| Title:
|
The efficiency of estimates in stationary autoregressive series (English) |
| Author:
|
Anděl, Jiří |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
15 |
| Issue:
|
1 |
| Year:
|
1970 |
| Pages:
|
18-30 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $X_1,\ldots,X_N$ be a finite random sequence with the expectation $EX_t=\alpha\varphi_t(1\leq t\leq N)$ and with the regular covariance matrix $\bold G$. The matrix $\bold G$ and the values of $\varphi_t$ are supposed to be known; $\alpha$ is an unknown parameter. The least squares estimate $\hat{\alpha}$ and the best linear unbiased estimate (BLUE) $\tilde{\alpha}$ of the parameter $\alpha$ are mentioned. The efficiency $\ell_N=var\ \hat{\alpha}/var\ \tilde{\alpha}$ is derived. The exact value of $\ell_N$ is given for cases when $X_1,\ldots,X_N$ is a finite part of the autoregressive series of the first and of the second order and $\varphi_t\equiv 1$ and $\varphi_t =t\ (1 \leq t\leq N)$ and for the autoregressive series of the $n$-th order with $\varphi_t\equiv 1$. The efficiency and the asymptotic efficiency of the BLUE $\tilde{\alpha}$ in cases when $\bold G$ is not true covariance matrix is also considered. () |
| MSC:
|
62F10 |
| MSC:
|
62M10 |
| idZBL:
|
Zbl 0205.46204 |
| idMR:
|
MR0258216 |
| DOI:
|
10.21136/AM.1970.103264 |
| . |
| Date available:
|
2008-05-20T17:46:54Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103264 |
| . |
| Reference:
|
[1] U. Grenander M. Rosenblatt: Statistical analysis of stationary time series.New York, 1957. MR 0084975 |
| Reference:
|
[2] J. Hájek: On linear statistical problems in stochastic processes.Czech. Math. J. 12 (87), 1962, 404-444. MR 0152090 |
| Reference:
|
[3] E. J. Hannan: Анализ временных рядов.Москва 1964. Zbl 0116.11402 |
| Reference:
|
[4] T. A. Magness J. В. McGuire: Comparison of least squares and minimum variance estimates of regression parameters.Ann. Math. Stat. 33, 1962, 462-470. MR 0141201, 10.1214/aoms/1177704573 |
| Reference:
|
[5] G. S. Watson: Linear least squares regression.Ann. Math. Stat. 38, 1967, 1679-1699. Zbl 0155.26801, MR 0219206, 10.1214/aoms/1177698603 |
| . |
