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

  • Authors:

  • Carlos A. Coelho

  • Affiliations:

  • Mathematics Department, Institute of Agronomy, Lisbon University of Technology, 1349-017 Lisboa, Portugal

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2004

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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 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.