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A matrix version of Chernoff inequality

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  • Wei, Zhengyuan
  • Zhang, Xinsheng

Abstract

An interesting result from the point of view of upper variance bounds is the inequality of Chernoff [Chernoff, H., 1981. A note on an inequality involving the normal distribution. Annals of Probability 9, 533-535]. Namely, that for every absolutely continuous function g with derivative g' such that , and for standard normal r.v. [xi], . Both the usefulness and simplicity of this inequality have generated a great deal of extensions, as well as alternative proofs. Particularly, Olkin and Shepp [Olkin, I., Shepp, L., 2005. A matrix variance inequality. Journal of Statistical Planning and Inference 130, 351-358] obtained an inequality for the covariance matrix of k functions. However, all the previous papers have focused on univariate function and univariate random variable. We provide here a covariance matrix inequality for multivariate function of multivariate normal distribution.

Suggested Citation

  • Wei, Zhengyuan & Zhang, Xinsheng, 2008. "A matrix version of Chernoff inequality," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1823-1825, September.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:13:p:1823-1825
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    References listed on IDEAS

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    1. Willink, R., 2005. "Normal moments and Hermite polynomials," Statistics & Probability Letters, Elsevier, vol. 73(3), pages 271-275, July.
    2. Chen, Louis H. Y., 1982. "An inequality for the multivariate normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 12(2), pages 306-315, June.
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