Parameter estimation

Ewens (1972, 1990) has shown that the number of types observed in a finite sample can serve as a sufficient statistics for the parameter θ. This can be seen

Guess and Ewens (1972) have explored estimating θ.

Ewens (1972) showed that with n = 250, E (K_n) ≈ 4.1 with variance about 2.9, while with θ = 0.4, E(K_n) ≈ 3.5 with variance about 2.4. These numbers are relevant to our example in Chapter 12. See his Tables I and II for more detail.

To interpret the parameter θ in the frequency spectrum, we introduce sequen­tial sampling into the relationship between the sample sizes and the number of different types of agents contained in the samples. Suppose we take two samples. The probability that they are of the same type is given by

Thus, the larger the value of θ, the smaller the probability that two samples are of the same type. In this sense, θ represents correlatedness of samples. For k > 1, we compute

This is the probability that first k samples are all of the same type. The next expression,

is the probability that the (k + 1)st draw is a new type; hence 1 - θ/(θ + k) = k/(θ + k) is the probability that the (k + 1)st draw is one of the types already drawn.