A measure of variability based on the harmonic mean, and its use in approximations (Q1062377)

Given a random variable X and a function g, crude approximations to the mean and variance of g(X) may be obtained by Taylor series arguments. The variance of X, \(\sigma^ 2\), is a key quantity in approximating the bias and variance of g(X). For X positive and g rapidly decreasing, the bias and variance of g(X) should be relatively insensitive to the tail behavior of X, and \(\sigma^ 2\) should therefore not play an important role. In practice, when \(\sigma^ 2\) is very large the approximations for rapidly decreasing functions are often poor. The author assumes that X is positive with finite \(EX^{-1}\) and EX, and g is completely monotone with \(g(0)<\infty\). He then considers \(c^ 2=1- (EX EX^{-1})^{-1}\) as a measure of variability and shows that \(0\leq [Eg(X)-g(EX)]/g(0)\leq c^ 2\) and \(Var[g(X)]/g(0)\leq c^ 2\).