I Fought the (distribution) Law (and the Law did not win)

A few days ago, I was asked if we should spend a lot of time to choose the distribution we use, in GLMs, for (actuarial) ratemaking. On that topic, I usually claim that the family is not the most important parameter in the regression model. Consider the following dataset

```> db <- data.frame(x=c(1,2,3,4,5),y=c(1,2,4,2,6))
> plot(db,xlim=c(0,6),ylim=c(-1,8),pch=19)```

To visualize a regression model, use the following code

```> nd=data.frame(x=seq(0,6,by=.1))
+ prd1=predict(reg,newdata=nd,se.fit = TRUE,type="response")
+ y1=prd1\$fit
+ y1_upp=prd1\$fit+prd1\$residual.scale*1.96*
prd1\$se.fit
+ y1_low=prd1\$fit-prd1\$residual.scale*1.96*
prd1\$se.fit
+ polygon(c(nd\$x,rev(nd\$x)),c(y1_upp,
rev(y1_low)),col="light green",angle=90,
density=40,border=NA)
+ lines(nd\$x,y1,col="red",lwd=2)
+ }```

For instance, with a Poisson regression (with a log link function) we get

```> plot(db)
+ data=db)
```

while, with a Gaussian regresion (but still with a log link function), we get

```> plot(db)
+ data=db)

If we just care about the expected value of our prediction, the output is more or less the same

```> plot(db)
> lines(nd\$x,predict(reg1,newdata=nd,
+ type="response"),col="red",lwd=1.5)
> lines(nd\$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)```

So, indeed, forget about the (distribution) law when running a GLM. Not convinced? Consider – on the same dataset – a Poisson regression (with an identity link function this time)

```> plot(db)
+ data=db)
```

while, with a Gaussian regresion (but still with an identity link function), we get

```> plot(db)
+ data=db)

Again, if we just plot the expected value of our prediction, the output is more or less the same

```> plot(db)
> lines(nd\$x,predict(reg1,newdata=nd,
+ type="response"),col="red",lwd=1.5)
> lines(nd\$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)```

So clearly, the simplistic message you should not care too much about the (distribution) law seems to be valid…

Exposé à Rennes 1

Exposé demain matin au workshop de micro-économie, à Rennes. Je présenterais les travaux récents, avec Emmanuel Flachaire, sur le thème Kernel Based Estimation of Inequality Indices and Risk Measures. Les slides sont en ligne.