Let X1, , Xn be i.i.d. Poisson with mean . Consider estimating g() = e by

Let X1, ··· , Xn be i.i.d. Poisson with mean λ. Consider estimating g(λ) = e−λ by the estimator Tn = e−X¯n . Find an approximation to the bias of Tn; specifically, find a function b(λ) satisfying Eλ(Tn) = g(λ) + n−1 b(λ) + O(n−2

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as n → ∞. Such an expression suggests a new estimator Tn −n−1b(λ), which has bias O(n−2). But, b(λ) is unknown. Show that the estimator Tn − n−1b(X¯n) has bias O(n−2)