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Approximation For The Expectation Of A Function Of The Sample Mean, Rasul A. Khan
Approximation For The Expectation Of A Function Of The Sample Mean, Rasul A. Khan
Mathematics and Statistics Faculty Publications
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 …