Question

Suppose the random variable X follows the Poisson P(m) PDF, and that you have a random sample X1, X2,...,Xn from it. (a)What is the Cramer-Rao Lower Bound on the variance of any unbiased estimator of the parameter m? (b) What is the maximum likelihood estimator ofm?(c) Does the variance of the MLE achieve the CRLB for all n?

Answer 1

a)

Here,

The CRLB is given by:

Now,

Now,

So, --------(1)

Using Central limit theorem we know, -------------(2)

So,

Thus, CRLB =

b)

Equating (1) to 0, we get,

which is the MLE of m.

c)

Now, {From (2)}

So, it achieves the CRLB.