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

Mixed Poisson Models for Claim Numbers 43
W = ̂ − 0 ̂̂ − 0
which is approximately 2 dim distributed under H 0, in large samples. The test statistic can
be interpreted as a measure of the distance between the maximum likelihood estimator ̂
and the hypothesized value 0 . The Wald test leads to the rejection of H 0 in favor of H 1 if
̂ is too far from 0 . Note that the Wald test suffers from the same problems as likelihood
ratio tests when 0 lies on the boundary of the parametric space.
Sometimes the calculation of the expected information is difficult, and we may use the
observed information instead.
Score Tests
Using the asymptotic theory, we have that U is approximately or0 distributed.
Therefore we can test H 0 = 0 versus H 1 ≠ 0 with the statistic
Q = U 0 I −1 0 U 0
which is approximately 2 dim distributed under H 0, in large samples.
The advantage of the score test is that the calculation of the maximum likelihood estimator
̂ is bypassed. Moreover, it remains applicable even if 0 lies on the boundary of the
parametric space.
Vuong Test
The Chi-square approximation to the distribution of the likelihood ratio test statistic is valid
only for testing restrictions on the parameters of a statistical model (i.e., H 0 and H 1 are
nested hypotheses). With non-nested models, we cannot make use of likelihood ratio tests for
model comparison. In this case, information criteria like AIC or (S)BIC are useful, as well
as the Vuong test for non-nested models. Recall that the AIC (Akaike Information Criteria),
is given by
AIC =−2L̂ + 2 dim
and the BIC (Bayesian Information Criteria), is given by
BIC =−2L̂ + lnn dim
Both criteria are equal to minus two times the maximum log-likelihood, penalized by a
function of the number of observations and sample size.
Vuong (1989) proposed a likelihood ratio-based statistic for testing the null hypothesis
that the competing models are equally close to the true data generating process against the
alternative that one model is closer. Consider two statistical models given by the probability
mass functions p· and q· with dim = dim, and define the likelihood ratio
statistic for the model p· against q· as
k∑
max
LR̂ n ̂ n = f k ln pk̂ n
qk̂ n
k=0