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摘要
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 Var{broken vertical bar g(xi)broken vertical bar} < infinity, and for standard normal r.v. xi, Var(broken vertical bar g(xi)broken vertical bar) <= E{(g '(xi)(2)}. 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.
