ABSTRACT
Suppose X1, X2, . . . , Xn are independent, identically distributed random variables with unknown mean μ. If we wish to estimate some parameter θ = h(μ) of the distribution, then a natural estimator is h(Xn), where Xn is the sample mean. To evaluate the asymptotic performance of h(Xn) we might find the limiting moments of h(Xn) as the sample size n →∞. For example, we could look for an expansion of the form
E[h(Xn)] = a0 + a1 n
+ a2 n2
+ · · · + am nm
+ o ( n−m
) (4.1)
as n →∞. Such an expansion can be used to study convergence of h(Xn) to h(μ) through the moments of the distribution. We would also want to have similar sorts of expansions for the variance of h(Xn) such as
Var[h(Xn)] = b1 n
+ b2 n2
+ · · · + bm nm
+ o ( n−m
) . (4.2)
The constant term will be zero under mild regularity because the variance goes to zero as n →∞. Another way to evaluate the asymptotic performance of h(Xn) is to consider its limiting distribution as n →∞. Since Xn converges almost surely to μ, the limit distribution of h(Xn) is degenerate with mass one on h(μ). However, we can try to centre and scale h(Xn) so that a nondegenerate distribution is the result. We might centre and scale so that √
n [h(Xn)− h(μ)] d=⇒ N (0, σ2) (4.3) for some σ2 > 0 as n → ∞. The normality of the limit distribution will follow from the local linearity of h(x) close to x = μ and the central limit approximation to Xn.
