The moments and distribution for an estimate of the Shannon information measure and its application to ecology

Abstract

This dissertation deals primarily with the moments and distribution H̅ = - Σ n_i/N logn_i/N. Some techniques of obtaining multivariate moments, in particular multinomial moments, are given. I

The approach used In obtaining the moments of H̅ was through the probability generating function of the multinomial distribution. A series of rather simple mathematical operations will produce the E(H̅) as an Integral and Var(H̅) as a double Integral. These Integrale are evaluated exactly thus giving the exact mean and variance of H̅.

The mean and variance Is also given In series form. The series for the mean of H̅ appears to be divergent. Several charts are given which Indicate the percent error incurred when the series are used.

The combinatorial approach was used In finding the asymptotic distribution of H̅. The IBM 1130 and the IBM 360 model 65 were used to do this work. The results is that H̅ is asymptotically normal In the general case and H̅ is asymptotically chi-square In the equiprobable case.

Tables are given for the mean and variance of H̅ in the general case and In the equiprobable case.

Two methods are given for finding multivariate moments. The Q-Product Method due to Shenton, Bowman, and Reinfelds [36th Session of the International Statistical Institute,, 1967] and the Small Sample Method. There is every indication that these methods can be completely automated. A table of the first fourteen binomial moments is given and a table through order six of the multinomial moments is given.