Asymptotic approximation of statistics based on the sample of negative binomial distribution

Abstract

The paper proves the theorem on the limit distribution of negative binomial random sums. The results of modeling negative binomial random sums using the R programming language are demonstrated. As is well-known, in classical problems of mathematical statistics sample volume is a known parameter. In these problems, the statistics of random sums usually converges to the distribution of Gauss. Additionally, it can be stated that mathematical expectations of independent random terms do not affect the structural features of the limit distribution itself, except mathematical expectations of the limit distribution. In case of random sums, where the sample volume is also a statistic, the limit distribution of random sums with the sums of terms with zero mathematical expectation from the limit distribution of random sums with the sums of terms with non-zero mathematical expectation may be different. The formulated theorem shows that if the accumulation of terms in random sums has the character of a negative binomial distribution, the structure of the limit  distribution depends on the symmetry of the random terms. To estimate the convergence rate, Lemma proved that negative binomial random sums are mixed Poisson random sums with a mixing gamma distribution. Therefore, accuracy of asymptotic models of negative binomial random sums can be estimated by estimating the convergence rate of mixed Poisson sums.