Segmented classification analysis with a class of rectangle-screened elliptical populations

In many practical situations, a statistical practitioner often faces a problem of classifying an object from one of the segmented (or screened) populations where the segmentation was conducted by a set of screening variables. This paper addresses this problem, proposing and studying yet another optimal rule for classification with segmented populations. A class of q-dimensional rectangle-screened elliptically contoured (RSEC) distributions is considered for flexibly modeling the segmented populations. Based on the properties of the RSEC distributions, a parametric procedure for the segmented classification analysis (SCA) is proposed. This includes motivation for the SCA as well as some theoretical propositions regarding its optimal rule and properties. These properties allow us to establish other important results which include an efficient estimation of the rule by the Monte Carlo expectation–conditional maximization algorithm and an optimal variable selection procedure. Two numerical examples making use of utilizing a simulation study and a real dataset application and advocating the SCA procedure are also provided.     

On the sample correlation coefficient in the truncated bivariate normal population

The truncated bivariate normal distribution (TBVND) with truncation in both variables on the left is studied here. The behaviour of the sample correlation coefficient is assessed through its moments when the sample is from such a population. Some inequalities established by Rao et al. (1968) are extended     

Maximum correlation for the generalized Sarmanov bivariate distributions

In order to improve the correlation of the traditional Sarmanov distribution, a ‘generalized’ version was introduced earlier by Bairamov et al. (2001). The extent of the improvement in correlation, however, was never investigated in the literature. In this note we compare the two Sarmanov models regarding their maximum correlation. Several examples are given. It is shown that unlike the traditional Sarmanov, the generalized one always has a correlation approaching one regardless of the marginals, as long as the marginals are of the same type. When they are not of the same type, however, the correlation has an upper bound strictly less than one. We find conditions under which the upper bound is attained. Finally, we investigate the rates of convergence to the maximum correlation for the generalized Sarmanov bivariate distributions.     

Hypothesis testing on the correlation coefficient

A simple and accurate test on the value of the correlation coefficient in normal bivariate populations is here proposed. Its accuracy compares favourably with any previous approximations.     

Extreme behaviour for bivariate elliptical distributions

The authors examine the asymptotic behaviour of conditional threshold exceedance probabilities for an elliptically distributed pair (X, Y) of random variables. More precisely, they investigate the limiting behaviour of the conditional distribution of Y given that X becomes extreme. They show that this behaviour differs between regularly and rapidly varying tails.     

Notes on the MLE of correlation coefficient in meta analysis

We study the properties of two approximations to the MLE of the correlation coefficient based on estimates from several studies in meta analysis. Our work is based on an approximation to the density of a function of the sample product-moment estimate due to Dclury, Hsu, and Kraemer. Regarding this approximation, we point out and correct some mistakes in the literature.     

Confidence intervals for the correlation from a bivariate normal

Improved confidence intervals are given for the correlation coefficient of the bivariate normal distribution. These are based on Cornish–Fisher expansions for the distribution, density and quantiles of the sample correlation.     

On singular elliptical models

The probability density function (pdf) ofsingular elliptical distributions is represented as an integralseries of singular normal distributions. Explicit formulas for the pdf and the cdf of the generalized Chi-square distribution are derived under singular elliptical assumptions extending the result of Díaz-García [(2002). Singular elliptical distribution: density and applications. Commun. Stat.—Theory Methods 31:665–681]. Applications are given of the proposed result for singular mixedmodels.     

A procedure for testing the equality of k dependent correlation coefficients

A procedure is proposed for testing the equality of k dependent correlation coefficients. The procedure is simulated utilizing Monte Carlo techniques; and, a method for post hoc probing is also suggested.     

The noncentral bivariate chisquared distribution and extensions

The distribution of certain correlated noncentral chisquared variates P, Q, is termed the noncentral bivariate chisquared distribution. Moment generating functions of the distributions of (P, Q), (P+Q) and other quadratic forms have been obtained. A relationship to the linear case of the noncentral Wishart distribution is indicated. Convolution properties and applications are presented.     

Inferences on correlation coefficients of bivariate log-normal distributions

This article considers inference on correlation coefficients of bivariate log-normal distributions. We developed generalized confidence intervals and hypothesis tests for the correlation coefficients, and extended the results to compare two independent correlations. Simulation studies show that the suggested methods work well. Two practical examples are used to illustrate the application of the proposed methods.     

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).     

Estimating the correlation coefficient in the presence of correlated observations from a bivariate normal population

Estimation of the correlation coefficient between two variates (p) in the presence of correlated observations from a bivar iate normal population is considered The estimated maximum likelihood estimator (EMLE), an estimate based on the maximum likelihood estimator (MLE), is proposed and studied for the estimation of p For the large sample case , approximate expressions foi the variance and the bias of the Pearson estimate of the correlation coefficient are derived. These expressions suggests that the Pearson’s estimator possesses high mean square error (MSE) in estimating ρ in comparison to the MLE The MSE is particularly high when the observations within clusters aie highly correlated. The Pearson’s estimate, the MLE, and the EMLE aie evaluated in a simulation study This study shows that the proposed EMLE pefoims bettei than the Pearson’s correlation coefficient except when the number of clusters is small.     

Vector correlation for elliptical distributions

A measure of multivariate correlation between two sets of vectors is considered when the underlying joint distribution is a member of the class of elliptical distributions. Its asymptotic distribution is derived under different situations and these results are used to test hypotheses on vector correlation when the underlying joint distribution is non-normal.     

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.     

Ultrastructural elliptical models

Dolby's (1976) ultrastructural model with no replications is investigated within the class of the elliptical distributions. General asymptotic results are given for the sample covariance matrix S in the presence of incidental parameters. These results are used to study the asymptotic behaviour of some estimators of the slope parameter, unifying and extending existing results in the literature. In particular, under some regularity conditions they are shown to be consistent and asymptotically normal. For the special case of the structural model, some asymptotic relative efficiencies are also reported which show that generalized least squares and the method of moment estimators can be highly inefficient under nonnormality.     

A new class of bivariate lifetime distributions

This paper introduces a new class of bivariate lifetime distributions. Let {X_i}_{i ? 1} and {Y_i}_{i ? 1} be two independent sequences of independent and identically distributed positive valued random variables. Define T₁ = min?(X₁, …, X_M) and T₂ = min?(Y₁, …, Y_N), where (M, N) has a discrete bivariate phase-type distribution, independent of {X_i}_{i ? 1} and {Y_i}_{i ? 1}. The joint survival function of (T₁, T₂) is studied.