| Title:
|
The linear model with variance-covariance components and jackknife estimation (English) |
| Author:
|
Kudeláš, Jaromír |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
39 |
| Issue:
|
2 |
| Year:
|
1994 |
| Pages:
|
111-125 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\theta^*$ be a biased estimate of the parameter $\vartheta$ based on all observations $x_1$, $\dots$, $x_n$ and let $\theta_{-i}^*$ ($i=1,2,\dots,n$) be the same estimate of the parameter $\vartheta$ obtained after deletion of the $i$-th observation. If the expectation of the estimators $\theta^*$ and $\theta_{-i}^*$ are expressed as $$ \align \mathrm{E}(\theta^*)&=\vartheta+a(n)b(\vartheta) \\ \mathrm{E}(\theta_{-i}^*)&=\vartheta+a(n-1)b(\vartheta)\qquad i=1,2,\dots,n, \endalign $$ where $a(n)$ is a known sequence of real numbers and $b(\vartheta)$ is a function of $\vartheta$, then this system of equations can be regarded as a linear model. The least squares method gives the generalized jackknife estimator. Using this method, it is possible to obtain the unbiased estimator of the parameter $\vartheta$. (English) |
| Keyword:
|
Jackknife estimator |
| Keyword:
|
least squares estimator |
| Keyword:
|
linear model |
| Keyword:
|
estimator of variance-covariance components |
| Keyword:
|
consistency |
| MSC:
|
62F10 |
| MSC:
|
62J10 |
| idZBL:
|
Zbl 0797.62057 |
| idMR:
|
MR1258187 |
| DOI:
|
10.21136/AM.1994.134248 |
| . |
| Date available:
|
2009-09-22T17:43:18Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134248 |
| . |
| Reference:
|
[1] W. Kruskal: When are Gauss-Markov and least square estimators identical? A coordinate free approach.Ann. Math. Statistics 39 (1968), 70–75. MR 0222998, 10.1214/aoms/1177698505 |
| Reference:
|
[2] G. H. Lavergne, J. R. Mathieu: The jackknife method and the Gauss-Markov estimation.Probability and Math. Statistics 8 (1987), 111–116. MR 0928124 |
| Reference:
|
[3] R. C. Miller, jr.: An unbalanced jackknife.Ann. Math. Statistics 2 (1974), 880–891. Zbl 0289.62042, MR 0356353, 10.1214/aos/1176342811 |
| Reference:
|
[4] M. H. Quenouille: Approximate test of correlation in time-series.J. Roy. Statist. Soc. Ser. B 11 (1949), 68–84. MR 0032176 |
| Reference:
|
[5] M. H. Quenouille: Notes on bias in estimation.Biometrika 43 (1956), 353–360. Zbl 0074.14003, MR 0081040, 10.1093/biomet/43.3-4.353 |
| Reference:
|
[6] C. R. Rao: Linear statistical inference and its applications.J. Wiley, 1973. Zbl 0256.62002, MR 0346957 |
| Reference:
|
[7] F. Štulajter: Consistency of linear and quadratic least squares estimators in regression models with covariance stationary errors.Applications of Mathematics 36(2) (1991), 149–155. MR 1097699 |
| Reference:
|
[8] J. W. Tukey: Variances of variance components: II. The unbalanced single classification.Ann. Math. Statist. 28 (1957), 43–56. MR 0084974, 10.1214/aoms/1177707036 |
| . |
