Generalized Near-Integer Gamma Research Articles

Improved near-exact distributions for the product of independent Generalized Gamma random variables

The Generalized Gamma distribution is an important distribution in Statistics since it has as particular cases many well known and important distributions and also due to its very interesting modeling properties, which makes it an attractive tool. The distribution of the product of independent Generalized Gamma distributions is investigated. Most of the results available for this distribution are based on Meijer-G or H functions which may still be very difficult to handle. Therefore, near-exact distributions which are based on the Generalized Near-Integer Gamma distribution and which have density and cumulative distribution functions easily implementable and computationally appealing are developed. Numerical studies with computationally intensive analyses are carried out to study the accuracy of these approximations in different scenarios. Also computational modules are provided for the implementation of these approximations. Finally, an example of application to quality control in microbiology is provided.

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.

Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions

In this paper the authors show how through the use of the characteristic function of the negative logarithm of the likelihood ratio test (l.r.t.) statistic to test circular symmetry it is possible to obtain highly manageable expressions for the exact distribution of such statistic, when the number of variables, $$p$$ , is odd, and highly manageable and accurate approximations for an even $$p$$ . For the case of an even $$p$$ , two kinds of near-exact distributions are developed for the l.r.t. statistic which correspond, for the logarithm of the l.r.t. statistic, to a Generalized Near-Integer Gamma distribution or finite mixtures of these distributions. Numerical studies conducted in order to assess the quality of these new approximations show their impressive performance, namely when compared with the only available asymptotic distribution in the literature.

A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis

In this paper we first show how the exact distributions of the most common likelihood ratio test (l.r.t.) statistics, that is, the ones used to test the independence of several sets of variables, the equality of several variance-covariance matrices, sphericity, and the equality of several mean vectors, may be expressed as the distribution of the product of independent Beta random variables or the product of a given number of independent random variables whose logarithm has a Gamma distribution times a given number of independent Beta random variables. Then, we show how near-exact distributions for the logarithms of these statistics may be expressed as Generalized Near-Integer Gamma distributions or mixtures of these distributions, whose rate parameters associated with the integer shape parameters, for samples of size n, all have the form (n−j)/n for j=2,…,p, where for three of the statistics, p is the number of variables involved, while for the other one, it is the sum of the number of variables involved and the number of mean vectors being tested. What is interesting is that the similarities exhibited by these statistics are even more striking in terms of near-exact distributions than in terms of exact distributions. Moreover all the l.r.t. statistics that may be built as products of these basic statistics also inherit a similar structure for their near-exact distributions. To illustrate this fact, an application is made to the l.r.t. statistic to test the equality of several multivariate Normal distributions.

Near-exact distributions for the independence and sphericity likelihood ratio test statistics

In this paper we show how, based on a decomposition of the likelihood ratio test for sphericity into two independent tests and a suitably developed decomposition of the characteristic function of the logarithm of the likelihood ratio test statistic to test independence in a set of variates, we may obtain extremely well-fitting near-exact distributions for both test statistics. Since both test statistics have the distribution of the product of independent Beta random variables, it is possible to obtain near-exact distributions for both statistics in the form of Generalized Near-Integer Gamma distributions or mixtures of these distributions. For the independence test statistic, numerical studies and comparisons with asymptotic distributions proposed by other authors show the extremely high accuracy of the near-exact distributions developed as approximations to the exact distribution. Concerning the sphericity test statistic, comparisons with formerly developed near-exact distributions show the advantages of these new near-exact distributions.

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.

Development and comparative study of two near-exact approximations to the distribution of the product of an odd number of independent Beta random variables

Using the concept of near-exact approximation to a distribution we developed two different near-exact approximations to the distribution of the product of an odd number of particular independent Beta random variables (r.v.'s). One of them is a particular generalized near-integer Gamma (GNIG) distribution and the other is a mixture of two GNIG distributions. These near-exact distributions are mostly adequate to be used as a basis for approximations of distributions of several statistics used in multivariate analysis. By factoring the characteristic function (c.f.) of the logarithm of the product of the Beta r.v.'s, and then replacing a suitably chosen factor of that c.f. by an adequate asymptotic result it is possible to obtain what we call a near-exact c.f., which gives rise to the near-exact approximation to the exact distribution. Depending on the asymptotic result used to replace the chosen parts of the c.f., one may obtain different near-exact approximations. Moments from the two near-exact approximations developed are compared with the exact ones. The two approximations are also compared with each other, namely in terms of moments and quantiles.

The generalized near-integer Gamma distribution: a basis for ‘near-exact’ approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables

In this paper the concept of near-exact approximation to a distribution is introduced. Based on this concept it is shown how a random variable whose exponential has a Beta distribution may be closely approximated by a sum of independent Gamma random variables, giving rise to the generalized near-integer (GNI) Gamma distribution. A particular near-exact approximation to the distribution of the logarithm of the product of an odd number of independent Beta random variables is shown to be a GNI Gamma distribution. As an application, a near-exact approximation to the distribution of the generalized Wilks Λ statistic is obtained for cases where two or more sets of variables have an odd number of variables. This near-exact approximation gives the exact distribution when there is at most one set with an odd number of variables. In the other cases a near-exact approximation to the distribution of the logarithm of the Wilks Lambda statistic is found to be either a particular generalized integer Gamma distribution or a particular GNI Gamma distribution.