Second order Chebyshev–Edgeworth and Cornish–Fisher expansions for distributions of statistics constructed from samples with random sizes

In practice, we often encounter situations where a sample size is not defined in advance and can be a random value. In the present paper second order Chebyshev–Edgeworth and Cornish–Fisher expansions based of Student's $t$- and Laplace distributions and their quantiles are derived for samples with random size of a special kind, using general transfer theorem, which allows to construct asymptotic expansions for distributions of randomly normalized statistics from the distributions of the considered non-randomly normalized statistics and of the random size of the underlying sample. Recently, interest in Cornish–Fisher expansions has increased because of study in risk management. Widespread risk measure Value at Risk (VaR) substantially depends on the quantiles of the loss function, which is connected with description of investment portfolio of financial instruments.