Bivariate Gamma Distribution Research Articles

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n1, …, nk)th degree, then all the mixed moments up to the order (n1, …, nk) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.

A Monte Carlo Investigation of the Robustness ofT2

Abstract The robustness of T2 for samples of size 5, 10 and 20 from several bivariate distributions is investigated. Samples are presented from bivariate normal, uniform, exponential, gamma, lognormal and double exponential distributions. Related observations on the one sample t and paired t are also given. Highly skewed distributions resulted in too many extreme values of T2 Other distributions gave conservative results. The use of the t-test and non-simultaneous techniques gave large overall levels of significance.

A Note on Positive Dependence and the Structure of Bivariate Distributions

Let X and Y be random variables whose marginal distributions $\mu ( \cdot )$ are identical. If their joint distribution function admits a series expansion in appropriate sets of functions with nonnegative coefficients, it is shown that \[ P\{ X \in A,Y \in A\} \geqq P\{ X \in A\} P\{ Y \in A\} \] for every $\mu $-measurable set A. This property is demonstrated for a variety of known two-dimensional distributions whose marginal measures are either discrete or absolutely continuous with respect to Lebesgue measure. Examples include bivariate Gaussian, exponential, gamma, chi-square, binomial, Poisson, negative binomial, and hypergeometric distributions.

Statistical Inference with Bivariate Gamma Distributions

Statistical Inference with Bivariate Gamma Distributions

The Canonical Correlation Coefficients of Bivariate Gamma Distributions

The Canonical Correlation Coefficients of Bivariate Gamma Distributions

Statistical inference with bivariate gamma distributions

In some experimental situations where the use of a covariate could increase the precision of the experiment the distributions of the test variate and the covariate are highly non-normal. Some of these cases can be analysed by using a bivariate Г distribution which is here defined. Methods of testing for a change in the scale parameter in one margin, using the other variate as a control, are described and the correponding extension to cross-over experiments briefly outlined.

Parameter estimation for a generalized gamma distribution : E. W. Stacy and G. A. Mihram, Technometrics, August (1965), pp. 349–358

A univariate gamma distribution is one of the most commonly adopted statistical distributions in hydrological frequency analysis. A bivariate gamma distribution constructed from specified gamma marginals may be useful for representing joint probabilistic properties of multivariate hydrological events such as floods and storms. This article presents a review of various bivariate gamma distribution models that are constructed from gamma marginals. Advantages and limitations of each of these models are pointed out. Applicability of a few bigamma distributions whose gamma marginal distributions have different scale and shape parameters is investigated. The dependence of these models is directly or indirectly measured via the Pearson's product-moment correlation coefficient. The scale and shape parameters of the models are estimated from their marginal distributions by the method of moments. Results indicate that these bigamma distribution models will be useful for describing the joint probability distribution of two correlated random variables with gamma marginals.

The bivariate gamma distribution and the random walk problem

The relation between the random walk problem in one dimension and the joint distribution of two correlated gamma variates of parameter ½ is firstly established. It is shown that the two variates which represent respectively the squares of distances from the origin (properly normalised) after P steps and P+Q (=N) steps in one dimension are a pair of correlated gamma variates with parameter ½. This is used as a heuristic principle to set up similar bivariate gamma distribution with parameters 1 by examining the corresponding problem of random walk in two dimensions. Finally, the results are extended to the case of generaln-dimensional random walk where it is shown to lead to the bivariate gamma distribution with parametern/2.