# Score-based inference for heterogeneous models

A heterogeneous model, with its – in most cases – analytically intractable likelihood, appears to be very “dark” for inference. On the other hand, the basic model is “enlightened” (its likelihood admits a closed-form). A digression may explain the method retained by the author.

## A short story

A man walks in a dark night. At some moment, he notices another man, bent to the ground, near a street lamp. He asks him: “what are you looking for The man near the street lamp (he is insane): “I am looking for my keys.” The passer-by: “did you lose them here?” The insane: “no, I lost them there, in the dark.” The passer-by: “so, why are you looking for them here?” The insane: “don’t be stupid! Because there is light here, of course!

### A more formal presentation

The pseudo-true value is here the limit of the m.l.e. on the basic model, whereas the data generating process includes heterogeneity with respect to this model. The idea is the following: we try to go from (tJ~, L) to the = (toll’ to 2), a consistent estimator of 9. Here, tJ~ is the m.l.e. for the basic model, and L is the Lagrangian with respect to 92, computed for the = (tJ?, O)6. This Lagrangian is the vector of residuals quoted in the preceding section.

are consistent estimators of 91 and 92 in the heterogeneous model. The parameters of the heterogeneous model are then consistently estimated from the m.l.e. on the basic model, which corresponds to the null hypothesis (no hidden information relevant for the explanation of the endogenous variables), and from related residuals. An important thing to notice is that is not bound to belong to 8 2, This property allows these estimators to be asymptotically normal (and efficient) under the null, although they converge in that case towards 0, which belongs to the boundary of 8 2, The author thinks that this property is not a drawback. Extreme estimators, obtained from

the maximization of an objective function (e.g. an explicit likelihood, or a likelihood approximated numerically or by simulation) will be obtained at the boundary of the parameter space if the heterogeneous model does not fit the data. In that case, prediction through the heterogeneous model is as impossible as with estimators obtained outside the parameter space. With the preceding method, the prediction approach.

This condition is easy to interpret because it can be expressed in terms of over dispersion, relative over dispersion, positive apparent contagion for residuals, etc. Besides, the probability that t)2 belongs to 8 2 can be consistently estimated under the null (see Piquet, 1997c).

### Examples of consistent estimators for heterogeneous models

We give examples for the heterogeneous models quoted in section 3.3.2. The estimators given here are explicit, consistent, asymptotically normal, and asymptotically efficient at the null. Remember that the null hypothesis is related to the basic model (no hidden information, relevant for the explanation of the severity variables

### Last word

a result that is not surprising if we think in terms of pseudo-true values. In the following tables are given numerical values for the predictors o~ the heterogeneity component (“bonus-malus” coefficients), related to three levels of At. The predictors for the second model are estimated from simulations. In France, the annual frequency of claims related to third-party liability in automobile insurance is equal to 0.07 on average.