Journal of Mathematical Sciences

, Volume 244, Issue 5, pp 811–839

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

• G. Christoph
• M. M. Monakhov
• V. V. Ulyanov
Article

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.

## References

1. 1.
B. V. Gnedenko, “An estimate of the distribution of the unknown parameters with a random number of independent observations,” Proc. Tbilisi Math. Inst., AN GSSR, 92, 146–150 (1989).
2. 2.
B. V. Gnedenko and V. Yu. Korolev, Random Summation, Limit Theorems, and Applications, CRC Press (1996).Google Scholar
3. 3.
V. E. Bening, N. K. Galieva, and V. Yu. Korolev, “Asymptotic expansions for the distribution functions of statistics constructed from samples with random sizes,” Informatics and Its Applications, IPI RAN, 7, 75–83 (2013).Google Scholar
4. 4.
V. E. Bening, N. K. Galieva, and V. Yu. Korolev, “On rate of convergence in distribution of asymptotically normal statistics based on samples of random size,” Ann. Math. Inform., 39, 17–28 (2012).
5. 5.
V. E. Bening and V. Yu. Korolev, “Some statistical problems related to the Laplace distribution,”Informatics and Its Applications, IPI RAN, 2, No. 2, 19–34 (2008).Google Scholar
6. 6.
V. E. Bening and V. Yu. Korolev, “On an application of the Student distribution in the theory of probability and mathematical statistics,” Teor. Veroyatn. Primen., 49, 417–435 (2004).
7. 7.
C. Schluter and M. Trede, “Weak convergence to the Student and Laplace distributions,” J. Appl. Probab., 53, 121–129 (2016).
8. 8.
E. A. Cornish and R. A. Fisher, “Moments and cumulants in the specification of distributions,” Rev. Inst. Internat. Statist., 4, 307–320 (1937).
9. 9.
G. W. Hill and A. W. Davis, “Generalized asymptotic expansions of Cornish–Fisher type,” Ann. Math. Statist., 39, 1264–1273 (1968).
10. 10.
V. V. Ulyanov, “Cornish–Fisher Expansions,” in: M. Louric (ed.), International Encyclopedia of Statistical Science, Springer, Berlin (2011), pp. 312–315.
11. 11.
S. Jaschke, “The Cornish–Fisher expansion in the context of delta-gamma-normal approximations,” J. Risk, 4, 33–52 (2002).
12. 12.
Y. Fujikoshi, V. V. Ulyanov, and R. Shimizu, Multivariate Statistics: High-Dimensional and Large-Sample Approximations, Wiley Series in Probability and Statistics, John Wiley & Sons, Hoboken, NJ (2010).Google Scholar
13. 13.
V. V. Ulyanov, M. Aoshima, and Y. Fujikoshi, “Non-asymptotic results for Cornish–Fisher expansions,” J. Math. Sci., 218, 84–91 (2016).
14. 14.
N. L. Johnson, A. W. Kemp, and S. Kotz, Univariate Discrete Distributions. 3rd ed., Wiley Series in Probability and Statistics, John Wiley & Sons, Hoboken, NJ (2005).Google Scholar
15. 15.
A. A. Markov, M. M. Monakhov, and V. V. Ulyanov, “Generalized Cornish–Fisher expansions for distributions of statistics based on samples of random size,” Informatics and Its Applications, IPI RAN, 10, No. 2, 84–91 (2016).Google Scholar
16. 16.
A. Buddana and T. J. Kozubowski, “Discrete Pareto distributions,” Econ. Qual. Control, 29, 143–156 (2014).
17. 17.
S. S. Wilks, “Recurrence of extreme observations,” J. Austral. Math. Soc., 1, 106–112 (1959).
18. 18.
O. O. Lyamin, “On the rate of convergence of the distributions of certain statistics to the Laplace distribution,” Vestnik Moskov. Univ., Ser. XV, 3, 30–37 (2010).
19. 19.
G. Christoph and W. Wolf, Convergence Theorems With a Stable Limit Law, SeriesMathematical Research, Akademie Verlag (1992).Google Scholar
20. 20.
G. Nemes, “Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal,” Proc. Roy. Soc. Edinburgh, 145, 571–596 (2015).
21. 21.
A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1. Elementary Functions, Gordon and Breach, New York (1992).Google Scholar
22. 22.
V. V. Petrov, Limit Theorems of Probability Theory. Sequences of Independent Random Variables, Clarendon Press, Oxford (1995).Google Scholar