Moments of the product and ratio of two correlated chi-square variables

The exact probability density function of a bivariate chi-square distribution with two correlated components is derived. Some moments of the product and ratio of two correlated chi-square random variables have been derived. The ratio of the two correlated chi-square variables is used to compare variability. One such application is referred to. Another application is pinpointed in connection with the distribution of correlation coefficient based on a bivariate t distribution.      

Simple expressions for a bivariate chisquare distribution

The joint distribution of the estimated variances from a correlated bivariate normal distribution has a long history. However, its joint probability density function, conditional moments and product moments are only known as infinite series. In this paper, simpler expressions, mostly finite sums of elementary functions, are derived for these properties. Expressions are also derived for the joint moment generating function and the joint characteristic function.     

Bivariate Kumaraswamy distribution: properties and a new method to generate bivariate classes

In this paper, we introduce a bivariate Kumaraswamy (BVK) distribution whose marginals are Kumaraswamy distributions. The cumulative distribution function of this bivariate model has absolutely continuous and singular parts. Representations for the cumulative and density functions are presented and properties such as marginal and conditional distributions, product moments and conditional moments are obtained. We show that the BVK model can be obtained from the Marshall and Olkin survival copula and obtain a tail dependence measure. The estimation of the parameters by maximum likelihood is discussed and the Fisher information matrix is determined. We propose an EM algorithm to estimate the parameters. Some simulations are presented to verify the performance of the direct maximum-likelihood estimation and the proposed EM algorithm. We also present a method to generate bivariate distributions from our proposed BVK distribution. Furthermore, we introduce a BVK distribution which has only an absolutely continuous part and discuss some of its properties. Finally, a real data set is analysed for illustrative purposes.     

A bivariate distribution with gamma and beta marginals with application to drought data

The first known bivariate distribution with gamma and beta marginals is introduced. Various representations are derived for its joint probability density function (pdf), joint cumulative distribution function (cdf), product moments, conditional pdfs, conditional cdfs, conditional moments, joint moment generating function, joint characteristic function and entropies. The method of maximum likelihood and the method of moments are used to derive the associated estimation procedures as well as the Fisher information matrix, variance–covariance matrix and the profile likelihood confidence intervals. An application to drought data from Nebraska is provided. Some other applications are also discussed. Finally, an extension of the bivariate distribution to the multivariate case is proposed.     

Stochastic Monotonicity and Conditioning in the Limit

Suppose that {( X _n , Y _n )} is a sequence of pairs of cector-valued stochastic variables which converges weakly to ( X , Y ), and that { y _n } converges to y . Sufficient conditions for the conditional distribution of X _n given Y = y are given in terms of stochastic monotonicity. Conditions, which guarantee that also moments of the conditional distributions converge to the moments of the ones of the limit, are also derived.     

On generalized order statistics from linear exponential distribution and its characterization

In this paper, recurrence relations for single and product moments of generalized order statistics (gOSs) from linear exponential distribution (LE) are derived and characterizations of this distribution based on the conditional moments of the gOSs are given.     

A comment on approximating the x2 distribution in the equiprobable case

In an article appearing in this journal, Smith, et al. (1979) reported that a beta approximation of the X² distribution in the equiprobable case did not perform well. This brief note points out that they used the wrong moments and range in their approximation. It is suggested that when this problem is cor-rected, a beta approximation performs better than the chi-square or adjusted chi-square when k ≥ 3 and n is not too small.     

Criterion and selection of optimal subsets in the linear regression,two stage procedures

In many practical situation the regression analysis with stochastic regressors is used. The estimations of this model are often influenced by a high degree of multicollinearity. For avoidance of this fact a criterion and a procedure for the selection of an optimal subset for regression will be derived on the base of the partition of the moments of the conditional normal distribution of the regressand under the condition of the regressors. Further two stage procedures improving the result of the subset regression. based also on the partition of the conditional moments will be given.     

A new compound class of log-logistic Weibull–Poisson distribution: model,properties and applications

A new class of distributions called the log-logistic Weibull–Poisson distribution is introduced and its properties are explored. This new distribution represents a more flexible model for lifetime data. Some statistical properties of the proposed distribution including the expansion of the density function, quantile function, hazard and reverse hazard functions, moments, conditional moments, moment generating function, skewness and kurtosis are presented. Mean deviations, Bonferroni and Lorenz curves, Rényi entropy and distribution of the order statistics are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators and width of the confidence interval for each parameter and finally applications of the model to real data sets are presented to illustrate the usefulness of the proposed distribution.     

On some study of skew-t distributions

Abstract

In this note, through ratio of independent random variables, new families of univariate and bivariate skew-t distributions are introduced. Probability density function for each skew-t distribution will be given. We also derive explicit forms of moments of the univariate skew-t distribution and recurrence relations for its cumulative distribution function. Finally we illustrate the flexibility of this class of distributions with applications to a simulated data and the volcanos heights data.     

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.     

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.     

Weighted bivariate logarithmic series distributions

In this paper, we are interested in the weighted distributions of a bivariate three parameter logarithmic series distribution studied by Kocherlakota and Kocherlakota (1990). The weighted versions of the model are derived with weight W(x,y) = x^[r] y^[s]. Explicit expressions for the probability mass function and probability generating functions are derived in the case r = s = l. The marginal and conditional distributions are derived in the general case. The maximum likelihood estimation of the parameters, in both two parameter and three parameter cases, is studied. A procedure for computer generation of bivariate data from a discrete distribution is described. This enables us to present two examples, in order to illustrate the methods developed, for finding the maximum likelihood estimates.     

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.     

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).     

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.     

An approximation to percentiles of a variable of the bivariate normal distribution when the other variable is truncated,with applications

A Cornish-Fisher expansion is used to approximate the per-centiles of a variable of the bivariate normal distribution when the other variable is truncated. The expression is in terms of the bivariate cumulants of a singly truncated bivariate normal distribution. The percentiles are useful in the problem of personnel selection where we use a screening variable to screen applicants for employment and a correlated performance variable to screen employees for rehiring. This paper provides a bivariate cumulants table for determining the cutoff score of the performance variable. The following two problems are also con¬sidered: (1) determine the proportion of applicants who would have been successful had no screening been applied, and (2) determine the proportion of individuals being rejected byscreening who would have been successful had they been hired, The variable that is used to measure job performance and the variable that measures the outcome of an aptitude test are assumed to be jointly normally distributed with correlation ρ     

A note on some characterizations of the hyperexponential distribution

In this paper, the hyperexponential distribution is characterized using relationships between two moments of order statistics. Also another characterization is introduced by considering the conditional expected spacings of order statistics.     

A backward construction and simulation of correlated Poisson processes

In this paper, we consider a generalisation of the backward simulation method of Duch et al. [New approaches to operational risk modeling. IBM J Res Develop. 2014;58:1–9] to build bivariate Poisson processes with flexible time correlation structures, and to simulate the arrival times of the processes. The proposed backward construction approach uses the Marshall–Olkin bivariate binomial distribution for the conditional law and some well-known families of bivariate copulas for the joint success probability in lieu of the typical conditional independence assumption. The resulting bivariate Poisson process can exhibit various time correlation structures which are commonly observed in real data.     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.