Estimation, Evaluation, and Selection of Actuarial Models

36 CHAPTER 2. MODEL ESTIMATION
Exercise 43 (*) You have observed the following five claim severities: 11.0, 15.2, 18.0, 21.0, and
25.8. Determine the maximum likelihood estimate of µ for the following model:
f(x) = √ 1 ·
exp − 1 (x − µ)2¸
, x,µ > 0.
2πx 2x
Exercise 44 (*) A random sample of size 5 is taken from a Weibull distribution with τ =2.Two
of the sample observations are known to exceed 50 and the three remaining observations are 20, 30,
and 45. Determine the maximum likelihood estimate of θ.
Exercise 45 (*) Phil and Sylvia are competitors in the light bulb business. Sylvia advertises that
her light bulbs burn twice as long as Phil’s. You were able to test 20 of Phil’s bulbs and 10 of
Sylvia’s. You assumed that both of their bulbs have an exponential distribution with time measured
in hours. You have separately estimated the parameters as ˆθ P = 1000 and ˆθ S = 1500 for Phil
and Sylvia respectively, using maximum likelihood. Using all thirty observations, determine ˆθ ∗ ,the
maximum likelihood estimate of θ P restricted by Sylvia’s claim that θ S =2θ P .
Exercise 46 (*) A sample of 100 losses revealed that 62 were below 1000 and 38 were above 1000.
An exponential distribution with mean θ is considered. Using only the given information, determine
the maximum likelihood estimate of θ. Now suppose you are also given that the 62 losses that were
below 1000 totalled 28,140 while the total for the 38 above 1000 remains unknown. Using this
additional information, determine the maximum likelihood estimate of θ.
Exercise 47 (*) The following values were calculated from a random sample of 10 losses:
10X
j=1
10X
j=1
10X
x −2
j
= 0.00033674
x −0.5
j
= 0.34445
j=1
x j = 31, 939
10X
j=1
10X
j=1
10X
j=1
x −1
j
=0.023999
x 0.5
j = 488.97
x 2 j = 211, 498, 983.
Losses come from a Weibull distribution with τ =0.5 (so F (x) =1− e −(x/θ)0.5 ). Determine the
maximum likelihood estimate of θ.
Exercise 48 (*) For claims reported in 1997, the number settled in 1997 (year 0) was unknown,
the number settled in 1998 (year 1) was 3 and the number settled in 1999 (year 2) was 1. The
number settled after 1999 is unknown. For claims reported in 1998 there were 5 settled in year
0, 2 settled in year 1, and the number settled after year 1 is unknown. For claims reported in
1999 there were 4 settled in year 0 and the number settled after year 0 is unknown. Let N be
the year in which a randomly selected claim is settled and assume that it has probability function
Pr(N = n) =p n =(1− p)p n ,n=0, 1, 2,.... Determine the maximum likelihood estimate of p.