Loss Models: From Data to Decisions, 4th Edition

14.7 Exercises

14.1 Assume that the binomial parameter m is known. Consider the mle of q.

(a) Show that the mle is unbiased.

(b) Determine the variance of the mle.

(c) Show that the asymptotic variance as given in Theorem 13.5 is the same as that developed in part (b).

(d) Determine a simple formula for a confidence interval using (10.4) on page 197 that is based on replacing q with in the variance term.

(e) Determine a more complicated formula for a confidence interval using (10.3) that is not based on such a replacement. Proceed in a manner similar to that used in Example 12.8 on page 228.

14.2 Use (14.5) to determine the mle of β for the geometric distribution. In addition, determine the variance of the mle and Verify that it matches the asymptotic variance as given in Theorem 13.5.

14.3 A portfolio of 10,000 risks produced the claim counts in Table 14.9.

Table 14.9 Data for Exercise 14.3.

No. of claims	No. of policies
0	9,048
1	905
2	45
3	2
4+	0

(a) Determine the mle of λ for a Poisson model and then determine a 95% confidence interval for λ.

(b) Determine the mle of β for a geometric model and then determine a 95% confidence interval for β.

(c) Determine the mle of r and β for a negative binomial model.

(d) Assume that m = 4. Determine the mle of q of the binomial model.

(e) Construct 95% confidence intervals for q using the methods developed in parts ...