Parameter Constraints for Finitized Distributions: Conjectures and Empirical Results

• Activity Number: 248
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Martin Levy*+ and James J. Cochran
• Companies: University of Cincinnati and The University of Alabama
• Keywords: power series family of distributions ; moments ; finitization ; variate simulation ; Taylor series expansion ; discrete probability distributions

Abstract

Moment preserving finitization transforms a discrete distribution into a distribution with smaller support of specified size, n; moments of an order n finitization coincide with those of the first n moments of the parent distribution. A property of a finitization via the Negative Taylor Series Method is that the maximum feasible parameter size (mfps), i.e., largest parameter yielding a proper distribution, is generally smaller than that of the parent distribution, and is a function of the finitization point. Determination of the mfps is essential for applications of finitizations, such as in fast variate generation. Based on extensive numerical studies we conjecture that: 1) The mfps for the finitized distribution is determined uniquely as a root obtained by setting the (n-1)st finitized probability to zero; 2) The mfps monotonically decreases as a function of increasing n except for the Poisson case where the mfps is constant and equal to 1 for all n, and 3) the Poisson is unique in that regard. The above hold for infinitely supported power series distributions, but other properties hold for the finitely supported ones, such as the binomial case.