Estimating distributions from samples with random size

The article is concerned with the estimating problem of a distribution function and the limiting behavior of the distance between the empirical and theoretical laws, namely, integrated square errors and Smirnov and Kolmogorov statistics by the samples with random size. We suppose that this random size and the initial sample are independent random variables and this random variable has the generalized negative binomial distribution. We find limiting distributions for integrated square errors of kernel distribution function estimators by the samples with random size. It is shown that for samples with random size the limiting distribution of the Smirnov and Kolmogorov statistics has more heavier tails than the Weibull and Kolmogorov distribution function for samples with the fixed size. We propose an asymptotic expansion approach to naturally balance the asymptotic distribution and random sample size. The problem of sequential estimation of the shift parameter of the uniform distribution is considered. The negative binomial distribution (samples size $\nu $) arises naturally here from a statistical experiment of performing a series of independent trials.