Statistical Methods for Insurance

AS305

Statistical Methods For Insurance

May 2015

Tutorial 6

Question 1

Find an equation which is satisfied by the MLE of qx for Special Case A, full data, under the hyperbolic

assumption. Solve this equation for qx if all deaths occur just before age x+ 1 (i. e., if si = 1 for all deaths).

Question 2

Intuitively, if there are no deaths in our sample we might expect to find qx = 0. Similarly, if the entire

sample of nx persons dies, we might expect to find qx = 1. It is obvious that these results are obtained for

the partial data and the full data, linear distribution, situations. Are they obtained for the full data,

exponential distribution, situation?

Question 3

For special case A full data situation, with the usual notation:

(a) Express the likelihood given by

in terms of qx under the exponential assumption.

(b) Express the log-likelihood for the likelihood in part (a).

(c) Solve this log-likelihood for the MLE of qx.

Question 4

Over the estimation interval (x, x+ 1] it is given that spx = 1 – (3s2 (qx)2)/2 for 0≤ s ≤ 2/3, and that spx = 1

– s(qx)2 for 2/3≤ s < 1. If nx = 300 and there are two observed deaths, one at x+.40 and one at x+. 75, find

the MLE of qx.

Question 5

Show that the Special Case C, partial data, uniform distribution likelihood, given by

, leads to Estimator

where

.

Question 6

Table below gives the data for a sample of five lives over the estimation interval (x,x+1].

i

ri

ti

i

1

0.0

1.0

0

2

0.0

.6

1

3

0.0

.7

1

4

0.5

.9

1

5

0.5

1.0

0

Find the MLE of qx under each of (a) the exponential distribution, and (b) the uniform distribution.

Question 7

A sample of 105 individuals were alive at age x. Five died before age x+l, and 10 others were enders at

age x+l/2. If the force of mortality is constant over (x, x+l,) find the maximum likelihood estimator of qx’

Question 8

Refer to the Schwartz-Lazar moment estimator , of Section 6.2.6.

(a) Of the cx group, what is the probability of...