Estimation, Evaluation, and Selection of Actuarial Models

56 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
distribution is obtained from the inverse of the matrix with rsth element
·
∂ 2 ¸ ·
∂ 2
¸
I(θ) rs = −E l(θ) = −nE ln f(X; θ)
∂θ s ∂θ r ∂θ s ∂θ r
· ∂
= E l(θ) ∂ ¸ · ∂
l(θ) = nE ln f(X; θ) ∂
¸
ln f(X; θ) .
∂θ r ∂θ s ∂θ r ∂θ s
The first expression on each line is always correct. The second expression assumes that the likelihood
is the product of n identical densities. This matrix is often called the information matrix. The
information matrix also forms the Cramér—Rao lower bound. That is, under the usual conditions,
no unbiased estimator has a smaller variance than that given by the inverse of the information.
Therefore, at least asymptotically, no unbiased estimator is more accurate than the mle.
Example 3.30 Estimate the covariance matrix of the maximum likelihood estimator for the lognormal
distribution. Then apply this result to Data Set B.
The likelihood function and its logarithm are
nY
·
1
L(µ, σ) =
x j σ √ 2π exp − (ln x j − µ) 2 ¸
2σ 2
l(µ, σ) =
j=1
j=1
nX
− ln x j − ln σ − 0.5ln(2π) − 1 µ ln xj − µ 2
.
2 σ
j=1
The first partial derivatives are
∂l
nX
∂µ = ln x j − µ
σ 2
The second partial derivatives are
∂ 2 l
∂µ 2 = − n σ 2
∂ 2 l
nX
∂σ∂µ = −2
and ∂l
∂σ = −n σ + nX
j=1
∂ 2 l
∂σ 2 = n nX
σ 2 − 3
ln x j − µ
σ 3
j=1
j=1
(ln x j − µ) 2
σ 4 .
(ln x j − µ) 2
σ 3 .
The expected values are (ln X j has a normal distribution with mean µ and standard deviation σ)
E[∂ 2 l/∂µ 2 ] = −n/σ 2
E[∂ 2 l/∂σ∂µ] = 0
E[∂ 2 l/∂σ 2 ] = −2n/σ 2 .
Changing the signs and inverting produces an estimate of the covariance matrix (it is an estimate
because Theorem 3.29 only provides the covariance matrix in the limit). It is
· ¸
σ2 /n 0
0 σ 2 .
/2n