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

Risk Classification 69
2.3.8 Wald Confidence Intervals
The asymptotic variance-covariance matrix ̂
of the maximum likelihood estimator ̂ of
the regression coefficients vector is the inverse of the Fisher information matrix. This
matrix can be estimated by
̂̂
=
( ) n∑ −1
˜x i˜x T ̂ i i where ̂ i = d i expŝcore i
i=1
We know from (1.49) that provided the sample size is large enough ̂− is approximately
or0 ̂̂ distributed. It is thus possible to compute confidence intervals at level 1− for
each of the j s. These intervals are of the form
[̂j − z /2̂̂j
̂
]
j + z /2̂̂j
(2.8)
where ̂ 2̂j
is the estimated variance of ̂ j , given by the element j j of ̂̂.
Remark 2.3 (Confidence Intervals for the j s: the Likelihood Ratio Method) The
confidence interval (2.8) is based on the large sample properties of the maximum likelihood
estimator ̂. Other methods for constructing such a confidence interval are available. One
such method is based on the profile likelihood for j that is defined as
j j = max
0 j−1 j+1 p
If ̂ is the maximum likelihood estimator of , we have that 2 ( L̂ − L j j ) is
approximately 1 2 provided j is the true parameter value. A confidence interval at level
1 − for j is then given by
{ ∣ ∣∣Lj
j j ≥ L̂ − 1 }
2 2 11−
where 2 11− is the 1 − th quantile of the 2 1 distribution.
2.3.9 Testing for Hypothesis on a Single Parameter
It is often interesting to check the validity of the null hypothesis H 0 : j = 0 against the
alternative H 1 : j ≠ 0. If the jth explanatory variable is dichotomous (think for instance
of gender), then failing to reject H 0 suggests that this variable is not relevant to explaining
the expected number of claims. If the jth explanatory variable is coded by means of a set
of binary variables, then the nullity of the regression coefficient associated with one of the
binary variables means that the corresponding level can be grouped with the reference level.
In such a case, equality between the regression coefficients should also be tested to decide
about the optimal grouping of the levels. Hypotheses involving a set of regression parameters
will be examined in Section 2.3.13.