# Régression de Poisson, et biais minimal

Lors du prochain cours d’actuariat, nous allons finir les arbres de régression, et introduire la régression de Poisson. Les transparents sont en ligne ici,

Je vais présenter la régression en Poisson, en faisant un parallèle avec la régression logistique, la session suivante portera sur la généralisation obtenue avec les modèles linéaires généralisés. Sur la régression de Poisson, je suggère de lire Frees (2010) chapitre 12 (p 343-361), Greene (2012), section 18.3 (p 802-828) ou encore de Jong Heller (2008) chapitre 6. Sur les méthodes de biais minimal, de Jong Heller (2008), section 1.3 et l’article de Sholom Feldblum, http://www.casact.org/…. Sur le passage de ces dernières méthodes (introduites par Robert Bailey dans les années 60, http://www.casact.org/… et http://www.casact.org/…), je recommande la lecture de l’article de Ben Zehnwirth, Ratemaking From Bailey and Simon (1960) to Generalized Linear Regression Models, en ligne sur http://www.casact.org/…

Comme annoncé au premier cours, j’essaye de mettre en ligne les transparents au fur et à mesure, mais j’avais pris l’habitude d’écrire au tableau ces dernières années. Il faut donc que je tape tout. Pour le devoir un courriel sera envoyé d’ici la fin de semaine à tous les groupes qui se sont inscrits.

# Words are important (when dealing with statistics). But still.

In statistics, it might be difficult to know what a symbol stands for. For instance, $\widehat{\theta}$ can either be a real value, i.e. the value taken by a statistics from a given sample. But it can also be a random variable, assuming that the sample is now a collection of i.i.d. random variables. We can usually distinguish $x_i$‘s (values from a given sample) and $X_i$‘s (the underlying random variables, i.e. $x_i=X_i(\omega)$ for some $\omega\in\Omega$). But notations might confusing, and it is hard to distinguish random variables, and values taken by random variables (or realizations). But usually, if we look at the context, one can figure out what symbols stand for.

But sometimes, it is difficult to get a proper definition, not for some symbols, but for words. And most of the time common words. Recently, I wrote a short paper, claiming that it was difficult to model the number of bodily injuries related to car accident, since it is difficult to define death. Actually, the definition of dead did change a few years ago (as weird as it might sound), which did cause a rupture of some series.

I recently had a similar story, discussing with a pharmacist in Montréal who said to me “you French are known to be the world’s champion in terms of drug consumption“, see e.g.

• The French are Europe’s champion medicine-takers” in http://economist.com/…, mentioning “heavy drug-consumption culture
• The data show that drug consumption in France remains one of the largest in Europe” in http://bizcovering.com/…
• France has one of the largest drug markets in the world and the drug consumption per capitahttp://ispor.org/… (among so many articles)

I do not think I am a drug addict (I might be – like most of my colleagues – a coffee addict, but as Paul Erdős  – or more probably Alfréd Rényi – said once, “a mathematician is a device for turning coffee into theorems“). The main problem here is the notion of “consumption“. The economics interpretation is simply that someone buys a product or a service (see http://dictionary.reference.com/…). There is also the food-related interpretation, where consuming means ingesting, i.e. eating or drinking (see http://dictionary.cambridge.org/…).

So pill and drug “consumption” is ambiguous: is it the number of pills purchased, or ingested (actually consumed), or prescribed? The first thing one should remember is that the Social Security in France refunds (almost) all medications prescribed by a doctor. So it is uncommon to leave the office of a doctor without a prescription, at least of aspirin: a visit to the doctor is usually, in France, the opportunity to stock some over-the-counter drugs. The second thing is that there is a major difference between France and North America when we go to the pharmacy. In Montréal for instance, if I have a prescription for 12 pills, then the pharmacist does give me exactly 12 pills (from a big pot). In France, pills are sold in prepackaged boxes, so if the box contains 10 pills, I will get 2 boxes, just to be sure I’ll get my 12 pills. From a medical point of view, I will consume my 12 pills, but from an economic perspective, I will consume 20. So comparing statistics is extremely difficult, not because the the maths, but because it is difficult to define (even simple) concepts.