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

42 Actuarial Modelling of Claim Counts
the time, the test is performed with j = 1, so that we compare the full model to a simpler
one with one parameter less.
Let ˜ be the maximum likelihood estimator under H 0 , and let ̂ be the maximum likelihood
estimator under H 1 . The likelihood ratio test is based on the ratio of the likelihoods between
a full and a restricted (or reduced) nested model with fewer parameters. The restricted model
must be nested within (i.e., be a subset of) the full model. The likelihood ratio test statistic is
T = 2ln ̂ (
)
˜ = 2 L̂ − L˜
The evidence against H 0 will be strong when T is large.
The Chi-square distribution plays a prominent role in likelihood ratio tests. Recall that the
Gamma distribution with = /2 and = 1/2 for some positive integer is known as the
Chi-square distribution with degrees of freedom (which is denoted as 2 ), with associated
probability density function
fx = x/2−1 exp−x/2
( )
2 2
/2
x>0
If X ∼ 2 then its mean is , and its variance 2. It is useful to recall that the 2 distribution
is closely related to the Normal distribution. Specifically, the 2 arises as the distribution of
the sum of independent squared or0 1 random variables.
Under H 0 , the test statistic T is approximatively Chi-square distributed with degrees
of freedom equal to the number of parameters in the full model minus the number of
parameters in the restricted model (that is, with j degrees of freedom) when the sample size
n is sufficiently large (and additional mild regularity conditions are fulfilled). Note that the
likelihood ratio test requires us to perform two maximum likelihood estimations, one under
H 0 and another one under H 1 . When the largest model H 1 is misspecified (that is, the data
have not been generated by this probability model), the likelihood ratio statistic is no longer
necessarily Chi-square distributed under H 0 .
Unfortunately, there are cases where regularity conditions do not hold for T to be
approximately j
2 distributed under H 0 . In particular this happens when a constrained
parameter is on the boundary of the parameter space, e.g., testing Poisson versus Negative
Binomial. Here Poisson is a particular case of Negative Binomial when the latter has a
parameter on its boundary space. In this case, the limiting distribution of the statistic T
becomes a mixture of Chi-square distributions. We refer the reader to Titterington ET AL.
(1985) for more details about these situations.
Wald Tests
The Wald test provides an alternative to the likelihood ratio test that requires the estimation
of only the full model, not the restricted model. The logic of the Wald test is that if the
restrictions are correct then the unrestricted parameter estimates should be close to the value
hypothesized under the restricted model.
The Wald test is based on the distribution of a quadratic form of the weighted sum of
squared Normal deviates, a form that is known to be Chi-square distributed. Specifically,
using (1.49), we can test H 0 = 0 versus H 1 ≠ 0 with the statistic