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 Zap. Nauchn. Sem. POMI, 2017, Volume 466, Pages 167–207 (Mi znsl6549)

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

G. Christopha, M. M. Monakhovb, V. V. Ulyanovbc

a Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Germany
b Lomonosov Moscow State University, Moscow, Russia
c Russian State University for the Humanities, Moscow, Russia

Abstract: 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.

Key words and phrases: Chebyshev–Edgeworth expansions, Cornish–Fisher expansions, samples with random sizes, Laplace distribution, Student's $t$-distribution.

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Document Type: Article
UDC: 519.2

Citation: G. Christoph, M. M. Monakhov, V. V. Ulyanov, “Second order Chebyshev–Edgeworth and Cornish–Fisher expansions for distributions of statistics constructed from samples with random sizes”, Probability and statistics. Part 26, Zap. Nauchn. Sem. POMI, 466, POMI, St. Petersburg, 2017, 167–207

Citation in format AMSBIB
\Bibitem{ChrMonUly17} \by G.~Christoph, M.~M.~Monakhov, V.~V.~Ulyanov \paper Second order Chebyshev--Edgeworth and Cornish--Fisher expansions for distributions of statistics constructed from samples with random sizes \inbook Probability and statistics. Part~26 \serial Zap. Nauchn. Sem. POMI \yr 2017 \vol 466 \pages 167--207 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6549}