Address the questions below well Image transcription text The claim amounts for a certain insurance company follow a Pareto distribution with probability distribution function given

Address the questions below well Image transcription text The claim amounts for a certain insurance company follow a Pareto distribution with
probability distribution function given by:
o 250" (250 + x) (i)
Show that the maximum likelihood estimator for o is given by:
a =
[4]
[ In(250 + x; ) – nin 250
A random sample of 150 claims gives > In(250 + x; ) = 869 .
(ii)
Given that the estimator is asymptotically unbiased, obtain a 95% confidence
interval for the parameter of.
[5] Image transcription text The random variable X has a generalised Pareto distribution with density function,
f(x;a) given by:
f(x;a) = a(a+1)(1,000)"x(1,000 + x)-0-2
x>0
A sample of 100 values of X gives the following information:
100
log (1,000 + x;) = 750
i=l
(i)
Calculate the maximum likelihood estimate o of a.
[6]
(ii)
Estimate the standard deviation of a .
[3]
[Total 9] Image transcription text Example 9.2.1. The following values were obtained during an investigation into the mortality of a.
large group of lives: “initial” age exposed deaths
to risk I E 3′: 35 295,551 232
35 312,91? 295
3? 323,333 395
33 342,131 425
39 359,555 45′?
49 359,?91 59’1"Ir
41 359,521 552
42 335,513 T33
43 331,993 334
44 334,421 954
45 393,451 1,139 Test whether the mortality of this group of lives follows a. given table of which the following is an
extract: age I 93,
35 9.99913
35 9.99 1991
3? 9.99 119?
33 9.99 1232
39 9.99 1339
49 9.99 15 51
41 9.99 1343
42 9.99 19 T3
43 9.99 2239
44 9.99 25 29
45 9.99 2343
45 9.9932 1 5
4′? 9.9935 23