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A Matrix Variance Inequality for k-Functions

Norhayati Rosli, and Wan Muhamad Amir W Ahmad, (2007) A Matrix Variance Inequality for k-Functions. Matematika, 23 (1). pp. 1-8. ISSN 01278274

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Official URL: http://www.fs.utm.my/matematika/images/stories/matematika/20072311.pdf

Affiliations

Kolej Universiti Kejuruteraan dan Teknologi Malaysia, Fakulti Kejuruteraan Kimia & Sumber Asli
Kolej Universiti Sains dan Teknologi Malaysia, Faculty of Science and Technology, Departments of Mathematics

Abstract

In this paper a course of solving variational problem is considered. [2] obtained what appears to be specialized inequality for a variance, namely, that for a standard normal variable X , V ar[g(x)] ¸ E[g0(x)]2 . However both of the simplicity and usefulness of the inequality has generated a plethora of extensions, as well as alternative proofs. [5] had focused on a result of two random variables for the normal and gamma distribution. They obtained the result of normal distribution with k functions, without proving and the proof is presented here. This paper also extend the result obtained by [5] to the k functions for the gamma distribution.

Item Type:Journal
Keywords:Normal Distribution, Gamma Distribution, Laguerre Family, Hermite Polynomials
Subjects:Q Science, Computer Science
ID Code:5482

[1] C. Stein, Estimation of the mean of a multivariate normal distribution, Ann. Statist. 9 (1981), 1135-1151.

[2] H. Chernoff, A note on an inequality involving the normal distribution, Ann. Probab. 9 (1981), 533-535.

[3] H.J. Branscamp & E.H. Lieb, On extensions of the Brunn-Minkowski and Prekopa Leinder theorems, including inequalities for log concave functions and with an application to the diffusion equation, J. Funct. Anal. 22 (1976), 366-389.

[4] Hwang, Chii-Ruey, Sheu, Shueen-Jyi, A generalization of Chernoff-inequality via stochastic analysis, Probab. Theory Related Fields 75 (1987), 149-157.

[5] I. Olkin & L. Shepp, A matrix variance inequality, J. Statist. Plan. Inference 130 (2005), 351-358.

[6] L.H.Y. Chen, An inequality for the multivariate normal distribution, J. Multivariate Anal. 12 (1982), 306-315.

[7] L.H.Y. Chen, The central limit theorem and Poincare-type inequalities, Ann. Probab. 16 (1988), 300-304.

[8] S. Borovkov & C. Houdre, Converse Poincare-type inequalities for convex functions, Satistic Probab. Lett. 42 (1997), 34-37.

[9] S. Karlin, A general class of variance inequalities, Multivariate Analysis: Future Directions. Elsevier Science Publishers, New York, 1993.

[10] T. Cacoullos, On upper and lower bounds for the variance of a function of a random variable, Ann. Probab. 10 (1982), 799-809.

[11] T. Cacoullos & V. Papathanasiou, Characterizations of distributions by generalizations of variance bounds and simple proofs of the C. L. T. J., Statist. Plann. Inference 63 (1997), 157-171.

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