Some Simple Method of Estimation for the Parameters of the Discrete Stable Distribution with the Probability Generating Function

Abstract

In this article, we develop a method to estimate the two parameters of the discrete stable distribution. By minimizing the quadratic distance between transforms of the empirical and theoretical probability generating functions, we obtain estimators simple to calculate, asymptotically unbiased, and normally distributed. We also derive the expression for their variance-covariance matrix. We simulate several samples of discrete stable distributed datasets with different parameters, to analyze the effect of tuncation on the right tail of the distribution.

Generalized definitions of discrete stability

“…The non-existence of a closed form formula of the probability mass function and non-existence of moments implies that the classical parameter estimation procedures such as maximum likelihood and method of moments cannot be applied. Marcheselli et al (2008) and Doray et al (2009) suggested some methods of parameter estimation of the discrete stable family based on the empirical characteristic function or on the empirical probability generating function.…”

Simulated Minimum Hellinger Distance Inference Methods for Count Data

“…If we can simulate one observation from the mixing distribution of Y which gives a realized value t and if it is not difficult to draw one observation from the distribution with LT ( ) t s κ then combining these two steps, we would be able to obtain one observation from the new distribution created by the PM operator. Consequently, simulated methods of inferences offer alternative methods to inferences methods based on matching selected points of the empirical pgf with its model counterpart or other related methods, see Doray et al [9] for regression methods using selected points of the pgfs. For these methods there is some arbitrariness on the choice of points which make it difficult to apply.…”