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

Mixed Poisson Models for Claim Numbers 39
Recall that having a n × n positive definite matrix M and a real vector , the random
vector X = X 1 X 2 X n T is said to have the multivariate Normal distribution with mean
and variance-covariance matrix M if its probability density function is of the form
f X x =
1
√
(−
2n detM exp 1 )
2 x − T M −1 x − x ∈ n (1.48)
Henceforth, we indicate that the random vector X has the multivariate Normal distribution
with probability density function (1.48) as X ∼ or M. A convenient characterization
of the multivariate Normal distribution is as follows: X ∼ or M if, and only if, any
random variable of the form ∑ n
i=1 iX i with ∈ R n , has the univariate Normal distribution.
Coming back to the properties of the maximum likelihood estimator ̂, we have that
̂ is approximately or ̂
distributed (1.49)
that is, the distribution function of ̂ can be approximated by integrating the Normal
probability density function
f S =
1
√
2 dim det̂
(
exp
− 1 2 S − T −1 S −
̂
)
S ∈ dim
Attribute (1.49) says that maximum likelihood estimators converge in distribution to a
Normal with mean equal to the population value of the parameter and variance-covariance
matrix equal to the inverse of the information matrix. This means that regardless of the
distribution of the variable of interest the maximum likelihood estimator of the parameters
will have a multivariate Normal distribution. Thus, a variable may be Poisson distributed,
but the maximum likelihood estimate of the Poisson mean will be asymptotically Normally
distributed, and likewise for any distribution. Note however that in the Poisson case, the exact
distribution of the maximum likelihood estimator of the parameter derived in Example 1.2
can easily be derived from the stability of the Poisson family under convolution.
Invariance
A natural question is how the parameterization of a likelihood affects the resulting
inference. Maximum likelihood has the property that any transformation of a parameter
can be estimated by the same transformation of the maximum likelihood estimate of
that parameter. This provides substantial flexibility in how we parameterize our models
while guaranteeing that we will get the same result if we start with a different
parameterization.
The invariance property can be stated formally as follows: If = t, where t· is a
one-to-one transformation, then the maximum likelihood estimator of is t̂. In particular,
the maximum likelihood estimator of pk is simply pk̂, that is,
̂pk = pk̂