Inference for Sichel's Compound Poisson Distribution

Abstract

In a long-tailed compound Poisson distribution introduced by Sichel, the probabilities can only readily be calculated if one of the parameters λ has half-integer values, that is, λ = m + 1/2, m integer. The maximum likelihood estimation of this parameter is compared with an approximate method derived from a grid of half-integer values. It is found that the approximate method, which avoids excessive special computation, yields inferences indistinguishable from those based on maximum likelihood and on the score statistic for testing values of λ. Thus a major numerical impediment to use of Sichel's distribution is removed.

Key Words:

• Approximate likelihood
• Compound Poisson distribution
• Modified Bessel functions of the second and third kinds
• Score statistic