NONNULL DISTRIBUTION OF LIKELIHOOD RATIO CRITERION FOR TESTING MULTISAMPLE SPHERICITY IN THE COMPLEX CASE

In this paper, nonnull moments of the likelihood ratio statistic for testing multisample sphericity in the complex case have been derived in series involving zonal polynomials. The nonnull asymptotic distribution of the statistic is also derived for certain alternatives.     

Asymptotic expansion of the nonnull distribution of likelihood ratio statistic for testing multisample sphericity

Asymptotic expansion of the nonnull distribution of the likelihood ratio statistic for testing muitisample sphericity in q multinormal populations is derived for the alternatives close to the null hypothesis.     

On testing structure of mean vector of compound symmetric gaussian model

In this paper the moments of the likelihood ratio statistic for testing the structure of mean vector of a compound symmetric Gaussian model, have been derived by using the orthogonal transformation of variables. Then the distribution of the test statistic is studied.     

A new bivariate beta distribution

A new bivariate beta distribution capable of providing better fits than all its competitors is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and conditional moments. The method of maximum likelihood is used to derive the associated estimation procedure. Applications to six bivariate data sets are illustrated.     

Bimatrix variate gamma-beta distributions

Two bimatrix distributions with beta and gamma marginals are introduced. Various properties (including product moments of determinants and traces, entropies, marginal distributions) are derived. Parameter estimation by the method of maximum likelihood is discussed. The performance and efficiencies of the maximum likelihood estimators and the associated confidence intervals are assessed by simulation. The efficiencies are compared versus those for the maximum likelihood estimators and the associated confidence intervals based on matrix variate gamma distributions. A discussion of possible applications of the bimatrix distributions is given.     

Matrix variate Macdonald distribution

Abstract

In this article, we generalize the univariate Macdonald distribution to the matrix case and give its derivation using matrix variate gamma distribution. We study several properties such as cumulative distribution function, marginal distribution of submatrix, triangular factorization, moment generating function, and expected values of several functions of the Macdonald matrix. Some of these results are expressed in terms of special functions of matrix arguments and zonal polynomials.     

The influence of an older sibling on preschoolers’ lie‐telling behavior

In the present study, children’s (2‐ to 5‐years old) lie‐telling was examined in relation to theory of mind (first‐order false belief understanding), executive functioning (measuring inhibitory control in conjunction with working memory), and presence of siblings in the home (no siblings vs. siblings; younger siblings vs. older siblings).  Lie‐telling was observed using a temptation resistance paradigm. Overall, of the 152 (74.9%) children who peeked at the toy, 73 (48%) lied during the temptation resistance paradigm. Children with higher scores on measures of first‐order false belief understanding, and measures that relied on inhibitory control, were more likely to lie compared to their truthful counterparts.  Additionally, children with older siblings were more likely to lie to the research assistant, and this relationship was independent of performance on cognitive tasks. Overall, results demonstrate that having an older sibling has an independent, direct effect on the development of young children’s lie‐telling abilities, irrespective of cognitive ability. These findings support the argument that lie‐telling is a behavior that is facilitated by both cognitive and social factors.     

On Exact Statistical Properties of Multidimensional Indices Based on Principal Components,Factor Analysis,MIMIC and Structural Equation Models

Recent empirical literature has seen many multidimensional indices emerge as well-being or poverty measures, in particular indices derived from principal components and various latent variable models. Though such indices are being increasingly and widely employed, few studies motivate their use or report the standard errors or confidence intervals associated with these estimators. This paper reviews the different underlying models, reaffirms their appropriateness in this context, examines the statistical properties of resulting indices, gives analytical expressions of their variances and establishes certain exact relationships among them.      

Non-central bivariate beta distribution

Let U, V and W be independent random variables, U and V having a gamma distribution with respective shape parameters a and b, and W having a non-central gamma distribution with shape and non-centrality parameters c and δ, respectively. Define X = U/(U + W) and Y = V/(V + W). Clearly, X and Y are correlated each having a non-central beta type 1 distribution, X ~ NCB1 (a,c;d){X \sim {\rm NCB1} (a,c;\delta)} and Y ~ NCB1 (b,c;d){Y \sim {\rm NCB1} (b,c;\delta)} . In this article we derive the joint probability density function of X and Y and study its properties.