Calculating a bias given the MOM and MLE

Question

Question: A company manufactured objects, starting from 1 to N. One of the objects is selected, and serial number is 888. We are asked to find things like the MOM, MLE etc. which I managed without problem.

MOM estimator $\hat{N}$ = $2\bar{X} - 1$, and MLE estimator $\hat{N} = \bar{X}$.

Now, what I want to do is to find the bias. I know that variance of MOM is 4 * Var($\bar{X}$), and variance of MLE is just Var($\bar{X}$). The formula, from what I understand is $[E(\hat{N})-N_o]^2$, but other than that I do not know how to start.

Answer given is that MOM estimator is unbiased, and MLE estimator has bias $\frac{N-1}{2}$.

Answer 1

I would say that the bias is $E[\hat N - N] = E[\hat N] - N$

Since $E[X]=\frac{N+1}{2}$, you have

• If $\hat N = 2X-1$, then $E[\hat N]=N$ and the bias is $N-N=0$ so the MOM is an unbiased estimator
• If $\hat N = X$, then $E[\hat N]=\frac{N+1}{2}$ and the bias is $\frac{N+1}{2}-N=-\frac{N-1}{2}$ so the MLE is biased downwards

This is the German tank problem with a single sample