# Maximum likelihood estimates for multivariate distributions

September 22, 2012
By

(This article was first published on Freakonometrics - Tag - R-english, and kindly contributed to R-bloggers)

Consider our loss-ALAE dataset, and – as in Frees & Valdez (1998) – let us fit a parametric model, in order to price a reinsurance treaty. The dataset is the following,

```> library(evd)
> data(lossalae)
> Z=lossalae
> X=Z[,1];Y=Z[,2]```

The first step can be to estimate marginal distributions, independently. Here, we consider lognormal distributions for both components,

```> Fempx=function(x) mean(X<=x)
> Fx=Vectorize(Fempx)
> u=exp(seq(2,15,by=.05))
> plot(u,Fx(u),log="x",type="l",
+ xlab="loss (log scale)")
> Lx=function(px) -sum(log(Vectorize(dlnorm)(
+ X,px[1],px[2])))
> opx=optim(c(1,5),fn=Lx)
> opx\$par
[1] 9.373679 1.637499
> lines(u,Vectorize(plnorm)(u,opx\$par[1],
+ opx\$par[2]),col="red")```

The fit here is quite good,

For the second component, we do the same,

```> Fempy=function(x) mean(Y<=x)
> Fy=Vectorize(Fempy)
> u=exp(seq(2,15,by=.05))
> plot(u,Fy(u),log="x",type="l",
+ xlab="ALAE (log scale)")
> Ly=function(px) -sum(log(Vectorize(dlnorm)(
+ Y,px[1],px[2])))
> opy=optim(c(1.5,10),fn=Ly)
> opy\$par
[1] 8.522452 1.429645
> lines(u,Vectorize(plnorm)(u,opy\$par[1],
+ opy\$par[2]),col="blue")```

It is not as good as the fit obtained on losses, but it is not that bad,

Now, consider a multivariate model, with Gumbel copula. We’ve seen before that it worked well. But this time, consider the maximum likelihood estimator globally.

```> Cop=function(u,v,a) exp(-((-log(u))^a+
+ (-log(v))^a)^(1/a))
> phi=function(t,a) (-log(t))^a
> cop=function(u,v,a) Cop(u,v,a)*(phi(u,a)+
+ phi(v,a))^(1/a-2)*(
+ a-1+(phi(u,a)+phi(v,a))^(1/a))*(phi(u,a-1)*
+ phi(v,a-1))/(u*v)
> L=function(p) {-sum(log(Vectorize(dlnorm)(
+ X,p[1],p[2])))-
+ sum(log(Vectorize(dlnorm)(Y,p[3],p[4])))-
+ sum(log(Vectorize(cop)(plnorm(X,p[1],p[2]),
+ plnorm(Y,p[3],p[4]),p[5])))}
> opz=optim(c(1.5,10,1.5,10,1.5),fn=L)
> opz\$par
[1] 9.377219 1.671410 8.524221 1.428552 1.468238```

Marginal parameters are (slightly) different from the one obtained independently,

```> c(opx\$par,opy\$par)
[1] 9.373679 1.637499 8.522452 1.429645
> opz\$par[1:4]
[1] 9.377219 1.671410 8.524221 1.428552```

And the parameter of Gumbel copula is close to the one obtained with heuristic methods in class.

Now that we have a model, let us play with it, to price a reinsurance treaty. But first, let us see how to generate Gumbel copula… One idea can be to use the frailty approach, based on a stable frailty. And we can use Chambers et al (1976) to generate a stable distribution. So here is the algorithm to generate samples from Gumbel copula

```> alpha=opz\$par[5]
> invphi=function(t,a) exp(-t^(1/a))
> n=500
> x=matrix(rexp(2*n),n,2)
> angle=runif(n,0,pi)
> E=rexp(n)
> beta=1/alpha
> stable=sin((1-beta)*angle)^((1-beta)/beta)*
+ (sin(beta*angle))/(sin(angle))^(1/beta)/
+ (E^(alpha-1))
> U=invphi(x/stable,alpha)
> plot(U)```

Here, we consider only 500 simulations,

Based on that copula simulation, we can then use marginal transformations to generate a pair, losses and allocated expenses,

```> Xloss=qlnorm(U[,1],opz\$par[1],opz\$par[2])
> Xalae=qlnorm(U[,2],opz\$par[3],opz\$par[4])```

In standard reinsurance treaties – see e.g. Clarke (1996) – allocated expenses are splited prorata capita between the insurance company, and the reinsurer. If $X$ denotes losses, and $Y$ the allocated expenses, a standard excess treaty can be has payoff

$\begin{cases} 0\cdot \boldsymbol{1}(X\leq R)\\ \displaystyle{\left(X-R+ \frac{X-R}{X}\cdot Y\right) \right )}\cdot\boldsymbol{1}(R

where $L$ denotes the (upper) limit, and $R$ the insurer’s retention. Using monte carlo simulation, it is then possible to estimate the pure premium of such a reinsurance treaty.

```> L=100000
> R=50000
> Z=((Xloss-R)+(Xloss-R)/Xloss*Xalae)*
+ (R<=Xloss)*(Xloss<L)+
+ ((L-R)+(L-R)/R*Xalae)*(L<=Xloss)
> mean(Z)
[1] 12596.45```

Now, play with it… it is possible to find a better fit, I guess…

To leave a comment for the author, please follow the link and comment on their blog: Freakonometrics - Tag - R-english.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...

If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Tags: , , , , , , , , , , , , , ,

Comments are closed.

## Sponsors

Contact us if you wish to help support R-bloggers, and place your banner here.

# Never miss an update! Subscribe to R-bloggers to receive e-mails with the latest R posts.(You will not see this message again.)

Click here to close (This popup will not appear again)