Estimation, Evaluation, and Selection of Actuarial Models

118 APPENDIX A. SOLUTIONS TO EXERCISES
The first two second partial derivatives do not contain x j and so the expected value is equal to the
indicated quantity. For the final second partial derivative, E( ¯X) =E(X) =αθ. Therefore,
"
#
n Γ(α)Γ00 (α)−Γ 0 (α) 2
I(α, θ) =
Γ(α)
nθ −1
2
nθ −1 nαθ −2 .
The derivatives of the gamma function are available in some better computer packages, but are not
available in Excel. Using numerical derivatives of the gamma function yields
· ¸
I(ˆα, ˆθ) 82.467 0.0078091
=
0.0078091 0.0000016958
and the covariance matrix is · 0.021503 −99.0188
−99.0188 1, 045, 668
Numerical second derivatives of the likelihood function (using h 1 =0.00005 and h 2 =0.25) yield
· ¸
· ¸
I(ˆα, ˆθ) 82.467 0.0078091
0.021502 −99.0143
=
and covariance matrix
.
0.0078091 0.0000016959
−99.0143 1, 045, 620
The confidence interval for α is 0.55616 ± 1.96(0.021502) 1/2 or 0.55616 ± 0.28741 and for θ is
2561.1 ± 1.96(1, 045, 620) 1/2 or 2561.1 ± 2004.2.
Exercise 68 The density function is f(x|θ) =θ −1 , 0 ≤ x ≤ θ. The likelihood function is
¸
.
L(θ) = θ −n , 0 ≤ x 1 ,...,x n ≤ θ
= 0, otherwise.
As a function of θ, the likelihood function is sometimes 0, and sometimes θ −n . In particular, it is
θ −n only when θ isgreaterthanorequaltoallthexs. Equivalently, we have
L(θ) = 0, θ < max(x 1 ,...,x n )
= θ −n , θ ≥ max(x 1 ,...,x n ).
Therefore, the likelihood function is maximized at ˆθ =max(x 1 ,...,x n ). Note that the calculus
technique of setting the derivative equal to zero does not work here because the likelihood function
is not continuous (and therefore not differentiable) at the maximum. From Examples 3.7 and 3.10
we know that this estimator is asymptotically unbiased and consistent and we have its variance
without recourse to Theorem 3.29. According to Theorem 3.29, we need
l(θ) = −n ln θ, θ ≥ max(x 1 ,...,x n )
l 0 (θ) = −nθ −1 , θ ≥ max(x 1 ,...,x n )
l 00 (θ) = nθ −2 , θ ≥ max(x 1 ,...,x n ).
Then E[l 00 (θ)] = nθ −2 because with regard to the random variables, nθ −2 is a constant and therefore
its expected value is itself. The information is then the negative of this number and must be
negative.