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Some extensions of the Kantorovich inequality and statistical applications

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  • Khatri, C. G.
  • Rao, C. Radhakrishna

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.

Suggested Citation

  • Khatri, C. G. & Rao, C. Radhakrishna, 1981. "Some extensions of the Kantorovich inequality and statistical applications," Journal of Multivariate Analysis, Elsevier, vol. 11(4), pages 498-505, December.
    • Handle: RePEc:eee:jmvana:v:11:y:1981:i:4:p:498-505
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    Cited by:

    1. Liu, Shuangzhe & Leiva, Víctor & Zhuang, Dan & Ma, Tiefeng & Figueroa-Zúñiga, Jorge I., 2022. "Matrix differential calculus with applications in the multivariate linear model and its diagnostics," Journal of Multivariate Analysis, Elsevier, vol. 188(C).

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