The Skew Normal Distribution and Beyond
Regular readers of this column will recall that I have promoted the use of building loss reserve models using incurred data instead of paid data. While I still think that is appropriate, I suspect that I have been unfair. My original conclusion was based on using one particular model. It could be the case that some other model may work with paid data. A likely candidate might be a model with a payment year trend, which has been championed by the likes of Ben Zehnwirth for years now. One problem with such a model is that a payment year trend makes sense only with incremental paid data. Cumulative data contain settled claims that are unaffected by inflation. The problem with incremental paid data is that, at least occasionally, they contain negative claim amounts. This is a problem that many (including me) have ignored. We might consider using a normal distribution instead of the lognormal or the overdispersed Poisson distribution, but our data are skewed.
I was talking about this problem with Frank Schmid at last year’s CLRS, and he suggested using what he called the skew t distribution. Faced with the problem above, I decided to look into it.
It turns out that a special case of the skew t distribution, called the skew normal distribution, has been written about extensively.1 Wikipedia has a nice summary of it and there is even a skew normal R package, called “sn,” available on CRAN. One of the nicer articles I found on an Internet search is one by Frühwirth-Schnatter and Pyne titled “Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions.”2
The random variable X has a skew normal distribution if X=μ+ω•δ•Z+ω•√1-δ2•ε
where Z has a half normal distribution, i.e., a standard normal distribution truncated at zero, and ε has a standard normal (0,1) distribution. For reasons that will be made clear below, I prefer the hierarchical formulation X ~ normal (μ+ω•δ•Z,ω•√1-δ2).
In looking at either expression for the skew normal distribution, one can see that when δ = 0, the skew normal becomes a normal (μ, ω) distribution. As δ approaches one, the distribution gets more skewed and becomes a half normal distribution when δ = 1. Figure 1 plots the density functions for μ = 0, ω = 15 and two values of δ close to one.
It should be apparent that the coefficient of skewness can never exceed the coefficient of skewness of the half normal distribution, which is equal to 0.995. As it turns out, this constraint is important. I have fit stochastic loss reserve models with the skew normal distribution and found that, for most triangles, the coefficient of skewness was very close to its theoretical limit. This suggests that a distribution with a higher coefficient of skewness is needed.
The formulation of the skew normal distribution described by Frühwirth-Schnatter and Pyne suggests an alternative. Simply replace the half normal distribution with another skewed distribution, such as the lognormal distribution. Here is one way to do that. Define
X ~ normal(Z, δ), where Z ~ lognormal(μ,σ).
Let’s call this distribution the mixed lognormal-normal (ln-n) distribution with parameters given by δ, μ and σ. The density of X is calculated by numerically integrating3 out the Z.
where fX is the density function for a normal distribution and fZ is the density function for a lognormal distribution.
Figure 2 plots the density functions for μ = 2, σ = 0.6, and two different values of δ. The R code that produced the figures will be distributed with the Web version of this article.
I am suggesting that the distributions like those in Figure 2 will fit the (possibly negative) incremental paid losses in a loss development triangle. I will talk more about that in a future column.
1 One of the more active scholars on the skew normal distribution is Adelchi Azzalini, a statistics professor at the Università di Padova in Italy. His skew normal website is at http://azzalini.stat.unipd.it/SN/.
3 I have been using the MCMC software JAGS to fit stochastic loss reserve models, so I haven’t had to calculate the density function in practice.