On Upper and Lower Bounds for the Variance of a Function of a Random Variable

Zero biasing in one and higher dimensions, and applications

“…is a density function for a distribution which satisfies (2). Though Definition 1.1 is stated in terms of random variables, it actually defines a transformation on a class of distributions, and we will use the language interchangeably.…”

“…Rather than changing the distribution of X to satisfy the right hand side of (2), in [4] the existence of a function w is postulated such that EXf (X) = σ 2 Ew(X)f (X). Based on an idea in [2], the use of the w function is extended in [3] to a multivariate case for independent mean zero variables X 1 , . .…”

Unified Variance Bounds and a Stein-Type Identity

“…It follows that 0 ≤ g(x) ≤ 1 and g(−x) = g(x) for all x. Moreover, g is a.c. with derivative g (x) (outside the set {0, ±a 1 We have shown in Lemma 2.1(iii) that X * is the only r.v. satisfying the identity (2.2), and thus, an equivalent definition of X * could be given via the covariance identity; the latter approach is due to Goldstein and Reinert ( [11], Definition 1.1), who proved that such a transformation is uniquely defined by this identity.…”

“…Upper and lower variance bounds of g(X) for an arbitrary r.v. X were considered in [1] and [3] (see also [4]- [7] and references therein). Both upper and lower variance bounds may be obtained as by-products of the Cauchy-Schwarz inequality.…”

“…The following Lemma summarizes and unifies these bounds in terms of the r.v. X * ; in effect, (3.1) is a Chernoff-type, [8], upper bound; (3.2) is a Cacoullos-type, [14], [15], lower bound as obtained in [1] and [3], in terms of a function w (see also (3.4) below). …”