Estimation of Simpson's Diversity When Counts Follow a Poisson Distribution

Abstract

Most of the exact distributional properties available for Gini's (1912) index of diversity [more commonly referred to as Simpson's (1949) index] are derived assuming an underlying multinomial distribution of species frequencies, with simple random sampling. In one sense this is the least restrictive model in that it assumes only that individuals fall into s categories, or species, at random with probabilities Pi, i = 1, . .. , s. However, the model also assumes that the total sample size is fixed. Questions arise as to the validity of using these distributional properties in practical situations in which total sample size is random. The fixed sample size restriction may be relaxed by assuming that each species frequency follows an underlying Poisson distribution with mean Ai, i = 1, . . ., s. In fact, the joint distribution of independent species frequencies, conditioned on a fixed total sample size, is multinomial with pi = Au/Z Ai, i = 1, . . ., s. In this paper a method for computing the moments of the index under the Poisson assumptions is given and its asymptotic distribution is derived. The adequacy of the Pearson curve fit to the distribution of the index is examined with respect to confidence interval estimation. The results are used to assess the effect of incorrectly assuming a fixed sample size when in fact it is random. Quadrat sampling is introduced because it is more commonly used in practice, and it allows comparison of confidence interval estimation results with two other commonly used methods.