# 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

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