Let X¯ n be the mean of a random sample of size n from a distribution with mean μ and variance σ2. Under some conditions it is shown that Ef(X¯ n ) = f(μ) + (σ2/2n) f″(μ) + O(n −2), and var(f(X¯ n )) = (σ2/n) (f′(μ))2 + O(n −2), where f is a continuous function with a suitable growth condition. This complements a result of Lehmann [(1991). Theory of Point Estimation. Wadsworth, California] and Cramér [(1946). Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J.] for wider application. An illustrative example is given to show an application where the usual approximations do not apply.
Khan, Rasul A., "Approximation for The Expectation of A Function of The Sample Mean" (2004). Mathematics Faculty Publications. 163.
This is an Author’s Accepted Manuscript of an article published in Statistics 2004, available online: http://www.tandfonline.com/10.1080/02331880310001655635