Estimation of subparameters by IPM method

Year 2016, Volume: 20 Issue: 2, 259 - 264, 01.08.2016

Nesrin Güler , Melek Eriş

https://doi.org/10.16984/saufenbilder.35292

Abstract

In this study, a general partitioned linear model  is considered to determine the best linear unbiased estimators (BLUEs) of subparameters  and . Some results are given related to the BLUEs of subparameters by using the inverse partitioned matrix (IPM) method based on a generalized inverse of a symmetric block partitioned matrix which is obtained from the fundamental BLUE equation.

 

Keywords

BLUE, generalized inverse; general partitioned linear model

IPM yöntemi ile alt parametrelerin tahmini

Abstract

Bu çalışmada,  ve  alt parametrelerinin en iyi lineer yansız tahmin edicilerini (BLUE’ larını) belirlemek için bir  genel parçalanmış lineer modeli ele alınmıştır. Temel BLUE denkleminden elde edilen simetrik blok parçalanmış matrisin bir genelleştirilmiş tersine dayanan parçalanmış matris tersi (IPM) yöntemi kullanılarak alt parametrelerin BLUE’ ları ile ilgili bazı sonuçlar verilmiştir.

---

Keywords

BLUE, genelleştirilmiş ters, genel parçalanmış lineer model

References

• C. R. Rao, “Unified theory of linear estimation”, Sankhyā , Ser. A 33:371-394, 1971. [Corrigendum (1972), 34, p. 194 and p. 477.].
• M. Nurhanen and S. Puntanen, “Effect of deleting an observation on the equality of the OLSE and BLUE”, Linear Algebra and its Applications, vol. 176, pp. 131-136, 1992.
• H. J. Werner and C. Yapar, “Some equalities for estimations of partial coefficients under a general linear regression model”, Linear Algebra and its Applications, 237/238, 395-404, 1996.
• P. Bhimasankaram and R. Saharay, “On a partitioned linear model and some associated reduced models”, Linear Algebra and its Applications, vol. 264, pp. 329-339, 1997.
• J. Gross and S. Puntanen, “Estimation under a general partitioned linear model”, Linear Algebra and its Applications, vol. 321, pp. 131-144, 2000.
• B. Zhang, B. Liu and C. Lu, “A study of the equivalence of the BLUEs between a partitioned singular linear model and its reduced singular linear models”, Acta Mathematica Sinica, English Series, vol. 20, no.3, pp.557-568, 2004.
• Y. Tian and S. Puntanen, “On the equivalence of estimations under a general linear model and its transformed models”, Linear Algebra and its Applications, vol. 430, pp. 2622-2641, 2009.
• Y. Tian, “On properties of BLUEs under general linear regression models”, Journal of Statistical Planning and Inference, vol. 143, pp. 771-782, 2013.
• J. K. Baksalary, and P. R. Pordzik, “InversePartitioned-Matrix method for the general GaussMarkov model with linear restrictions”, Journal of Statistical Planning and Inference, vol. 23, pp. 133143, 1989.
• C. R. Rao, Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Le Cam, L. M., Neyman, J., eds. Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability: Berkeley, California, 1965/1966. Vol. 1. Univ. Of California Press, Berkeley, 355-372, 1967.
• G. Zyskind, “On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models” The Annals of Mathematical Statistics, vol. 38, pp. 1092-1109, 1967.
• C. R. Rao, “Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix”, Journal of Multivariate Analysis, vol. 3, pp. 276-292, 1973.
• I. S. Alalouf and G. P. H. Styan, “Characterizations of estimability in the general linear model” The Annals of Statistics, vol. 7, pp. 194-200, 1979.
• S. Puntanen, G.P.H. Styan and J. Isotalo, “Matrix tricks for linear statistical models”, Our Personal Top Twenty, Springer, Heidelberg, 2011.
• C. R. Rao, “A note on the IPM method in the unified theory of linear estimation”, Sankhyā, Ser.A., vol. 34, pp. 285-288, 1972.
• H. Drygas, “A note on the Inverse-PartitionedMatrix method in linear regression analysis”, Linear Algebra and its Applications, vol. 67, pp. 275-277, 1985.
• F. J. Hall and C. D. Meyer, “Generalized inverses of the fundamental bordered matrix used in linear estimation”, Sankhyā, Ser. A., vol. 37, pp. 428438, 1975. [Corrigendum (1978), 40, p. 399.]
• D. A. Harville, “Matrix algebra from a statisticians perspective”, Springer, New York, 1997.
• J. Isotalo, “Puntanen, S., Styan, G. P. H., A useful matrix decomposition and its statistical applications in linear regression”, Commun. Statist. Theor. Meth., vol. 37, pp. 1436-1457, 2008.
• R. M. Pringle and A. A. Rayner, “Generalized Inverse Matrices with Applications to Statistics”, Hafner Publishing Company, New York, 1971.
• C. R. Rao, “Some recent results in linear estimation”, Sankhyā, Ser. B, vol. 34, pp. 369-378, 1972.
• C. R. Rao, “Linear Statistical Inference and its Applications”, 2nd Ed., Wiley, New York, 1973.
• H. J. Werner, C. R. Rao’s IPM method: a geometric approach. New Perspectives in Theoretical and Applied Statistics (M. L. Puri, J. P. Vilaplana & W. Wertz, eds.). Wiley, New York, 367-382, 1987.