# Second-Order Chebyshev–Edgeworth and Cornish–Fisher Expansions for Distributions of Statistics Constructed from Samples with Random Sizes

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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, we derive second-order Chebyshev–Edgeworth and Cornish–Fisher expansions based of Student’s t- and Laplace distributions and their quantiles for samples with random size of a special kind. This derivation uses a general transfer theorem, which allows us to construct asymptotic expansions for distributions of randomly normalized statistics from the distributions of the considered nonrandomly 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 quantiles of the loss function, which is connected with description of investment portfolio of financial instruments.

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