Refer to Exercise 9.109.a Find the MLE of N.b Show that E(

Chapter 9, Problem 110SE

(choose chapter or problem)

Refer to Exercise 9.109.

a Find the MLE \(\hat{N}_{2}\)  of \(N\).

b Show that \(E\left(\hat{N}_{2}\right))\) is approximately \([n /(n+1)] N\). Adjust \(\hat{N}_{2}\) to form an estimator \(\hat{N}_{3}\) that is approximately unbiased for \(N\).

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

                                                                \(\frac{n N^{2}}{(n+1)^{2}(n+2)}\)

d Show that for large \(N\) and \(n>1, V\left(\widehat{N}_{3}\right)<V\left(\widehat{N}_{1}\right)\).

Equation Transcription:

Text Transcription:

Hat N_2

N

E (hat N_2)

[n /(n+1)] N

Hat N_2

Hat N_3

N

Hat N_3

N

n N^2/(n+1)^2(n+2)

N and n>1, V(widehatN_3)<V(widehatN_1)