     

Goodness-of-fit tests for centralized Wishart processes

Abstract

In this paper we present several goodness-of-fit tests for the centralized Wishart process, a popular matrix-variate time series model used to capture the stochastic properties of realized covariance matrices. The new test procedures are based on the extended Bartlett decomposition derived from the properties of the Wishart distribution and allows to obtain sets of independently and standard normally distributed random variables under the null hypothesis. Several tests for normality and independence are then applied to these variables in order to support or to reject the underlying assumption of a centralized Wishart process. In order to investigate the influence of estimated parameters on the suggested testing procedures in the finite-sample case, a simulation study is conducted. Finally, the new test methods are applied to real data consisting of realized covariance matrices computed for the returns on six assets traded on the New York Stock Exchange.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Multivariate Matsumoto–Yor property is rather restrictive

Matsumoto and Yor [2001. An analogue of Pitman's 2M-X$2M-X$ theorem for exponential Wiener functionals. Part II: the role of the GIG laws. Nagoya Math. J. 162, 65–86] discovered an interesting invariance property of a product of the generalized inverse Gaussian (GIG) and the gamma distributions. For univariate random variables or symmetric positive definite random matrices it is a characteristic property for this pair of distributions. It appears that for random vectors the Matsumoto–Yor property characterizes only very special families of multivariate GIG and gamma distributions: components of the respective random vectors are grouped into independent subvectors, each subvector having linearly dependent components. This complements the version of the multivariate Matsumoto–Yor property on trees and related characterization obtained in Massam and Weso?owski [2004. The Matsumoto–Yor property on trees. Bernoulli 10, 685–700].     

Stochastic orderings related to ranking in an nth order record process

Let X₁, X₂, …, X_m be successive observations on m objects, numbered 1,2, …, m. If X₁ belongs to the n largest observations among X₁,X₂,…,X_i, object i is called a record, i = 1,2,…,m; n?1.In investigating the influence of the ranking of the objects on the expected number of records, a hierarchy of stochastic order relations between random variables arises.It is these order relations and their relationship with known stochastic orderings that are studied in this paper.     

Computing the moments of order statistics from independent nonidentically distributed Beta random variables

Recently Thomas and Samuel (Stat Pap 49:139–146, 2008) derived recurrence relations for the moments of order statistics arising form independent identically distributed Beta variables. In this paper the case of moments of independent nonidentically distributed Beta random variables is discussed. Another proof to the independent identical case is also presented.     

On a random series in a hilbert space

Let (S_n) be partial sums of a non-degenerate sequence of Identically and independently distributed random variables taking values in a separable Hilbert space. Then for 0 ≤ β ≤ 3/2, the series converges almost nowhere. For β > 3/2 this may not be true.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

Current records and record range with some applications

In a sequence of independent and identically distributed (iid) random variables, the upper (lower) current records and record range are studied. We derive general recurrence relations between the single and product moments for the upper and lower current records based on Weibull and positive Weibull distributions, as well as Pareto and negative Pareto distributions, respectively. Moreover, some asymptotic results for general current records are established. In addition, a recurrence relation and an explicit formula for the moments of record range based on the exponential distribution are given. Finally, numerical examples are presented to illustrate and corroborate theoretical results.     

基于偏正态随机效应模型的信度保费

信度模型是非寿险经验费率厘定的主要方法。传统的Buhlmann-Straub信度模型可以表示为随机截距模型,而随机截距模型假设随机效应服从正态分布。在实际的保险损失数据中,部分个体风险的损失可能远远高于总体平均水平,从而使得不同个体风险之间的风险差异呈现右偏特征。在这种情况下,Buhlmann-Straub模型有可能低估高风险的信度保费。本文在随机截距模型中假设随机效应服从偏正态分布,求得了偏正态随机效应假设下的信度保费。可以证明,Buhlmann-Straub信度保费是其特例。模拟分析和实证研究的结果都表明,偏正态随机效应假设下的信度模型可以更好地预测高风险的信度保费,从而改进传统信度模型的保费估计结果。     

Regression models for analyzing clustered binary and continuous outcomes under an assumption of exchangeability

Scientific experiments commonly result in clustered discrete and continuous data. Existing methods for analyzing such data include the use of quasi-likelihood procedures and generalized estimating equations to estimate marginal mean response parameters. In applications to areas such as developmental toxicity studies, where discrete and continuous measurements are recorded on each fetus, or clinical ophthalmologic trials, where different types of observations are made on each eye, the assumption that data within cluster are exchangeable is often very reasonable. We use this assumption to formulate fully parametric regression models for clusters of bivariate data with binary and continuous components. The regression models proposed have marginal interpretations and reproducible model structures. Tractable expressions for likelihood equations are derived and iterative schemes are given for computing efficient estimates (MLEs) of the marginal mean, correlations, variances and higher moments. We demonstrate the use the ‘exchangeable’ procedure with an application to a developmental toxicity study involving fetal weight and malformation data.     

a bayesian analysis of the multivariate mixed-linear model

The Bayesian analysis of the multivariate mixed linear model is considered. The exact posterior distribution for the fixed effects matrix and the error covariance matrix are obtained. The exact posterior means and variances of the Bayesian estimators for the covariance matrices of random effects are also derived. These posterior moments are computed without constrained optimization and numerical integration. The calculations are feasible for arbitrary models. Reasonable approximations for the posterior distributions for the covariance matrices associated with the random effects are obtained also. Results are illustrated with a numerical example.