Generalized Cornish–Fisher expansions for distributions of statistics based on samples of random size

Generalized Cornish–Fisher expansions are constructed for quantiles of sample mean for a sample of random size in terms of quantiles for the Laplace distribution and Student's $t$-test. In recent years, the interest in Cornish–Fisher expansions grew significantly in the context of research on risk management. The widespread risk measure Value at Risk, or VaR, is, in fact, the quantile of the loss function. The authors use the general transfer theorem that makes it possible to obtain asymptotic expansions for the distribution functions of statistics based on samples of random size by asymptotic expansions for the distribution function of the random sample size and asymptotic expansions for the distribution functions of statistics based on nonrandom samples. A computational experiment was performed to illustrate the obtained Cornish–Fisher expansions.