Financial data are often thick-tailed and exhibit skewness. The versatile Generalized Tukey Lambda (GTL) distribution is able to capture varying degrees of skewness in thin- or thick-tailed data. Such versatility makes the GTL distribution potentially useful in the area of financial risk measurement. Moreover, for GTL-distributed random variables, the familiar risk measures of Value at Risk (VaR) and Expected Shortfall (ES) may be expressed in simple analytical forms. It turns out that, both analytically and through Monte Carlo simulations, GTL’s VaR and ES differ significantly from other flexible distributions. The asymptotic properties of the maximum likelihood estimator of the GTL parameters are also examined. In order to study risk in financial data, the GTL distribution is inserted into a GARCH model. This GTL-GARCH model is estimated with data on daily returns of GE stock, demonstrating that, for certain data sets, GTL may capture risk measurements better than other distributions. 1 1 Online supplementary materials consist of appendices with proofs and additional Monte Carlo results, data used in this study, an R script for fitting GTL densities by maximum likelihood, and an R script for estimation of the GTL-GARCH model (see Appendix A). 
Linking Tukey’s legacy to financial risk measurement