On upper bounds for the variance of functions of random variables
The upper bounds for the variance of a function g of a r.v. X obtained in this paper were originally motivated by \textit{H. Chernoff}'s inequality [Ann. Probab. 9, 533-535 (1981; Zbl 0457.60014)]: If g is absolutely continuous with derivative g' and X is N(0,1), then \(Var[g(X)]\leq E[g'(X)]^ 2\). \textit{L. H. Y. Chen} [J. Multivariate Anal. 12, 306-315 (1982; Zbl 0483.60011)], using the Cauchy-Schwarz (C-S) inequality obtained a multivariate extension: if \(X_ 1,...,X_ k\) are i.i.d. N(0,1) and g a function defined on \(R^ k\) with partial derivatives \(g_ 1,...,g_ k\), then \(Var[g(X_ 1,...,X_ k)]\leq \sum^{k}_{i=1}Eg^ 2_ i(X_ 1,...,X_ k).\) This paper improves the variance upper bounds given by the first author [Ann. Probab. 10, 799-809 (1982; Zbl 0492.60021)] for continuous or discrete r.v.'s, based on a similar use of the C-S inequality. The improvement relies on an appropriate use both of the C-S inequality and the Lagrange identity, resulting in the inequality \[ Var[g(X)]\leq \int^{\infty}_{-\infty}[g'(x)]^ 2\{\int^{\infty}_{x}(t- \mu)f(t)dt\}dx \] where, as in the above inequalities, equality holds iff g is linear. For a discrete r.v. X, g'(x) is replaced by \(\Delta g(x)=g(x+1)-g(x)\) and \(\int^{\infty}_{x}\) by \(\sum^{\infty}_{x=k+1}\). It is pointed out that the earlier bounds obtained by the first author coincide with the improved ones when \(E(X)=0\). A multivariate extension for any independent discrete or continuous r.v.'s is also given (cf. Chen, op. cit).
scientific article; zbMATH DE number 3913345