| Title:
|
Characterization of the multivariate Gauss-Markoff model with singular covariance matrix and missing values (English) |
| Author:
|
Oktaba, Wiktor |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
1998 |
| Pages:
|
119-131 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The aim of this paper is to characterize the Multivariate Gauss-Markoff model $(MGM)$ as in () with singular covariance matrix and missing values. $MGMDP2$ model and completed $MGMDP2Q$ model are obtained by three transformations $D$, $P$ and $Q$ (cf. ()) of $MGM$. The unified theory of estimation (Rao, 1973) which is of interest with respect to $MGM$ has been used. The characterization is reached by estimation of parameters: scalar $\sigma ^{2}$ and linear combination $\lambda ^{\prime }\bar{B}$ ( $\bar{B}=vecB)$ as in (), (), () as well as by the model of the form () (cf. Th. ). Moreover, testing linear hypothesis in the available model $MGMDP2$ by test function $F$ as in () and () is considered. It is known (Oktaba 1992) that ten quantities in models $MGMDP2$ and $MGMDP2Q $ are identical (invariant). They permit to say that formulas for estimation and testing in both models are identical (Oktaba et al., 1988, Baksalary and Kala, 1981, Drygas, 1983). An algorithm and the $UMGMBO$ program for calculations concerning estimation and testing in $MGM$ have been presented by Oktaba and Osypiuk (1993). (English) |
| Keyword:
|
multivariate Gauss-Markoff model |
| Keyword:
|
missing value |
| Keyword:
|
developed model |
| Keyword:
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available model |
| Keyword:
|
completed model |
| Keyword:
|
elementary transformation |
| Keyword:
|
BLUE |
| Keyword:
|
estimation |
| Keyword:
|
testing |
| Keyword:
|
consistency |
| Keyword:
|
invariant |
| MSC:
|
62H05 |
| MSC:
|
62J05 |
| idZBL:
|
Zbl 0937.62067 |
| idMR:
|
MR1609162 |
| DOI:
|
10.1023/A:1023215001376 |
| . |
| Date available:
|
2009-09-22T17:57:17Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134380 |
| . |
| Reference:
|
[1] Allan, F.E., Wishart, J.: A method of estimating the yield of a missing plot in field experimental work.Jour. Agr. Sci. 20 (1930), 399–406. 10.1017/S0021859600006912 |
| Reference:
|
[2] Baksalary, J.K., Kala, R.: Linear transformations preserving best linear unbiased estimators in a general Gauss-Markoff model.Ann. St. 9 (1981), no. 4, 913–916. MR 0619297, 10.1214/aos/1176345533 |
| Reference:
|
[3] Bartlett, M.S.: Some examples of statistical methods of research in agriculture and applied biology.J.R. Statist. Soc. Suppl. 4 (1937), 137–170. 10.2307/2983644 |
| Reference:
|
[4] Biggers, J.D.: The estimation of missing and mixed-up observations in several experimental designs.Biometrika 16 (1959), 91–105. MR 0103574, 10.1093/biomet/46.1-2.91 |
| Reference:
|
[5] Drygas, H.: Sufficiency and completeness in the general Gauss-Markoff model.Sankhyā 45A, (1983), 88–99. MR 0749356 |
| Reference:
|
[6] Fisher, R.A.: The Design of Experiments.Edinburgh (1960). |
| Reference:
|
[7] Hartley, H.O., Hocking, R.R.: The analysis of incomplete data.Biometrics 4 (1971), 783–823. 10.2307/2528820 |
| Reference:
|
[8] Little, R.J.A., Rubin, D.B.: Statistical Analysis with Missing Data.J. Wiley, New York, 1987. MR 0890519 |
| Reference:
|
[9] Oktaba, W.: Estimation in the general multivariate Gauss-Markoff model with the known covariance matrix.XIX Colloquium Metodol. Agrobiom, Warsaw, Polish Academy of Sciences, 1989, pp. 156–169. (Polish) |
| Reference:
|
[10] Oktaba, W.: Predictor of the vector of missing observations in the general multivariate Gauss-Markoff model.Probastat’91, Proceedings of the 10th Conference on Probability and Mathematical Statistics, Bratislava, Czechoslovakia, 1991, pp. 51–67. |
| Reference:
|
[11] Oktaba, W.: Invariants in estimation and testing in the available and completed multivariate generalized Gauss-Markoff models with missing values.XXII Colloquium Metodol. Agrobiom., Warsaw, Polish Academy of Sciences, 1992, pp. 125–139. (Polish) |
| Reference:
|
[12] Oktaba, W., Kornacki, A. and Wawrzosek, J.: Estimation and verification of hypotheses in some Zyskind-Martin models with missing values.Biom. J. 27, 7 (1985), 733–740. MR 0834248, 10.1002/bimj.4710270704 |
| Reference:
|
[13] Oktaba, W., Kornacki, A. and Wawrzosek, J.: Estimation of missing values in the general Gauss-Markoff model.Statistics 17, 2 (1986), 167–177. MR 0838709, 10.1080/02331888608801923 |
| Reference:
|
[14] Oktaba, W., Kornacki, A. and Wawrzosek, J.: Invariants linearly sufficient transformations of the general Gauss-Markoff model. Estimation and testing.Scand. J. Statist. 15 (1988), 117–124. MR 0968158 |
| Reference:
|
[15] Oktaba, W. and Osypiuk, Z.: Program for estimation and testing in the multivariate generalized Gauss-Markoff model with missing values and known covariance matrix.XXIII Colloquium Metodol. Agrobiom., Warsaw, Polish Academy of Sciences, 1993, pp. 192–211. (Polish) |
| Reference:
|
[16] Rao, C.R.: Linear Statistical Inference and its Applications.sec. ed. New York, 1973. Zbl 0256.62002, MR 0346957 |
| Reference:
|
[17] Wilkinson, G. N.: A general recursive procedure for analysis of variance.Biometrika 57 (1970), 19–46. Zbl 0193.17003, 10.1093/biomet/57.1.19 |
| Reference:
|
[18] Yates, F.: The analysis of replicated experiments when the fields results are incomplete.Emp. Jour. Exp. Agr. 1 (1933), 235–244. |
| Reference:
|
[19] Zyskind, G., Martin, F.B.: On best linear estimation and a general Gauss-Markoff theorem in linear models with arbitrary nonnegative covariance structure.SIAM J. Appl. Math. 17 (1969), 1190–1202. MR 0261758, 10.1137/0117110 |
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