A note on the distribution of the product of zero‐mean correlated normal random variables

The problem of finding an explicit formula for the probability density function of two zero‐mean correlated normal random variables dates back to 1936. Perhaps, surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero‐mean correlated normal random variables as a variance‐gamma random variable, from which an explicit formula for the probability density function is immediate.     

Admissible and minimax estimators of 363-1363-1363-1in the gamma distribution with truncated parameter space

In this paper we consider asmissible and minimax estimation of the parameter in the gamma distribution with truncated parameter space. We give a necessary and sufficient condition for minimaxity (Theorem 1) and obtain the classes of new minimax and asmissible estimators. The results of the paper can be applied to estimation of parameters in the normal, lognormal, Pareto, generalized gamma, generalized Laplace and other distributions.     

On decompositions of some random variables

The aim of this paper is to present several stochastic analogs of classical formulas for the gamma function. The obtained results provide representation of some random variables as finite or infinite products of independent random variables. Examples include generalized gamma, normal, beta and other distributions.     

Ordering results of the smallest and largest order statistics from independent heterogeneous exponentiated gamma random variables

In this paper, we discuss stochastic comparisons of lifetimes of series and parallel systems with heterogeneous exponentiated gamma components. The results established here are developed in two directions. First, when a system possibly has different shape and scale parameters and the matrix of those different parameters following the chain majorization order, we study the reversed hazard rate order of parallel systems and the usual stochastic order of series systems. Next, by using the concept of vector majorization, we establish the usual stochastic order of series systems and the reversed hazard rate order of parallel systems.     

The life-length of banknotes*

Abstract With the aid of the Bank's banknote sorting system the issue and subsequent withdrawal of f 25–banknotes on three varieties of paper have been recorded for two-and-a-half years. The aim was to measure the durability of the three paper varieties in circulation. The results of this second trial with f 25–banknotes confirm the statistical model developed previously for the first trial with f 100–banknotes. GRESHAM's Law is equally not applicable, neither to f 25–banknotes nor to f 100–banknotes. A two-parameter gamma distribution fits the cumulative fraction of banknotes withdrawn reasonably well.     

The exact density of a statistic related to the shape parameter of a gamma variate

The exact density of the statistic ln , where and denote, respectively, the arithmetic and the geometric means of a random sample from a two-parameter gamma distribution, is obtained in a computable form using the technique of the inverse Mellin transform. This statistic is related to the maximum likelihood estimator of the shape parameter of a gamma distribution.     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.     

Orthogonal expansions for the generalized Cramér-von Mises statistics

We provide a new proof for the representation of Cramér-von Mises statistics under (known) gamma and normal distributions. The new method uses orthogonal polynomials and provides an explicit form of the statistics from which the asymptotic distribution can be calculated.Acknowledgements This research was partially supported by FQM-331, FQM-270, BMF 2001-2378 and BMF2002-04525-C02-02. The authors are thankful to the referees for their suggestions and helpful comments.     

Flexible mixture modelling of stochastic frontiers

This paper introduces new and flexible classes of inefficiency distributions for stochastic frontier models. We consider both generalized gamma distributions and mixtures of generalized gamma distributions. These classes cover many interesting cases and accommodate both positively and negatively skewed composed error distributions. Bayesian methods allow for useful inference with carefully chosen prior distributions. We recommend a two-component mixture model where a sensible amount of structure is imposed through the prior to distinguish the components, which are given an economic interpretation. This setting allows for efficiencies to depend on firm characteristics, through the probability of belonging to either component. Issues of label-switching and separate identification of both the measurement and inefficiency errors are also examined. Inference methods through MCMC with partial centring are outlined and used to analyse both simulated and real data. An illustration using hospital cost data is discussed in some detail.

Analysis of Hybrid Life-tests in Presence of Competing Risks

The mixture of type-I and type-II censoring schemes, called the hybrid censoring scheme is quite common in life-testing or reliability experiments. In this paper, we consider the competing risks model in presence of hybrid censored data. Under this set up, it is assumed that the item may fail due to various causes and the corresponding lifetime distributions are independent and exponentially distributed with different scale parameters. We obtain the maximum likelihood estimators of the mean life of the different causes and derive their exact distributions. Using the exact distributions, all the moments can be obtained. Asymptotic confidence intervals and two bootstrap confidence intervals are also proposed. Bayes estimates and credible intervals of the unknown parameters are obtained under the assumptions of independent inverted gamma priors of the mean life of the different causes. Different methods have been compared using Monte Carlo simulations. Onereal data set has been analyzed for illustrative purposes. Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council.     

Three dual regression schemes for the Lukacs theorem

In the paper we study regressional versions of Lukacs' characterization of the gamma law. We consider constancy of regression instead of Lukacs' independence condition in three new schemes. Up to now the constancy of regressions of U=X/(X + Y) given V=X + Y for independent X and Y has been considered in the literature. Here we are concerned with constancy of regressions for X and Y while independence of U and V is assumed instead.     

Comparisons of Bayesian estimation procedures for two-way contingency tables without interaction

Bayesian and empirical Bayesian estimation methods are reviewed and proposed for the row and column parameters in two-way Contingency tables without interaction. Rasch's multiplicative Poisson model for misreadings is discussed in an example. The case is treated where assumptions of exchangeability are reasonable a priori for the unknown parameters. Two different types of prior distributions are compared, It appears that gamma priors yield more tractable results than lognormal priors.     

Approximation for the cumulative and inverse gamma distribution

The gamma distribution function can be expressed in terms of the Normal distribution and density functions with sufficient accuracy for most practical purposes.
The distribution function for the density x^Λ-1e^-x/μ/μ^ΛΓ(A) on 0 -R(Λ){(1 + 1/1 2Λ) φ(z) + 11 -z/4Λ^1/2+2(z²+ 2)/45Λ] φ(z) /3 Λ^1/2} where φ(z)≅1/[1 +_e^{-2z(√2/π+z2 /28)}] and φ(z) = e^{-z2 /2}/√2π are the Normal distribution and density functions, y is the appropriate root of y-y²/6+y³/36-y⁴/270= In (x/Λμ), z= Λ^1/2 y, and R( Λ) is the remainder term in Stirling's approximation for In Γ(Λ).     

Waiting time probabilities in the M/G/1 + M queue

We consider an M/G/1 queueing system where the customers may leave the queue if their services do not commence before an exponentially distributed random time. The (conditional) offered waiting time distribution is approximated by a gamma distribution via matching the first and second moments of the actual waiting time. A simulation study is conducted to assess the accuracy of the approximation and it reveals that the approximation performs satisfactorily under general conditions on service time distributions.     

A characterization of the gamma process by conditional moments

Summary The gamma process is determined by the form of conditional expectations and conditional variances. Also a new characterization of the gamma law is obtained and then applied to characterize the gamma process among the processes with independent increments.     

Compound gamma,beta and F distributions

Summary In this paper a compound gamma distribution has been derived by compounding a gamma distribution with another gamma distribution. The resulting compound gamma distribution has been reduced to the Beta distributions of the first kind and the second kind and to theF distribution by suitable transformations. This includes theLomax distribution as a special case which enjoys a useful property. Moment estimators for two of its parameters are explicitly obtained, which tend to a bivariate normal distribution. The paper contains expressions for a bivariate probability density function, its conditional expectation, conditional variance and the product moment correlation coefficient. Finally, all the parameters of the compound gamma distribution are explicitly expressed in terms of the functions of the moments of the functions of random variables in two different ways. This note is based on a technical report prepared by the author while he was with the Procter and Gamble Company.     

A new bivariate distribution in natural exponential family

We propose a new bivariate distribution following a GLM form i.e., natural exponential family given the constantly correlated covariance matrix. The proposed distribution can represent an independent bivariate gamma distribution as a special case. In order to derive the distribution we utilize an integrating factor method to satisfy the integrability condition of the quasi-score function. The derived distribution becomes a mixture of discrete and absolute continuous distributions. The proposal of our new bivariate distribution will make it possible to develop some bivariate generalized linear models. Further the discrete correlated bivariate distribution will also arise from an independent bivariate Poisson mass function by compounding our proposed distribution (Iwasaki and Tsubaki, 2002).Received March 2003     

Empirical evidence on the sign of the slope of the hazard rate from unemployment from a fixed effects model

Empirical evidence on the sign of the slope of the hazard rate from unemployment is obtained from a fixed effects model based on the gamma distribution for unemployment duration. The data used are pairs of unemployment spells for adult males in the control and experimental groups of the Denver Income Maintenance Experiment. The sample selection issue involved in selecting the pair of spells used in the empirical work is discussed. The empirical results suggest that for these samples the hazard function is monotone decreasing and support the assumptions that the first two spells of unemployment are identically distributed and can be modelled using the gamma distribution.