10 The size of claims (in units of 1,000) arising from a portfolio of house contents

insurance policies can be modelled using a random variable X with probability density

function (pdf) given by:

1 ( ) ,

a

X a

ac f x xc

x

+ =

where a > 0 and c > 0 are the parameters of the distribution.

(i) Show that the expected value of X is [ ] 1

ac E X

a = , for a >1. [2]

(ii) Verify that the cumulative distribution function of X is given by

( ) 1 ,

a

X

c Fx xc

x

= (and = 0 for x < c). [2] CT3 A20105 PLEASE TURN OVER

Suppose that for the distribution of claim sizes X it is known that c = 2.5, but a is

unknown and needs to be estimated given a random sample x1, x2, ..., xn.

(iii) Show that the maximum likelihood estimate (MLE) of a is given by:

1

log 2.5

n i

i

n

a

x

=

=

. [3]

(iv) Derive the asymptotic variance of the MLE a , and hence determine its

approximate asymptotic distribution. [4]

Consider a sample of 30 observations from this distribution, for which:

30

1

log( ) 32.9 i

i

x

=

= .

(v) Calculate the MLE a in this case, together with an approximate 95%

confidence interval for a. [5]

In the current year, claim sizes are assumed to follow the distribution of X with a = 6,

c = 2.5. Inflation for the following year is expected to be 5%.

(vi) Calculate the probability that the size of a claim arising from this portfolio in

the following year will exceed 4,000.

2 Consider the following three independent random samples from a normally

distributed population with unknown mean :

Sample 1:

19.9 20.4 20.3 22.3 16.7 18.7 20.5 19.0 20.1 16.4 21.5 21.4 17.8 22.5 15.2

For these data: n = 15, 2 292.7, 5,778.69 i i x x = =

Sample 2:

20.8 25.9 22.1 21.7 16.0 12.1 27.6 16.1 16.8 17.1 21.3 18.6 24.9 14.8 22.2

For these data: n = 15, 2 298.0, 6,192.32 i i x x = =

sample mean = 19.867, sample variance = 19.432

Sample 3:

20.6 18.5 21.5 16.9 21.5 21.2 20.9 22.4 14.5 22.0 20.2 17.0 20.3 23.0 19.3

18.9 20.6 20.9 15.3 21.5 16.8 18.5 21.6 16.8 20.4

For these data: n = 25, 2 491.1, 9,773.77 i i x x = =

sample mean = 19.644, sample variance = 5.275

Consider t-tests of the hypotheses H0: = 18 v H1: > 18.

(i) (a) Calculate the sample mean and variance for Sample 1.

(b) Carry out a t-test of the stated hypotheses using the Sample 1 data

(stating the approximate P-value) and show that H0 can be rejected at

the 1% level of testing.

[6]

(ii) (a) Carry out a t-test of the stated hypotheses using the Sample 2 data

(stating the approximate P-value and the conclusion clearly).

(b) Discuss the comparison of the results with those based on Sample 1

(include reasons for any difference or similarity in the test

conclusions).

[6]

(iii) (a) Carry out a t-test of the stated hypotheses using the Sample 3 data

(stating the approximate P-value and your conclusion clearly).

(b) Discuss the comparison of the results with those based on Sample 1

(include reasons for any difference or similarity in the test

conclusions)