Distributional Properties of Generalized Diversity Index Based on two-key Species Abundance Models

In the area of ecological research, the study of species diversity of a community or population seems to have been fully developed. However, the problem of how the distributions and expectations of the sample diversity indices are affected by population diversity has received little attention. This paper is concerned with methods of moments of improved generalized diversity index, N(α,β) due to Shamia's (2013) proposal which includes special cases. This improved index is a further generalization due to Good as described by Backowski et al., (1997). In this article, the first four central moments of N(α,β) are derived for both general species relative abundance models: the Broken-Stick model and Sequential-Breakage model within a range of (α,β) considered. This allows the determination of the skewness and kurtosis of N(α,β) and thus gives information about the behaviour of the distribution of the improved index. The results are applied for comparing the diversities of the communities based on the samples n>s, and they yielded certain desirable monotonicity properties for large samples. It can be also shown that such indices are asymptotically normally distributed.