LetX = (X1, . . . , Xn)Tdenote a random sample of size n on a discrete random variable X which hasprobability function f (x; θ), where θ is an unknown parameter.Suppose we wish to estimate f (k; θ), the probability that X is equal to agiven arbitrary (non-negative) integer k. We shall subsequently write f (k; θ)as the function g(θ), suppressing its dependence on the value k.Let U (X1) be a statistic defined as:U (X1) ={1, X1 = k,0, otherwise.(i) On letting T (X1, ..., Xn) denote a sufficient statistic for θ, show thatW (T ) = E{U (X1) | T (X1, ..., Xn)}is a statistic; that is, explain why it does not depend on θ.(ii) Show that W (T ) is an unbiased estimator of g(θ).(iii) Without assuming the result of any theorem, show that the variance ofW (T ) will be less than the variance of U unless U is constant given Twhere in this case W will have the same variance as U .(iv) Explain why W (T ) is the UMVU estimator of g(θ) in the case whereT is a complete, minimal sufficient statistic for θ.
Question
LetX = (X1, . . . , Xn)Tdenote a random sample of size n on a discrete random variable X which hasprobability function f (x; θ), where θ is an unknown parameter.Suppose we wish to estimate f (k; θ), the probability that X is equal to agiven arbitrary (non-negative) integer k. We shall subsequently write f (k; θ)as the function g(θ), suppressing its dependence on the value k.Let U (X1) be a statistic defined as:U (X1) ={1, X1 = k,0, otherwise.(i) On letting T (X1, ..., Xn) denote a sufficient statistic for θ, show thatW (T ) = E{U (X1) | T (X1, ..., Xn)}is a statistic; that is, explain why it does not depend on θ.(ii) Show that W (T ) is an unbiased estimator of g(θ).(iii) Without assuming the result of any theorem, show that the variance ofW (T ) will be less than the variance of U unless U is constant given Twhere in this case W will have the same variance as U .(iv) Explain why W (T ) is the UMVU estimator of g(θ) in the case whereT is a complete, minimal sufficient statistic for θ.
Solution
(i) A statistic is a function of the data that does not depend on any unknown parameters. Here, W(T) is defined as the conditional expectation of U(X1) given T(X1, ..., Xn). Since both U(X1) and T(X1, ..., Xn) are functions of the data, W(T) is also a function of the data and does not depend on θ. Therefore, W(T) is a statistic.
(ii) To show that W(T) is an unbiased estimator of g(θ), we need to show that E[W(T)] = g(θ). We have:
E[W(T)] = E[E[U(X1)|T(X1, ..., Xn)]] = E[U(X1)] = g(θ)
The second equality follows from the law of iterated expectations, and the third equality follows from the definition of U(X1) and g(θ).
(iii) To show that the variance of W(T) is less than the variance of U unless U is constant given T, we can use the law of total variance, which states that Var(Y) = E[Var(Y|X)] + Var(E[Y|X]) for any two random variables X and Y. Applying this to Y = U(X1) and X = T(X1, ..., Xn), we get:
Var(U) = E[Var(U|T)] + Var(E[U|T]) = E[Var(U|T)] + Var(W)
Since variances are always non-negative, we have Var(U) >= Var(W) with equality if and only if U is constant given T.
(iv) W(T) is the UMVU (Uniformly Minimum Variance Unbiased) estimator of g(θ) if T is a complete, minimal sufficient statistic for θ because of the Lehmann-Scheffe theorem. This theorem states that if T is a complete, minimal sufficient statistic and W(T) is an unbiased estimator of g(θ), then W(T) is the UMVU estimator of g(θ). We have already shown that W(T) is an unbiased estimator of g(θ), so if T is a complete, minimal sufficient statistic, then W(T) is the UMVU estimator of g(θ).
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