Modeling heavy-tails with two-piece Burr distributions via conditional values-at-risk

A novel asymmetric heavy-tailed distribution termed “the two-piece Burr distribution” with real line support is constructed using an established framework of two-sided (TS) distributions. Notably, the two-piece Burr distribution possesses a closed form expression for its cumulative distribution function (cdf). It is a seven-parameter family of distributions with a mode parameter, two parameters in each tail that separately control the heaviness of the left and right tails, in addition to a scale parameter in each tail. Keeping one tail parameter fixed in each of the tails, three novel five-parameter heavy-tailed subfamilies are considered herein termed the two-piece Pareto, two-piece loglogistic and the two-piece paralogistic distribution. A novel algorithm is constructed that solves for the five parameters of these subfamilies, given left and right tail quantiles, a modal value, and left and right tail CVaR values. That is, the algorithm allows for modeling specific tail heaviness beyond the specified quantiles. While valuable on its own right, that solution may be used as a starting point for a five or seven parameter maximum likelihood routine. The estimation of the parameters is demonstrated using both methods for two data sets: one related to daily returns of S&P500 closing prices; and the other related to crop yields data to evaluate the effect of climate change adaptations.