On the Stirling Distribution of the First Kind

Summary

A distribution of probabilities at positive integers t = n, n+1, …, proportional to F(t , n) θ^t/t!, 0 < θ < 1, is called the Stirling distribution of the first kind (SDFK) with parameters n and θ. The distribution is so named because it depends upon F(t , n), the Stirling numbers of the first kind. Patil and Wani (1965) have shown that the SDFK is the distribution of the sum of n independent and identically distributed random variables following the logarithmic series distribution. In this paper, some alternative derivations of the Patil and Wani’s result are given to further study the probabilistic structure of the SDFK. We show that with respect to the parameter n, the convolution of two independent SDFK’s is again a SDFK. Exact as well as approximate expressions of the distribution function of the SDFK are derived. Recurrence relations among the moments and the cumulants of the SDFK follow easily once we recognize that the distribution is a member of the class of power series distributions. Also, the minimum variance unbiased estimator of the probability function at any given point is derived. Several results of Patil and Wani (1965) follow as particular cases of ours when n = 1. If only θ is to be estimated, an easy graphical method is sketched to estimate it.