Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

38 Actuarial Modelling of Claim Counts
is called convergence in probability and is defined more formally as follows: A consistent
estimator T j for some parameter j computed from a sample of size n is one for which
lim
n↗ PrT j − j ≥c = 0
for all positive c. This will henceforth be denoted as T j → proba j as n ↗+. A consistent
estimator is thus an estimator that converges to the population parameter as the sample size
goes to infinity. Consistency is an asymptotic property.
Asymptotic Normality
Any estimator will vary across repeated samples. We must be able to calculate this
variability in order to express our uncertainty about a parameter value and to make statistical
inferences about the parameters. This variability is measured by the variance-covariance
matrix of the estimators. This matrix provides the variances for each parameter on the
main diagonal while the off-diagonal elements estimate the covariances between all pairs of
parameters.
The asymptotic variance-covariance matrix ̂
for maximum likelihood estimators ̂ is
the inverse of what is called the Fisher information matrix . Element ij of is given
by
[ ] [ ]
2
2
−E ln L =−nE ln pN
i j i 1
j
[
= nE ln pN
1 ]
ln pN
i 1
j
∑
= n pk ln pk ln pk
i j
k=0
Thus, ̂
= ( ) −1
.
An insight into why this makes sense is that the second derivatives measure the rate of
change in the first derivatives, which in turn determines the value of the maximum likelihood
estimate. If the first derivatives are changing rapidly near the maximum, then the peak of
the likelihood is sharply defined and the maximum is easy to see. In this case, the second
derivatives will be large and their inverse small, indicating a small variance of the estimated
parameters. If on the other hand the second derivatives are small, then the likelihood function
is relatively flat near the maximum and so the parameters are less precisely estimated. The
inverse of the second derivatives will produce a large value for the variance of the estimates,
indicating low precision of the estimates.
The distribution of ̂ is usually difficult to obtain. Therefore we resort to the following
asymptotic theory: Under mild regularity conditions (including that the true value of the
parameter must be interior to the parameter space, that the log-likelihood function must be
thrice differentiable, and that the third derivatives must be bounded) that are usually fulfilled,
the maximum likelihood estimator ̂ has approximately in large samples a multivariate
Normal distribution with mean equal to the true parameter and variance-covariance matrix
given by the inverse of the information matrix.