On Number of Unobserved Positive Integers Less Than the Maximum Sample From Gem Distribution

For a sample from GEM distribution considered as a random discrete distribution of positive integers, there are probably unobserved less positive integers than the maximum sample. The number of these integers follows a mixed Poisson distribution. Here, we provide a simple proof of the convergence in distribution of this number. Similarly, its asymptotic distribution is a mixed Poisson. We derived the upper bounds for the total variation between the finite and asymptotic distributions. These distributions are illustrated by examples; the related total values are provided in table 1.