Statistical Methods for Biodiversity Assessment

This thesis focuses on statistical methods for estimating the number of species which is a natural index for measuring biodiversity. Both parametric and nonparametric approaches are investigated for this problem. Species abundance models including homogeneous and heterogeneous model are explored for species richness estimation. Two new improvements to the Chao estimator are developed using the Good-Turing coverage formula.

Although the homogeneous abundance model is the simplest model, the species are collected with different probability in practice. This leads to overdispersed data, zero inflation and a heavy tail. The Poisson-Tweedie distribution, a mixed-Poisson distribution including many special cases such as the negative-binomial distribution, Poisson, Poisson inverse Gaussian, P\'{o}lya-Aeppli and so on, is explored for estimating the number of species. The weighted linear regression estimator based on the ratio of successive frequencies is applied \add{to data generated from} the Poisson-Tweedie distribution. There may be a problem with sparse data which provides zero frequencies for species seen $i$ times. This leads to the weighted linear regression not working. Then, a smoothing technique is considered for improving the performance of the weighted linear regression estimator. Both simulated data and some real data sets are used to study the performance of parametric and nonparametric estimators in this thesis.

Finally, the distribution of the number distinct species found in a sample is hard to compute. Many approximations including the Poisson, normal, COM-Poisson Binomial, Altham's multiplicative and additive-binomial and P\'{o}lya distribution are used for approximating the distribution of distinct species. Under various abundance models, Altham's multiplicative-binomial approximation performs well. Building on other recent work, the maximum likelihood and the maximum pseudo-likelihood estimators are applied with Altham's multiplicative-binomial approximation and compared with other estimators.