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

70 Actuarial Modelling of Claim Counts
Considering (1.52), the easiest test statistic for H 0 against H 1 is certainly
T =
̂j
̂̂j
which is approximately or0 1 under H 0 , provided the sample size is large enough.
Alternatively, T 2 is approximately Chi-square with one degree of freedom. Rejection of H 0
occurs when T is large in absolute values, or when T 2 is large. In this case, j is significantly
different from 0, and the associated characteristic has a significant impact in ratemaking.
Note that, as always with hypothesis testing, statistical significance is the resultant of two
effects: firstly, the distance between the true parameter value and the hypothesized value,
and secondly, the number of observations (or more precisely, the amount of information
contained in the data). Even a hypothesis that is approximately true (and useful as a working
hypothesis) will be rejected with a sufficiently large sample. Conversely, any hypothesis may
fail to be rejected (and accepted as a working hypothesis) as long as the actuary has only
scanty data at his disposal. The reader should keep this in mind in the numerical illustrations
worked out in this book.
If the explanatory variables are correlated (as it is usually the case in actuarial studies),
it becomes difficult to disentangle the effects of one explanatory variable from another, and
the parameter estimates may be highly dependent on which explanatory variables are used in
the model. If the explanatory variables are strongly correlated then the maximum likelihood
estimators will have a large variance. The actuary should then reduce the set of regressors.
2.3.10 Confidence Interval for the Expected Annual Claim Frequency
It is possible to build a confidence interval for the annual claim frequency. Recall that
the multivariate Normal distribution has the following useful invariance property. Let C
be a given n × n matrix with real entries and let b be a n-dimensional real vector. If
X ∼ or M then Y = CX + b is orC + b CMC T . The variance of the predicted
score, ŝcore i = ˜x T i ̂, is thus given by
which is estimated by
Vŝcore i = ˜x T i ̂˜x i
̂ Vŝcore i = ˜x T i ̂̂˜x i
As the maximum likelihood estimator ̂ is approximately Gaussian when the number of
policies is large, ŝcore i is also Gaussian and an approximate confidence interval at level
1 − for the annual claim frequency can be computed as
[
exp
(̂T˜x i − z /2
√˜x i T ̂̂˜x
)
i exp
(̂T˜x i + z /2
√˜x i T ̂̂˜x
)]
i