# Article

 Title: Asymptotic expansions of functions of statistics (English) Author: Hurt, Jan Language: English Journal: Aplikace matematiky ISSN: 0373-6725 Volume: 21 Issue: 6 Year: 1976 Pages: 444-456 Summary lang: English Summary lang: Czech Summary lang: Russian . Category: math . Summary: Let $\{T_n\}$ be a sequence of statistics such that $E\left|T_n-0\right|^{2(q+1)}=O(n^{-(q+1)})$. Let $g=g(t,n)$ be a real function defined on $R\times N$. In the paper it is shown that under some assumptions concerning $g$, the expectation $Eg(T_n,n)$ (the variance var $g(T_n,n)$) may be expressed in terms of the derivatives of $g$ and the moments $E(T_n-0)^j, j=1, \ldots, q(j=1,\ldots, 2q)$, the remainder term being $O(n^{-(q+1/2}) (O(n^{-(q+2/2)}))$. Similar results for vector $T'_n$s are also obtained. Applications in reliability theory are given. () MSC: 62E20 MSC: 62F99 MSC: 62N05 idZBL: Zbl 0354.62034 idMR: MR0418309 DOI: 10.21136/AM.1976.103669 . Date available: 2008-05-20T18:06:10Z Last updated: 2020-07-28 Stable URL: http://hdl.handle.net/10338.dmlcz/103669 . Reference: [1] Ångström K. H.: An asymptotic expansion of bias in a non-linear function of a set of unbiased characteristics from a finite sample.Skandinavisk Aktuarietidskrift 1958, 40-46, MR 0107321 Reference: [2] Cramér H.: Mathematical methods of statistics.Princeton Univ. Press, Princeton 1946. MR 0016588 Reference: [3] Hodges, Jr. J. L., Lehmann E. L.: Deficiency.Ann. Math. Statist. 41 (1970), 783-801. Zbl 0225.62063, MR 0272092, 10.1214/aoms/1177696959 Reference: [4] Jarník V.: Diferenciální počet II.NČSAV, Praha 1956. Reference: [5] Lomnicki Z. A., Zaremba S. K.: On the estimation of autocorrelation in time series.Ann. Math. Statist. 28 (1957), 140-158. Zbl 0081.14101, MR 0088855, 10.1214/aoms/1177707042 Reference: [6] Rao C. R.: Linear statistical inference and its applications.2nd ed., Wiley, New York 1973. Zbl 0256.62002, MR 0346957 Reference: [7] Riordan J.: Combinatorial identities.Wiley, New York 1968. Zbl 0194.00502, MR 0231725 Reference: [8] Zacks S., Even M.: The efficiencies in small samples of the maximum likelihood and best unbiased estimators of reliability functions.Journ. Amer. Stat. Assoc. 61 (1966), 1033-1051. Zbl 0151.23203, MR 0207147, 10.1080/01621459.1966.10482193 .

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