Estimation, Evaluation, and Selection of Actuarial Models

58 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
Changing the signs and inverting produces the same values as in (3.4). This is a feature of the
lognormal distribution that need not hold for other models. ¤
Sometimes it is not even possible to take the derivative. In that case an approximate second
derivative can be used. A reasonable approximation is
∂ 2 f(θ) . 1 = [f(θ + h i e i /2+h j e j /2) − f(θ + h i e i /2 − h j e j /2)
∂θ i ∂θ j h i h j
−f(θ − h i e i /2+h j e j /2) + f(θ − h i e i /2 − h j e j /2)]
where e i is a vector with all zeroes except for a one in the ith position and h i = θ i /10 v where v is
one-third the number of significant digits used in calculations.
Example 3.32 Repeat the previous example, using approximate derivatives.
Assume that there are 15 significant digits being used. Then h 1 = 6.1379/10 5 and h 2 =
1.3894/10 5 . Reasonably close values are 0.00006 and 0.00001. The first approximation is
∂ 2 l
∂µ 2 . =
l(6.13796, 1.3894) − 2l(6.1379, 1.3894) + l(6.13784, 1.3894)
(0.00006) 2
=
−157.713893081982 − 2(−157.713893049683) + (−157.713893054682)
(0.00006) 2
= −10.3604.
The other two approximations are
∂ 2 l/∂σ∂µ = . 0.0003
∂ 2 l/∂σ 2 . = −20.7208.
We see that here the approximation works very well. ¤
For grouped data, the process works the same way. In Loss Models on pages 59 and 66 it is
noted that the information can be obtained using only first partial derivatives. These still may
need to be numerically evaluated.
Exercise 67 Determine 95% confidence intervals for the parameters of exponential and gamma
models for Data Set B. The likelihood function and maximum likelihood estimates were determined
in Example 2.32.
Exercise 68 Let X have a uniform distribution on the interval from 0 to θ. Show that the maximum
likelihood estimator is ˆθ =max(X 1 ,...,X n ). Use Examples 3.7 and 3.10 to show that this
estimator is asymptotically unbiased and to obtain its variance. Show that Theorem 3.29 yields a
negative estimate of the variance and that item (ii) in the conditions does not hold.
The information matrix provides a method for assessing the quality of the maximum likelihood
estimators of a distribution’s parameters. However, we are often more interested in a quantity that