Informatics and Applications
2021, Volume 15, Issue 2, pp 7281
CHEBYSHEVEDGEWORTH EXPANSIONS FOR DISTRIBUTIONS OF GENERALISED HOTELLINGTYPE STATISTICS BASED ON RANDOM SIZE SAMPLES
Abstract
The general transfer theorem for the distribution function of asymptotically normal statistics was generalized on the Hotellingtype statistics case and analog of general transfer theorem for the distribution function of Hotellingtype statistics with random size was proved. It allowed to obtain the ChebyshevEdgeworth expansion for initial Hotellingtype statistics. The explicit form of the ChebyshevEdgeworth expansion was obtained for the case when the random sample size distribution is the negative binomial distribution shifted by 1. The limit distribution for this case was Fdistribution. The CornishFisher expansion was obtained for the special case of parameter of random sample size. The computational experiment was conducted and graphs were plotted for ChebyshevEdgeworth expansion illustration.
[+] References (9)
 Bening, V. E., N. K. Galieva, and V. Yu. Korolev. 2013. Asimptoticheskie razlozheniya dlya funktsiy raspredeleniya statistik, postroennykh po vyborkam sluchaynogo ob"ema
[Asymptotic expansions for the distribution functions of statistics constructed from samples with random sizes]. Informatika i ee Primeneniya  Inform. Appl. 7(2):7583.
 Bening, V. E., N. K. Galieva, and V. Yu. Korolev. 2012. Otsenki skorosti skhodimosti dlya funktsiy raspredeleniya asimptoticheski normal'nykh statistik, osnovannykh na vyborkakh sluchaynogo ob"ema [On rate of convergence in distribution of asymptotically normal statistics based on samples of random size]. Vestnik Tverskogo gos. unta. Ser. Prikladnaya matematika [Bull. of the Tverskoy State University. Ser. Appl. Math.] 17:5365.
 Markov, A. S., M. M. Monakhov, and V.V. Ulyanov. 2016. Razlozheniya tipa KornishaFishera dlya raspredeleniy statistik, postroennykh po vyborkam sluchaynogo razmera [Generalized CornishFisher expansions for distributions of statistics based on samples of random size]. Informatika
i ee Primeneniya  Inform. Appl. 10(2):8491.
 Christoph, G., M. M. Monakhov, and V.V. Ulyanov. 2017. Razlozheniya ChebyshevaEdzhvorta i KornishaFishera vtorogo poryadka dlya raspredeleniy statistik, postroyennykh po vyborkam sluchaynogo razmera [Second order ChebyshevEdgeworth and CornishFisher expansions for distributions of statistics constructed from samples with random sizes]. Zapiski nauchnykh seminarov
POMI [POMI Notes of Scientific Seminars] 466:167 207.
 Ulyanov, V. V, M. Aoshima, and Y. Fujikoshi. 2016. Non asymptotic results for CornishFisher expansions. J. Math. Sci. 218(3):363368.
 Fujikoshi, Y., V.V. Ulyanov, and R. Shimizu. 2005. L1norm error bounds for asymptotic expansions of multivariate scale mixtures and their applications to Hotelling's generalized T2. J. Multivariate Anal. 96:119.
 Fujikoshi, Y., V.V. Ulyanov, and R. Shimizu. 2010. Multivariate statistics: Highdimensional and largesample approximations. Wiley ser. in probability and statistics. Hoboken, NJ: Wiley. 568 p.
 Ulyanov, V.V. 2011. CornishFisher expansions. International encyclopedia of statistical science. Ed. M. Lovric. Berlin: Springer. 312315.
 Prudnikov, A. P, Yu. A. Brychkov, andO. I. Marichev. 1992. Integraly i ryady [Integrals and series]. Moscow: Nauka. Vol. 1.259 p.
[+] About this article
Title
CHEBYSHEVEDGEWORTH EXPANSIONS FOR DISTRIBUTIONS OF GENERALISED HOTELLINGTYPE STATISTICS BASED ON RANDOM SIZE SAMPLES
Journal
Informatics and Applications
2021, Volume 15, Issue 2, pp 7281
Cover Date
20210630
DOI
10.14357/19922264210211
Print ISSN
19922264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
generalised ChebyshevEdgeworth expansion; CornishFisher expansion; sample with random size; Fdisribution; Hotellingtype statstics
Authors
M. M. Monakhov
Author Affiliations
Moscow Center for Fundamental and Applied Mathematics, M. V. Lomonosov Moscow State University, 152 Leninskie Gory, GSP1, Moscow 119991, Russian Federation
