Geodesic
Asymptotic expansions of functions of statistics
Applications of Mathematics, Tome 21 (1976) no. 6, pp. 444-456

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

DOI MR   Zbl

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.
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.
DOI : 10.21136/AM.1976.103669
Classification : 62E20, 62F99, 62N05 
Hurt, Jan. Asymptotic expansions of functions of statistics. Applications of Mathematics, Tome 21 (1976) no. 6, pp. 444-456. doi: 10.21136/AM.1976.103669
@article{10_21136_AM_1976_103669,
     author = {Hurt, Jan},
     title = {Asymptotic expansions of functions of statistics},
     journal = {Applications of Mathematics},
     pages = {444--456},
     year = {1976},
     volume = {21},
     number = {6},
     doi = {10.21136/AM.1976.103669},
     mrnumber = {0418309},
     zbl = {0354.62034},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1976.103669/}
}
TY  - JOUR
AU  - Hurt, Jan
TI  - Asymptotic expansions of functions of statistics
JO  - Applications of Mathematics
PY  - 1976
SP  - 444
EP  - 456
VL  - 21
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1976.103669/
DO  - 10.21136/AM.1976.103669
LA  - en
ID  - 10_21136_AM_1976_103669
ER  - 
%0 Journal Article
%A Hurt, Jan
%T Asymptotic expansions of functions of statistics
%J Applications of Mathematics
%D 1976
%P 444-456
%V 21
%N 6
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1976.103669/
%R 10.21136/AM.1976.103669
%G en
%F 10_21136_AM_1976_103669

[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

[2] Cramér H.: Mathematical methods of statistics. Princeton Univ. Press, Princeton 1946. | MR

[3] Hodges, Jr. J. L., Lehmann E. L.: Deficiency. Ann. Math. Statist. 41 (1970), 783-801. | DOI | MR | Zbl

[4] Jarník V.: Diferenciální počet II. NČSAV, Praha 1956.

[5] Lomnicki Z. A., Zaremba S. K.: On the estimation of autocorrelation in time series. Ann. Math. Statist. 28 (1957), 140-158. | DOI | MR | Zbl

[6] Rao C. R.: Linear statistical inference and its applications. 2nd ed., Wiley, New York 1973. | MR | Zbl

[7] Riordan J.: Combinatorial identities. Wiley, New York 1968. | MR | Zbl

[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. | DOI | MR | Zbl

Cité par Sources :