Distributions of ratios of random variables from the power-quadratic exponential family and applications

We derive the closed form exact expression of the density of the ratio X ₁/X ₂, where X ₁ and X ₂ are independent and belong to the very versatile family of power-quadratic exponential distributions. The case of X ₂ as a gamma variable is studied in detail, and new applications found in different domains show the varied uses of the results presented.     

Sums, products and ratios of generalized beta variables

Recent papers by Professor T. Pham-Gia derived distributions of sums, products and ratios of independent beta random variables. In this paper, we extend professor Pham-Gia's results when X ₁ and X ₂ are independent random variables distributed according to two generalized beta distributions. For each of these distributions, we derive exact expressions for the densities of S=X ₁+X ₂, D=X ₁−X ₂, P=X ₁ X ₂, and R=X ₂/X ₁. The expressions turn out to involve the incomplete beta function as well as the hypergeometric functions of one and two variables.     

The distribution of the product of independent beta variables

We investigate the classic distribution and approximate distribution of the product of Beta variables which are independent. We show that the product of independent Beta variables is a Beta variable under the some assumptions. We also obtain the approximate distribution of the product of independent Beta variables.     

On linear combinations of independent exponential variables

The exact distribution of a linear combination of n indepedent negative exponential random variables , when the coefficients cf the linear combination are distinct and positive , is well-known. Recently Ali and Obaidullah (1982) extended this result by taking the coeff icients to be arbitrary real numbers. They used a lengthy geometric.

al approach to arrive at the result . This article gives a simple derivation of the result with the help of a generalized partial fraction technique. This technique also works when the variables involved are gamma variables with certain types of parameters. Results are presented in a form which can easily be programmed for computational purposes. Connection of this problem t o various problems in different fields is also pointed out.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

Sampling Methods for Wallenius' and Fisher's Noncentral Hypergeometric Distributions

Several methods for generating variates with univariate and multivariate Walleniu' and Fisher's noncentral hypergeometric distributions are developed. Methods for the univariate distributions include: simulation of urn experiments, inversion by binary search, inversion by chop-down search from the mode, ratio-of-uniforms rejection method, and rejection by sampling in the τ domain. Methods for the multivariate distributions include: simulation of urn experiments, conditional method, Gibbs sampling, and Metropolis-Hastings sampling. These methods are useful for Monte Carlo simulation of models of biased sampling and models of evolution and for calculating moments and quantiles of the distributions.     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

On the Characteristic Function for Asymmetric Exponential Power Distributions

The econometric literature has seen a surge of developments in the theory and applications of asymmetric exponential power distributions (AEPDs). Here, for the first time, we derive explicit closed form expressions for the characteristic function of AEPDs. The expressions involve the complex parameter Wright generalized hypergeometric function.     

Alpha power Weibull distribution: Properties and applications

In this paper, a new lifetime distribution is defined and studied. We refer to the new distribution as alpha power Weibull distribution. The importance of the new distribution comes from its ability to model monotone and non monotone failure rate functions, which are quite common in reliability studies. Various properties of the proposed distribution are obtained including moments, quantiles, entropy, order statistics, mean residual life function, and stress-strength parameter. The maximum likelihood estimation method is used to estimate the parameters. Two real data sets are used to illustrate the importance of the proposed distribution.     

Distributions of the product and ratio of Maxwell and Rayleigh random variables

The distributions of the product and ratio of independent random variables arise in many applied problems. These have been extensively studied by many researchers. In this paper, the distributions of the product | XY | and ratio have been derived, when X and Y are Maxwell and Rayleigh random variables and are distributed independently of each other. The associated cdfs, pdfs, kth moments, entropies, etc., have been given. To describe the possible shapes of the associated pdfs and entropies, the respective plots are provided. The percentage points associated with the cdfs of the product and ratio have been tabulated.     

Multivariate distributions involving ratios of normal variables

The papsr considers distributions of collections of ratios of normal variables, The derivation of the joint density is linked to SKI sting literature on absolute, incomplete or truncated moments of multinormals. The distribution function may be expressed as a sum of rectangular multi normal probabilities. When the coefficients of variation of the denominators are close to zero, then a simple transformation of the ratios is approximately inultinormal. An application to Bayesian analysis is included.     

Umvu estimators for the population mean and variance based on random effects models for lognormal data

Cross-classified data are often obtained in controlled experimental situations and in epidemiologic studies. As an example of the latter, occupational health studies sometimes require personal exposure measurements on a random sample of workers from one or more job groups, in one or more plant locations, on several different sampling dates. Because the marginal distributions of exposure data from such studies are generally right-skewed and well-approximated as lognormal, researchers in this area often consider the use of ANOVA models after a logarithmic transformation. While it is then of interest to estimate original-scale population parameters (e.g., the overall mean and variance), standard candidates such as maximum likelihood estimators (MLEs) can be unstable and highly biased. Uniformly minimum variance unbiased (UMVU) cstiniators offer a viable alternative, and are adaptable to sampling schemes that are typiral of experimental or epidemiologic studies. In this paper, we provide UMVU estimators for the mean and variance under two random effects ANOVA models for logtransformed data. We illustrate substantial mean squared error gains relative to the MLE when estimating the mean under a one-way classification. We illustrate that the results can readily be extended to encompass a useful class of purely random effects models, provided that the study data are balanced.     

A Family Which Integrates the Generalized Charlier Series and Extended Noncentral Negative Binomial Distributions

The generalized Charlier series distribution includes the binomial distribution, and the noncentral negative binomial distribution extends the negative binomial distribution. The present article proposes a family of counting distributions, which contains both the generalized Charlier series and extended noncentral negative binomial distributions. Compound and mixture formulations of the proposed distribution are given. The probability mass function is expressible in terms of the confluent hypergeometric function as well as the Gauss hypergeometric function. Recursive formulae for probability mass function have been studied by Panjer, Sundt and Jewell, Schröter, Sundt, and Kitano et al. in the context of insurance risk. This article explores horizontal, vertical, triangular, and diagonal recursions. Recursive formulae as well as exact expressions for descending factorial moments are studied. The proposed distribution allows overdispersion or underdispersion relative to a Poisson distribution. An illustrative example of data fitting is given.     

Laplace approximations to posterior moments and marginal distributions on circles,spheres, and cylinders

We extend recent work on Laplace approximations (Tierney and Kadane 1986; Tierney, Kass, and Kadane 1989) from parameter spaces that are subspaces of R^k to those that are on circles, spheres, and cylinders. While such distributions can be mapped onto the real line (for example, a distribution on the circle can be thought of as a function of an angle θ, 0 ? 0 ? 2π), that the end points coincide is not a feature of the real line, and requires special treatment. Laplace approximations on the real line make essential use of the normal integral in both the numerator and the denominator. Here that role is played by the von Mises integral on the circle, by the Bingham integrals on the spheres and hyperspheres, and by the normal-von Mises and normal-Bingham integrals on the cylinders and hypercylinders, respectively. We begin with a brief introduction to Laplace approximations and to previous Bayesian work on circles, spheres, and cylinders. We then develop the theory for parameter spaces that are hypercylinders, since all other shapes considered here are special cases. We compute some examples, which show reasonable accuracy even for small samples.     

On unbiased estimation of gini index and yntema-pietra index of lognormal distribution and their variances

Here we derive the uniformly minimum variance unbiased (best) estimator and a strongly consistent, asymptotically normal, unbiased estimator of each of Gini index and Yntema-Pietra index of lognormal distribution . These estimators are in terms of generalized hypergeometric functions ₁F₂. Further, the variances of these estimators and the best estimators of variances of best estimators are found out. They are in terms of Kempé de Fériet's hypergeometric functions.     

Distribution of the sum of independent generalized logarithmic series variables

Jain and Gupta (1973) have given a generalized logarithmic series distribution which, for β = 1, reduces to the logarithmic series distribution. In this note we obtain the distribution of the sum of independent generalized logarithmic series variables. This distribution conforms, in a special case, to the First-type Stirling distribution (Patil and Wani, 1965) and would be useful in estimation theory.     

On the type-B integral equation and the distribution of wilks statist for testing independence of several groups of variables

In this paper, WILKS'type-B integral equation is solved in the general form of a series of beta functions and a series of weighted gamma functions as proposed by WALD and BROOKNER 1941. The coefficients in both representations can be obtained by explicit recurrence relartions, therefore the results solve many distributional problems and have the fewest computational difficulties of any representation that has surfaced to date. The radius of convergence of the second series representation is given, whereas the convergence property of the first series representation is given, whereas the convergence property of the first series representation was studied by WALD and Brookner. The exact null distributions of WILKS' statistic A for testing the independence of several groups of variables and of V = -log A are given. The coefficients in all the series representration can be computed recursilvely and hence can be obtained easily with the help of modern computatinal facilities