To the order of approximation used in Sect. 11.6, show that the maximumlikelihood estimate, \(\hat{\alpha}\left(T_{n}ight)\), of the species-diversity parameter as a function of the cumulative species count \(T_{n}\), defines a martingale.

Data From Section 11.6

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The Ewens sampling formula (Ewens, 1972), is the static probabilistic description of an exchangeable process, which can be viewed as a sequence Y, Y2,... of species or types. In its original genetic form, Pn,a is the probability distribution of the number N of distinct alleles and the multiplicity of each type occurring in a sample of n individuals (technically haplotypes) taken from an infinite population that evolves neutrally with mutation rate a. This combinatorial stochastic process is a thing of uncommon mathematical beauty; it occurs in a surprisingly wide range of mathematical and scientific applications from linguistic studies to genetics to ecology and probabilistic number theory (Crane, 2016; Pitman, 2006; Tavar, 2021). Only two distributional facts are relevant to the present story. The first fact is that the number of distinct types in a sample of n objects is equal in distribution to the sum of n independent Bernoulli variables N~ X1 + X2+ ... + Xn, where X, is Bernoulli with parameter a/(i - 1 + a). The Bernoulli parameter is the probability that the ith specimen is a new type that is different from previous specimens 1, 2, ..., i - 1. The sequential description leading to this conclusion is called the Chinese restaurant process: see Exercises 11.9-11. Thus, the expected value is + 1 + 2+ = a(n + a) (), = a log(n) + 0(1), E(N) = 1+ +. + n - 1 + a' where is the derivative of the log-gamma function. A similar calculation shows that the variance is less than the mean, but only slightly, var(N) = a log(n) + 0(1). In fact all cumulants of all orders differ from the mean by O(1). To this order of approximation, the species count is Poisson with parameter a log(n). The second fact is that Pn,a is a one-parameter exponential family with canonical parameter log a, and canonical sufficient statistic N. Accordingly, the maximum- likelihood estimate is unique for N