An approximation to the distribution of the product of two dependent correlation coefficients

Many methodological studies depend on the product of two dependent correlation coefficients. However, the behavior of the distribution of the product of two dependent correlation coefficients is not well known. The distribution of sets of correlation coefficients has been well studied, but not the distribution of the product of two dependent correlation coefficients. The present study derives an approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I distribution. A simulation study is also conducted to assess the accuracy of the approximation.     

On an Application of Concomitants of Order Statistics in Characterizing a Family of Bivariate Distributions

In this article, we consider a family of bivariate distributions which includes the well-known Morgenstern family of bivariate distributions as its subclass. We identify some properties of concomitants of order statistics which characterize this generalized class of distributions. An application of the characterization result in modeling a bivariate distribution to a data is also explained.     

Role of concomitants of order statistics in determining parent bivariate distributions

In this paper, we establish the role of concomitants of order statistics in the unique identification of the parent bivariate distribution. From the results developed, we have illustrated by examples the process of determination of the parent bivariate distribution using a marginal pdf and the pdf of either of the concomitant of largest or smallest order statistic on the other variable. An application of the results derived in modeling of a bivariate distribution for data sets drawn from a population as well is discussed.     

A new generalized p-value approach for testing equality of coefficients of variation in k normal populations

A new generalized p-value method is proposed for testing the equality of coefficients of variation in k normal populations. Simulation studies show that the type I error probabilities are close to the nominal level. The proposed test is also compared with likelihood ratio test, modified Bennett's test and score test through Monte Carlo simulation, the results demonstrate that the generalized p-value method has satisfactory performance in terms of sizes and powers.     

Testing equality of correlation coefficients in two populations via permutation methods

The present paper investigates the asymptotic behaviour of a studentized permutation test for testing equality of (Pearson) correlation coefficients in two populations. It is shown that this test is asymptotically of exact level and has the same power for contiguous alternatives as the corresponding asymptotic test. As a by-product we specify the assumptions needed for the validity of the permutation test suggested in Sakaori (2002). A small simulation study compares the finite sample properties of the considered tests.     

Test for the equality of several correlation coefficients

We derive two C(α) statistics and the likelihood-ratio statistic for testing the equality of several correlation coefficients, from k ≥ 2 independent random samples from bivariate normal populations. The asymptotic relationship of the C(α) tests, the likelihood-ratio test, and a statistic based on the normality assumption of Fisher's Z-transform of the sample correlation coefficient is established. A comparative performance study, in terms of size and power, is then conducted by Monte Carlo simulations. The likelihood-ratio statistic is often too liberal, and the statistic based on Fisher's Z-transform is conservative. The performance of the two C(α) statistics is identical. They maintain significance level well and have almost the same power as the other statistics when empirically calculated critical values of the same size are used. The C(α) statistic based on a noniterative estimate of the common correlation coefficient (based on Fisher's Z-transform) is recommended.     

Testing the equality of correlation matrices when sample correlation matrices are dependent

We consider the situation in which a 2m×2m$2m\times 2m$ correlation matrix is naturally partitioned into a 2×2$2\times 2$ form, where each submatrix is m×m$m\times m$ and we wish to test for the equality of the two correlation matrices appearing on the diagonal of the partitioned form. Standard tests for the equality of correlation matrices do not apply since the corresponding sample correlation matrices are correlated. We obtain Wald statistics under the assumption of multivariate normality as well as extensions that apply to elliptical distributions.     

A permutation procedure for testing the equality of pattern hypotheses across groups involving correlation or covariance matrices

This paper describes a permutation procedure to test for the equality of selected elements of a covariance or correlation matrix across groups. It involves either centring or standardising each variable within each group before randomly permuting observations between groups. Since the assumption of exchangeability of observations between groups does not strictly hold following such transformations, Monte Carlo simulations were used to compare expected and empirical rejection levels as a function of group size, the number of groups and distribution type (Normal, mixtures of Normals and Gamma with various values of the shape parameter). The Monte Carlo study showed that the estimated probability levels are close to those that would be obtained with an exact test except at very small sample sizes (5 or 10 observations per group). The test appears robust against non-normal data, different numbers of groups or variables per group and unequal sample sizes per group. Power was increased with increasing sample size, effect size and the number of elements in the matrix and power was decreased with increasingly unequal numbers of observations per group.     

Error rate for testing a contrast after a significant F test

In statistical applications an experimenter often tests a particular contrast after a significant F test for the equality of means. This paper evaluates the overall error rate for this testing .procedure.     

Bivariate generalized gamma distributions of Kibble's type

In this paper, a new type of bivariate generalized gamma (BGG) distribution derived from the bivariate gamma distribution of Kibble [Two-variate gamma-type distribution. Sankh?a 1941;5:137–150] by means of a power transformation is presented. The explicit expressions of statistical properties of the BGG distribution are presented. The estimation of marginal and dependence parameters using the method of moments and the method of inference functions for margins are discussed, and their performance through a Monte Carlo simulation study is assessed. Finally, an example is given to illustrate the applicability of the distributions introduced here.     

A Single Form for t-Distributions and Symmetric Beta Distributions

A single parametric form is given for the symmetric distributions in the Pearson system with finite variance. In effect, these are Student's t-distributions with ν > 2 and all centered symmetric beta distributions. A different parametrization allows the inclusion of the t-distributions with ν ≤2 at the expense of symmetric beta distributions with a low shape parameter.     

Double sample tests for the equality of two means

A double sample (two stage) testing procedure is proposed as an alternative to the usual one stage     

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.     

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).     

A robust procedure for testing the equality of mean vectors of two bivariate populations with unequal covariance matrices

A robust test is developed for testing equality of the mean vectors of two bivariate (multivariate) populations when the variance-covariance matrices are not necessarily equal. The test is an extension of the univariate robust test given by Tiku and Singh (1981).     

Rényi statistics for testing equality of autocorrelation coefficients

The problem of testing for equality of autocorrelation coefficients of two populations in multivariate data when errors are autocorrelated is considered. We derive Rényi statistics defined as divergences between unrestricted and restricted estimated joint probability density functions and we show that they are asymptotically chi-square distributed under the null hypothesis of interest. Monte Carlo simulation experiments are carried out to investigate the behavior of Rényi statistics and to make comparisons with test statistics based on the approach of Bhandary [M. Bhandary, Test for equality of autocorrelation coefficients for two populations in multivariate data when the errors are autocorrelated, Statistics & Probability Letters 73 (2005) 333–342] for the problem under consideration. Rényi statistics showed to have significantly better behavior.     

Percentage points of LRC for testing equality of covariance matrices under intraclass correlation structure

The exact null distribution of the likelihood ratio test statistic for testing equality of covariance matrices of q compound symmetric Gaussian models (bivariate or trivariate) has been obtained and percentage points for q ? 5 have been computed. The inverse Mellin transform and calculus of residues have been used to derive these results.     

The ordering of negative quadrant dependence

A partial ordering is developed among negative quadrant dependent distributions with fixed marginals. Basic properties and closure under certain statistical operations are derived. Applications of the results in statistics and probability are given.     

A two-stage procedure with accelerated test for selecting the best exponential population

In this article, we discuss a two-stage procedure for selecting the largest location parameter among k(k≥2) two-parameter exponential populations(or products) from an accelerated test. The accelerated test will be conducted at a higher stress level than that of normal in the second stage. under certain assumptions between parameter and stress leveL, the two-stage selection procedure, which guarantees that the probability of correct selection is at least p^*, is proposed. At the end of the paper , we present some useful tables that serve as a guide for the needed sample size in the second stage.