Some extensions of the Kantorovich inequality and statistical applications

Khatri, C. G. ; Radhakrishna Rao, C. (1981) Some extensions of the Kantorovich inequality and statistical applications Journal of Multivariate Analysis, 11 (4). pp. 498-505. ISSN 0047-259X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/004725...

Related URL: http://dx.doi.org/10.1016/0047-259X(81)90092-0

Abstract

Kantorovich gave an upper bound to the product of two quadratic forms, (X'AX) (X'A-1X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X'AX| |X'A-1X| where X is n × k matrix such that X'X = Ik. In this paper we determine the bounds for the traces and determinants of matrices of the type X'AYY'A-1X, X'B2X(X'BCX)-1 X'C2X(X'BCX)-1 where X and Y are n × k matrices such that X'X = Y'Y = Ik and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Kantorovich Inequality; Inverse Cauchy Inequality; least Squares; Efficiency; Regression
ID Code:42489
Deposited On:04 Jun 2011 09:09
Last Modified:04 Jun 2011 09:09

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