Lambert W random variables and their applications in loss modelling

TL;DR

This paper explores Lambert W transformations to create skewed distributions from existing ones, focusing on their mathematical properties and practical application in insurance loss modeling, offering an alternative to traditional skewed distributions.

Contribution

It introduces Lambert W skewed distributions derived from normal and exponential distributions, analyzing their properties and demonstrating their effectiveness on real insurance data.

Findings

Lambert W skewed distributions can model skewness effectively.

These distributions exhibit heavy tails suitable for loss data.

They outperform some traditional loss models in empirical tests.

Abstract

Several distributions and families of distributions are proposed to model skewed data, think, e.g., of skew-normal and related distributions. Lambert W random variables offer an alternative approach where, instead of constructing a new distribution, a certain transform is proposed (Goerg, 2011). Such an approach allows the construction of a Lambert W skewed version from any distribution. We choose Lambert W normal distribution as a natural starting point and also include Lambert W exponential distribution due to the simplicity and shape of the exponential distribution, which, after skewing, may produce a reasonably heavy tail for loss models. In the theoretical part, we focus on the mathematical properties of obtained distributions, including the range of skewness. In the practical part, the suitability of corresponding Lambert W transformed distributions is evaluated on real insurance data. The results are compared with those obtained using common loss distributions.