Generalized Integer Gamma Distribution Research Articles

On the distribution of the likelihood ratio test of independence for random sample size — a computational approach

The test of independence of two groups of variables is addressed in the case where the sample size N is considered randomly distributed. This assumption may lead to a more realist testing procedure since in many situations the sample size is not known in advance. Three sample schemes are considered where N may have a Poisson, Binomial or Hypergeometric distribution. For the case of two groups with p1 and p2 variables, it is shown that when either p1 or p2 (or both) are even the exact distribution corresponds to a finite or an infinite mixture of Exponentiated Generalized Integer Gamma distributions. In these cases a computational module is made available for the cumulative distribution function of the test statistic. When both p1 and p2 are odd, the exact distribution of the test statistic may be represented as a finite or an infinite mixture of products of independent Beta random variables whose density and cumulative distribution functions do not have a manageable closed form. Therefore, a computational approach for the evaluation of the cumulative distribution function is provided based on a numerical inversion formula originally developed for Laplace transforms. When the exact distribution is represented through infinite mixtures, an upper bound for the error of truncation of the cumulative distribution function is provided. Numerical studies are developed in order to analyze the precision of the results and the accuracy of the upper bounds proposed. A simulation study is provided in order to assess the power of the test when the sample size N is considered randomly distributed. The results are compared with the ones obtained for the fixed sample size case.

Approximations for the Likelihood Ratio Statistic for Hypothesis Testing Between Two Beta Distributions

In this paper, the likelihood ratio to test between two Beta distributions is addressed. The exact distribution of the likelihood ratio statistic, for simple hypotheses, is obtained in terms of Gamma or Generalized Integer Gamma distributions, when the first or the second of the two parameters of the Beta distributions are equal and integers. In the remaining cases addressed, near-exact or asymptotic approximations are developed for the likelihood ratio statistic. Both the exact, asymptotic or near-exact representations are obtained using a logarithm transformation of the likelihood ratio statistic and by working with the corresponding characteristic function. The numerical studies illustrate the precision of the approximations developed. Simulations are developed to analyse the power and the reproducibility probability of the tests.

On the linear combination of independent logistic random variables

In this work the exact distribution of the linear combination of p independent logistic random variables is studied. It is shown that the exact distribution may be represented as a shifted infinite sum of independent random variables distributed as the difference of two independent Generalized Integer Gamma distributions. In addition, two near-exact approximations are developed for this distribution. Numerical studies are conducted to access the degree of precision and also the computational performance of these approximations. The developed methodology is used to derive near-exact approximations for the linear combination of independent generalized logistic random variables.

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.

On the distribution of linear combinations of independent Gumbel random variables

The distribution of linear combinations of independent Gumbel random variables is of great interest for modeling risk and extremes in the most different areas of application. In this paper we develop near-exact approximations for the distribution of linear combination of independent Gumbel random variables based on a shifted generalized near-integer gamma distribution and on the distribution of the difference of two independent generalized integer gamma distributions. These near-exact distributions are computationally appealing and numerical studies confirm their accuracy, as assessed by a proximity measure used in related studies. We illustrate the proposed approximations on applied problems in networks engineering, computational biology, and flood risk management.

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 Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.

The block sphericity test—exact and near‐exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its hth null moment, and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a generalized integer gamma distribution with the sum of a given number of independent logbeta distributions, taking the form of a single generalized integer gamma distribution when each set of variables has two variables. The development of near‐exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent‐induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near‐exact distributions proposed and to compare them with an asymptotic approximation on the basis of Box's method. Copyright © 2011 John Wiley & Sons, Ltd.

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.

The Exact and Near-Exact Distributions for the Wilks Lambda Statistic Used in the Test of Independence of Two Sets of Variables

Synoptic We develop the exact distribution of the Wilks Lambda statistic to test the independence of two sets of variables, both with an odd number of variables, under the form of an infinite mixture of Generalized Integer Gamma distributions. Based on truncations of the exact characteristic function, for the product of independent Beta random variables, we obtain near-exact distributions for such product and then by direct application of these results, and once again based on truncations, we develop near-exact distributions for the Wilks Lambda statistic. These near-exact distributions are finite mixtures of Generalized Integer Gamma and Generalized Near-Integer Gamma distributions. By construction, the two first moments of these approximations are equal to the exact moments. These distributions are manageable and relatively easy to implement computationally, allowing for the computation of near-exact quantiles which may indeed be regarded as virtually exact, given the good convergence properties of the series involved, mainly when the difference between the sample size and the overall number of variables involved is rather small. We assess the proximity between these near-exact di stributions and the exact distribution by using two measures based on the Berry-Esseen bounds.

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.

A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables: applications

In this paper we show first how the distribution of the logarithm of a random variable with a Beta distribution may be expressed either as a mixture of Gamma distributions or as a mixture of Generalized Integer Gamma (GIG) distributions and then how the exact distribution of the product of an odd number of independent Beta random variables whose first parameter evolves by 1/2 and whose second parameter is the half of an odd integer may be expressed as a mixture of GIG distributions. Some particularities of these mixtures are analysed. The results are then used to obtain the exact distribution of the logarithm of the Wilks Λ statistic to test the independence of two sets of variables, both with an odd number of variables, and the exact distribution of the logarithm of the generalized Wilks Λ statistic to test the independence of several sets of variables, in the case where two or three of them have an odd number of variables. A discussion of relative advantages and disadvantages of the use of the exact versus near-exact distributions is carried out.

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.

A Generalized Integer Gamma Distribution as an Asymptotic Replacement for a Logbeta Distribution & Applications

SYNOPTIC ABSTRACTThe use of the concept of near-exact distribution enables us to obtain distributions which are very close to the exact distribution but which are far more manageable. Indeed such near-exact distributions are usually far more accurate than any of the usual asymptotic distributions, being thus much useful in the computation of quantiles and as a tool in testing procedures. The use of the concept of near-exact distributions may be the solution to several approximation problems to distributions. In this paper it is shown how we may asymptotically replace the distribution of a random variable whose exponential has a Beta distribution by a particular Generalized Integer (GI) Gamma distribution. This result is then applied in obtaining the near-exact distribution of the product of an odd number of particular independent Beta random variables. A random variable with a distribution of this type is the generalized Wilks Lambda statistic, when two or more sets with an odd number of variables are considered. From the results obtained a near-exact distribution for the generalized Wilks Lambda is obtained under the form of a particular GI Gamma distribution.