| 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 |
| . |
