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

Abstract

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 L 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. r

On the distribution of linear combinations of independent Gumbel random variables

“…approximates the exact characteristic function Φ W ; the distribution of the random variable W is said to be a near-exact distribution of W (Coelho 2004). Based on the characterization of the exact distribution of W in Corollary 1, we take…”

“…Parenthetically, we further note that to be ensured that we accurately approximate the tail of the exact distribution, we need to keep increasing the precision parameter γ as we move towards higher quantiles; further details on the measure ∆ can be found in Grilo and Coelho (2007), Marques and Coelho (2008), and Coelho andMarques (2010, 2012). …”

“…The Generalized Near-Integer Gamma (GNIG) distribution of depth ( + 1) ∈ N, introduced by Coelho (2004), is defined as the distribution of Y = X + j=1 X j , where X is independent of j=1 X j , and X ∼ Gamma(ρ, l), with ρ ∈ R * + \N. We denote this by Y ∼ GNIG(r , λ , + 1), where r = (r, ρ) and λ = (λ, l), and the corresponding density and distribution functions are…”

“…Our near-exact distributions have links with phase-type approximations (Aldous and Shepp 1987;O'Cinneide 1990) and, as we discuss below, their accuracy can be controlled effectively through a precision parameter. Our first near-exact distribution is based on the Generalized Integer Gamma (GIG) and Generalized NearInteger Gamma (GNIG) distributions, which have a wealth of applications in multivariate analysis (Marques and Coelho 2008;Coelho andMarques 2010, 2012;Marques et al 2011;Coelho et al 2013). The GIG distribution corresponds to the distribution of the sum of independent Gamma random variables with integer shape parameters (Amari and Misra 1997;Coelho 1998), while the GNIG distribution corresponds to the distribution of the sum of a GIG random variable with an independent Gamma random variable with a non-integer shape parameter (Coelho 2004); further details on the GIG and GNIG distributions are given in Appendix A.…”

“…Our first near-exact distribution is based on the Generalized Integer Gamma (GIG) and Generalized NearInteger Gamma (GNIG) distributions, which have a wealth of applications in multivariate analysis (Marques and Coelho 2008;Coelho andMarques 2010, 2012;Marques et al 2011;Coelho et al 2013). The GIG distribution corresponds to the distribution of the sum of independent Gamma random variables with integer shape parameters (Amari and Misra 1997;Coelho 1998), while the GNIG distribution corresponds to the distribution of the sum of a GIG random variable with an independent Gamma random variable with a non-integer shape parameter (Coelho 2004); further details on the GIG and GNIG distributions are given in Appendix A. We show that the exact distribution of a positive linear combination of independent Gumbel random variables can be decomposed as the sum of two independent random variables: the first corresponding to a linear combination of independent logarithmized Gamma random variables, and the second to a shifted Generalized Integer Gamma (SGIG) random variable.…”