Goodness of Fit for Discrete Random Variables Using the Conditional Density

Abstract

Sufficient statistics, Goodness of fit, Conditional distributions,

Model Selection Using Conditional Densities

“…For a proof of this result, see González-Barrios et al (2006). First we notice that the observations are conditionally exchangeable, that is, given any permutation of the set 1 2 n P X 1 = x 1 X n = x n T n = t = P X 1 = x 1 X n = x n T n = t Downloaded by [University of Louisville] at 05:38 26 December 2014 for every t ≥ 0 and any vector x 1 x n such that n i=1 x i = t. Since T n is a sufficient statistic of in the power series distributions.…”

“…It is easy to observe, see González-Barrios et al (2006), that the number of arrangements of t n is considerably smaller than the number of ways in which we can obtain t by adding n non negative integers. The idea of using arrangements facilitates finding the values of the density of X 1 X 2 X n given any fixed value t of T n .…”

“…According to the well-known theory of sufficient statistics, the values off n do not depend on . Also, as noticed in González-Barrios et al (2006), see Theorem 2.1 below, we can find an exact expression for these densities in the case of the power series distributions.…”

“…However, he did not fully developed the idea in specific cases. More recently, the idea of using conditional densities in the case of goodness-of-fit was developed in González- Barrios et al (2006).…”

A Simulation-Free Exact Conditional Goodness-of-Fit Test for The Binomial Distribution

“…N. Henze (1996), a general simulation-based method for creating conditional exact goodness-of-t tests by R. A. Lockhart, F. J. OReilly and M. A. Stephens (2007), an interesting exhaustive enumeration based conditional exact test for the class of power series distributions by J. M. González-Barrios, F. OReilly and R. Rueda (2006), exact conditional Kolmogorov-Smirnov test for Poisson distribution by J. Frey (2012).…”