Lambert W Random Variables and Their Applications in Loss Modelling

Several distributions and families of distributions are proposed to model skewed data, e.g., with skew-normal and related distributions. Lambert W random variables offer an alternative approach in which, instead of constructing a new distribution, a certain transformation is proposed. Such an approach allows the construction of a Lambert W skewed version from any distribution. Here, we choose the Lambert W normal distribution as a natural starting point and include the 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 the corresponding Lambert W transformed distributions is evaluated on real insurance data. Finally, the results are compared with those obtained using common loss distributions.

Automated summary

This paper proposes an alternative approach to modeling skewed data by using Lambert W random variable transformations instead of constructing new distributions. The Lambert W normal distribution and Lambert W exponential distribution are chosen as starting points, and their suitability in practical scenarios is evaluated using real insurance data.