Estimation, Evaluation, and Selection of Actuarial Models

106 APPENDIX A. SOLUTIONS TO EXERCISES
negative first moment rather than the positive first moment. That is, equate the average reciprocal
to the expected reciprocal:
1
nX 1
= E(X −1 )=θ −1 .
n x j
j=1
This special method of moments estimate is identical to the maximum likelihood estimate.
For the inverse gamma distribution with α =2,
f(x|θ) = θ2 e −θ/x
x 3 Γ(2) , ln f(x|θ) =2lnθ − θx−1 − 3lnx
nX
nX
l(θ) = (2 ln θ − θx −1
j
− 3lnx j )=2n ln θ − nyθ − 3 ln x j
j=1
l 0 (θ) = 2nθ −1 − ny, ˆθ =2/y.
For Data Set B, ˆθ = 395.44 and the value of the loglikelihood function is −169.07. Themethodof
moments estimate solves the equation
1, 424.4 = θ
α − 1 =
θ
2 − 1 , ˆθ =1, 424.4
which differs from the maximum likelihood estimate.
For the inverse gamma distribution with both parameters unknown,
f(x|θ) =
j=1
θα e −θ/x
x α+1 Γ(α) , ln f(x|θ) =α ln θ − θx−1 − (α +1)lnx − ln Γ(α).
The likelihood function must be maximized numerically. The answer is ˆα =0.70888 and ˆθ = 140.16
and the loglikelihood value is −158.88. The methods of moments estimate is the solution to the
two equations
1, 424.4 =
13, 238, 441.9 =
θ
α − 1
θ 2
(α − 1)(α − 2) .
Squaring the first equation and dividing it into the second equation gives
6.52489 = α − 1
α − 2
which leads to ˆα =2.181 and then ˆθ =1, 682.2. This does not match the maximum likelihood
estimate (which had to be the case because the mle produces a model that does not have a mean).
Exercise 28 For the inverse exponential distribution, the cdf is F (x) =e −θ/x . Numerical maximization
yields ˆθ =6, 662.39 and the value of the loglikelihood function is −365.40. For the gamma
distribution, the cdf requires numerical evaluation. In Excel the function GAMMADIST(x, α, θ,true)
can be used. The estimates are ˆα = 0.37139 and ˆθ = 83, 020. The value of the loglikelihood
function is −360.50. For the inverse gamma distribution, the cdf is available in Excel as