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

Mixed Poisson Models for Claim Numbers 41
1.5.4 Hypothesis Tests
Sample Distribution of Individual Parameters
Standard hypothesis tests about parameters in maximum likelihood models are handled quite
easily, thanks to the asymptotic Normal distribution of the maximum likelihood estimator.
Specifically, we use the fact that
̂j − j
̂j
is approximately or0 1
where the standard deviation ̂j
of ̂ j is the square root of the jth diagonal element of
̂
= −1
Such tests will be useful in Chapter 2 to select the relevant risk factors.
In practice, often involves unknown parameters so that it is estimated by the jth
̂j
element of ̂̂j
̂̂
= ̂ −1
In such a case, ̂ j − j is approximately Student’s distributed with n − 1 degrees
/̂̂j
of freedom. This is the familiar z-score for a standard Normal variable developed in all
introductory statistics classes. The Normality of maximum likelihood estimates means that
our testing of hypotheses about the parameters is as simple as calculating the z-score and
finding the associated p-value from a table or by calling a software function.
The hypothesis test is based on Student’s t-distribution. However, because the maximum
likelihood properties are all asymptotic we are unable to address the finite sample distribution.
Asymptotically, the Student’s t-distribution converges to the Normal as the degrees for
freedom grow, so that using or0 1 p-values in the maximum likelihood test is the same
as the t-test as long as the number of cases is large enough. Specifically,
̂j − j
̂̂j
is approximately or0 1 (1.52)
if the sample size n is large enough.
In addition to the test of hypotheses about a single parameter, there are three classical
tests that encompass hypotheses about sets of parameters as well as one parameter at a time:
the likelihood ratio, Wald, and Lagrange multiplier tests. All are asymptotically equivalent,
but they differ in the ease of implementation depending on the particular case. Here, we will
present Wald, likelihood ratio and Vuong tests, as well as the Score test.
Likelihood Ratio Test
This test is based on a comparison of maximized likelihoods for nested models. Specifically,
the null hypothesis H 0 corresponds to a constrained model with dim − j parameters,
whereas the alternative H 1 corresponds to the full model with dim parameters. Most of