TY - JOUR

T1 - A Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

AU - Coelho, Carlos Manuel Agra

PY - 2012/1/1

Y1 - 2012/1/1

N2 - 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.

AB - 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.

KW - Characteristic function

KW - Small sample sizes

KW - Mixtures

KW - Independent random variables

KW - Sum of gamma random variables

KW - Proximity measures

U2 - 10.1080/03610926.2011.604146

DO - 10.1080/03610926.2011.604146

M3 - Article

SN - 0361-0926

VL - 41

SP - 2321

EP - 2341

JO - Communications In Statistics-Theory And Methods

JF - Communications In Statistics-Theory And Methods

IS - 13-14

ER -