Question

Problem 3: Suppose X1, X2, is a sequence of i.i.d. random variables having the Poisson distribution with mean λ. Let A,-X, (a) Is λη an unbiased estimator of λ? Explain your answer. (b) Is in a consistent estimator of A? Explain your answer 72

Answer 1

Answer:

Given that:

Suppose X1, X2,... is a sequence of i.i.d random variables having the Poisson distribution with mean $=\lambda$

To solve the question considering Xn as an order statistic, we need to know the total number of X's

(a)

$E(\hat{\lambda}_n)=E(X_n)=\lambda$

$Thus \hat{\lambda}_n is an unbiased estimator of \lambda$

(b)

To check consistency we find the Mean Square Error(MSE) of the estimator. Since the estimator is unbiased, the MSE is equal to the Variance of the estimator. Thus

$MSE(\hat{\lambda}_n) = Var(\hat{\lambda}_n)$

$=Var(X_n)$

$=\lambda$

Since, MSE doesnot approach zero as n approaches infinity, the estimator is not consistent.