Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

Gerd Christoph; Vladimir V. Ulyanov

doi:10.3390/math9070775

journal article Open Access Apr 02, 2021

Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

Gerd Christoph Vladimir V. Ulyanov

Mathematics Vol. 9 No. 7 pp. 775 · MDPI AG

View at Publisher Save 10.3390/math9070775

Abstract

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized T02 statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples.

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Details

Published: Apr 02, 2021
Vol/Issue: 9(7)
Pages: 775
License: View

Authors

G

Gerd Christoph

Department of Mathematics, Otto-von-Guericke University Magdeburg, 39016 Magdeburg, Germany

V

Vladimir V. Ulyanov

Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia

Funding

Russian Science Foundation Award: RSF grant No. 18-11-00132

Cite This Article

Gerd Christoph, Vladimir V. Ulyanov (2021). Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes. Mathematics, 9(7), 775. https://doi.org/10.3390/math9070775

Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

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