The exact noncentral distribution of the generalized multiple correlation matrix

Given p×n X N(βY, ∑?I), β, ∑ unknown, the noncentral multivariate beta density of the matrix L = [(YY′)^-1/2Y X′ (XX′)^-1XY′ (YY′)^-1/2] is desired. Khatri (1964) finds this density when β is of rank unity. The present paper derives the noncentral density of L and the density of the roots matrix of L for full rank β. The dual case density of L is also obtained. The derivations are based on generalized Sverdrup's lemma, Kabe (1965), and the relationship between primal and dual density of L is explicitly established.     

A further note on the bivariate normal distribution

It is indicated to what extent the conditional normality of the distribution of one comnonent of a bivariate random vector given the value of the other component together with a restricted type of conditional normality or the marginal normality for the other component is equivalent to the bivariate normality of this random vector.     

A property of coefficients of variation for ordered bivariate log-normal variables

This paper considers the maximum and minimum of a pair of log-normal variables with equal mean. It shows that either order statistic has a smaller coefficient of variation than the two original log-normal variables provided the latter are of equal variance. When the variances are unequal, as the variance ratio increases, the minimum (maximum), has a smaller coefficient of variation if the correlation coefficient of the log-normal variables is small (small) and the variances are large (small).     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.     

The bivariate yule distribution and some of its properties

In this paper, a bivariate extension of the YULE distribution is defined and some of its structural properties are examined. It is shown in particular, that it can be obtained in the context of a bivariate STER model and as the only distribution with tail probabilities satisfying certain conditions     

Mixture formulations of a bivariate negative binomial distribution with applications

This paper considers further mixture formulations of the bivariate negative binomial (BNB) distribution of Edwards and Gurland (1961) and Subrahmaniam (1966). These formulations and some known ones are applied (1) to obtain a bivariate generalized negative binomial (BGNB) distribution of Bhattacharya (1966), (2) to establish a connection between the accident-proneness models given by the BNB, BGNB and Bhattacharya's bivariate distributions, and (3) to compute the grade correlation and distribution function of the Wicksell-Kibble bivariate gamma distribution.     

Forms for the distribution of ss ilkptxcxtt statistic for bivariate sphericity

This paper examines the nonnuU distribution of the ellipticity statistic used for testing sphericity in a bivariate normal population. The distribution function is expressed as mixtures of F distribution functions, where the mixing probabilities are binomial and mgatiire binomial. A simple approximation to the distribution is suggested and examined     

The Correlated Bivariate Noncentral F Distribution and Its Application

This article proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pre-test test for the multivariate simple regression model (MSRM).     

Computation of bivariate gamma and inverted beta distribution functions

This paper considers the evaluation of the distribution functions of the bivariate gamma distribution of Wicksell and Kibble, and a bivariate inverted beta distribution. Simple expansions of the distribution functions in terms of marginal quantities and the negative binomial probabilities are derived.     

On the distribution of rin samples from the mixtures of bivariate normal populations

The distributions of some transformations of the sample correlation coefficient r are studied here, when the parent population is a mixture of two standard bivariate normals. The behavior of these transformations is assessed through the first four standard moments. It is shown that there is a close relationship between the behavior of the transformed variables and the lack of normality as evinced by the 'kurtosis' defined in the bivariate population     

Bias reduction for nonparametric correlation coefficients under the bivariate normal copula assumption with known detection limits

The authors derive the asymptotic mean and bias of Kendall's tau and Spearman's rho in the presence of left censoring in the bivariate Gaussian copula model. They show that tie corrections for left‐censoring brings the value of these coefficients closer to zero. They also present a bias reduction method and illustrate it through two applications.     

Canonical expansions,correlation structure,and conditional distributions of bivariate distributions generated by mixtures

A simple result concerning the canonical expansions of mixed bivariate distributions is considered. This result is then applied to analyze the correlation structures of the Bates-Neyman accident proneness model and its generalization, to derive probability inequalities based on the concept of positive dependence, and to construct a bivariate beta distribution with positive correlation coefficient applicable in computer simulation experiments. The mixture formulation of the conditional distribution of this class of mixed bivariate distributions is used to define and generate first-order autoregressive gamma and negative binomial sequences.     

On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: Robustness and the effect of outliers

The distribution of the sample correlation coefficient is derived when the population is a mixture of two bivariate normal distributions with zero mean but different covariances and mixing proportions 1 - λ and λ respectively; λ will be called the proportion of contamination. The test of ρ = 0 based on Student's t, Fisher's z, arcsine, or Ruben's transformation is shown numerically to be nonrobust when λ, the proportion of contamination, lies between 0.05 and 0.50 and the contaminated population has 9 times the variance of the standard (bivariate normal) population. These tests are also sensitive to the presence of outliers.