Some distribution properties of the sample species-diversity indices and their applications

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 the population diversity has received little attention. In this paper we show that if the diversities of the communities can be partially ordered through majorization as proposed by Solomon [1979, in Ecological Diversity in Theory and Practice, J. F. Grassle, G. P. Patil, W. K. Smith and C. Taillie (eds), 29-35, Fairland, Maryland: International Co-operative Publishing House], and if the sample sizes remain the same, then the sample diversity indices can be stochastically ordered when the samples are selected at random from the communities either with or without replacement. We also show that, when the sample size becomes large, the sample diversity indices are asymptotically normally distributed. These results can be applied for comparing the diversities of the communities on the basis of the samples, and they yield certain desirable monotonicity properties.