Acknowledgment of priority: Usage of the Lambert W function in statistics

TL;DR

This paper acknowledges prior work on the Lambert W function's use in statistics, clarifies its earlier overlooked applications, and discusses its roles in distribution derivation and data transformation.

Contribution

It clarifies the historical use of the Lambert W function in statistics, highlighting its applications in distribution derivation and data modeling.

Findings

Lambert W function used in deriving likelihood ratio test distributions.

The function is effective for modeling skewed data.

Historical oversight of Lambert W in statistical literature is corrected.

Abstract

In my 2011 Annals of Applied Statistics article [Goerg (2011)] I wrote that "Whereas the Lambert $W$ function plays an important role in mathematics, physics, chemistry, biology and other fields, it has not yet been used in statistics." This was incorrect. At the time of publication I was unaware of Stehl\'{\i}k (2003), who used the Lambert $W$ function to derive the exact distribution of the likelihood ratio test statistic. He has also used it in more recent work such as Stehl\'{\i}k (2006), Stehl\'{\i}k et al. (2010), or Stehl\'{\i}k (2014) amongst others. While Stehl\'{i}k's use of the Lambert $W$ function was focused on the distribution of the likelihood ratio test statistic, my work dealt with the modeling of skewed random variables and symmetrizing data using the Lambert $W$ function as a variable transformation.