Chapter 9, Problem 110SE

Refer to Exercise 9.109.

1. a Find the MLE ${\hat{N}}_2$ of N.
2. b Show that E( ${\hat{N}}_2$) is approximately [n/(n + 1)]N. Adjust ${\hat{N}}_2$ to form an estimator ${\hat{N}}_3$ that is approximately unbiased for N.
3. c Find an approximate variance for ${\hat{N}}_3$ by using the fact that for large N the variance of the largest sampled integer is approximately

   $\frac{n{N}^2}{{\left(n+1\right)}^2\left(n+2\right)}.$

4. d Show that for large N and n > 1, V( ${\hat{N}}_3$) < V( ${\hat{N}}_1$).

9.109. Suppose that n integers are drawnat random and with replacement from the integers 1, 2, . . . , N. That is, each sampled integer has probability 1/N of taking on any of the values 1, 2, . . . , N, and the sampled values are independent.

1. a Find the method-of-moments estimator ${\hat{N}}_1$ of N.
2. b Find E( ${\hat{N}}_1$) and V( ${\hat{N}}_1$).