Multivariate normal estimation: the case (n < p)

Estimation in the multivariate context when the number of observations available is less than the number of variables is a classical theoretical problem. In order to ensure estimability, one has to assume certain constraints on the parameters. A method for maximum likelihood estimation under constraints is proposed to solve this problem. Even in the extreme case where only a single multivariate observation is available, this may provide a feasible solution. It simultaneously provides a simple, straightforward methodology to allow for specific structures within and between covariance matrices of several populations. This methodology yields exact maximum likelihood estimates.     

Some properties of Lin–Wong divergence on the past lifetime data

Measures of statistical divergence are used to assess mutual similarities between distributions of multiple variables through a variety of methodologies including Shannon entropy and Csiszar divergence. Modified measures of statistical divergence are introduced throughout the present article. Those modified measures are related to the Lin–Wong (LW) divergence applied on the past lifetime data. Accordingly, the relationship between Fisher information and the LW divergence measure was explored when applied on the past lifetime data. Throughout this study, a number of relations are proposed between various assessment methods which implement the Jensen–Shannon, Jeffreys, and Hellinger divergence measures. Also, relations between the LW measure and the Kullback–Leibler (KL) measures for past lifetime data were examined. Furthermore, the present study discusses the relationship between the proposed ordering scheme and the distance interval between LW and KL measures under certain conditions.     

Conditional covariance penalties for mixed models

The prediction error for mixed models can have a conditional or a marginal perspective depending on the research focus. We introduce a novel conditional version of the optimism theorem for mixed models linking the conditional prediction error to covariance penalties for mixed models. Different possibilities for estimating these conditional covariance penalties are introduced. These are bootstrap methods, cross-validation, and a direct approach called Steinian. The behavior of the different estimation techniques is assessed in a simulation study for the binomial-, the t-, and the gamma distribution and for different kinds of prediction error. Furthermore, the impact of the estimation techniques on the prediction error is discussed based on an application to undernutrition in Zambia.     

Strong law of large numbers and complete convergence for non-identically distributed WOD random variables

In this paper, we establish the strong law of large numbers and complete convergence for non-identically distributed WOD random variables. We derive some new inequalities of Fuk–Nagaev type for the sums of non-identically distributed WD random variables. All these results further extend and refine previous ones.     

An algorithmic approach to construct D-optimal saturated designs for logistic model

In this paper, locally D-optimal saturated designs for a logistic model with one and two continuous input variables have been constructed by modifying the famous Fedorov exchange algorithm. A saturated design not only ensures the minimum number of runs in the design but also simplifies the row exchange computation. The basic idea is to exchange a design point with a point from the design space. The algorithm performs the best row exchange between design points and points form a candidate set representing the design space. Naturally, the resultant designs depend on the candidate set. For gain in precision, intuitively a candidate set with a larger number of points and the low discrepancy is desirable, but it increases the computational cost. Apart from the modification in row exchange computation, we propose implementing the algorithm in two stages. Initially, construct a design with a candidate set of affordable size and then later generate a new candidate set around the points of design searched in the former stage. In order to validate the optimality of constructed designs, we have used the general equivalence theorem. Algorithms for the construction of optimal designs have been implemented by developing suitable codes in R.     

The exponential distribution analog of the Grubbs–Weaver method

Abstract

Grubbs and Weaver (1947 Grubbs, F. E., and C. L. Weaver. 1947. The best unbiased estimate of population standard deviation based on group ranges. Journal of the American Statistical Association 42 (238):224–41. doi: 10.2307/2280652.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) suggest a minimum-variance unbiased estimator for the population standard deviation of a normal random variable, where a random sample is drawn and a weighted sum of the ranges of subsamples is calculated. The optimal choice involves using as many subsamples of size eight as possible. They verified their results numerically for samples of size up to 100, and conjectured that their “rule of eights” is valid for all sample sizes. Here we examine the analogous problem where the underlying distribution is exponential and find that a “rule of fours” yields optimality and prove the result rigorously.     

Multivariate normal mean–variance mixture distribution based on Birnbaum–Saunders distribution

A multivariate normal mean–variance mixture based on a Birnbaum–Saunders (NMVMBS) distribution is introduced and several properties of this new distribution are discussed. A new robust non-Gaussian ARCH-type model is proposed in which there exists a relation between the variance of the observations, and the marginal distributions are NMVMBS. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of this normal mean–variance mixture distribution is given. A simulation study and some real data are used to demonstrate the modelling strength of this new model.     

Tail asymptotic of discounted aggregate claims with compound dependence under risky investment

This paper considers the tail asymptotic of discounted aggregate claims with compound dependence under risky investment. The price of risky investment is modeled by a geometric Lévy process, while claims are modeled by a one-sided linear process whose innovations further obeying a so-called upper tail asymptotic independence. When the innovations are heavy tailed, we derive some uniform asymptotic formulas. The results show that the linear dependence has significant impact on the tail asymptotic of discounted aggregate claims but the upper tail asymptotic independence is negligible.     

Likelihood ratio tests for multivariate normality

This paper presents some powerful omnibus tests for multivariate normality based on the likelihood ratio and the characterizations of the multivariate normal distribution. The power of the proposed tests is studied against various alternatives via Monte Carlo simulations. Simulation studies show our tests compare well with other powerful tests including multivariate versions of the Shapiro–Wilk test and the Anderson–Darling test.