Penalized likelihood estimation for Pearson's family of distributions, with an application to financial market risk
Pearson’s family is a system of continuous univariate probability distributions whose log-density is a valid solution to a differential equation involving a ratio of quadratic polynomials with unknown coefficients. This rich class of distributions, which includes many classical models, can accommodate both skewness and flexible tail behavior. However, estimation of a Pearson density is challenging because a small variation in the parameters can induce a wild change in the shape of the solution. In this talk, I will show how these parameters and the corresponding solution can be estimated effectively through a penalized likelihood procedure incorporating Pearson’s differential equation. The approach relies on a parameter cascading method initially developed in the context of functional data analysis. Simulations and an illustration involving the S&P 500 index will show that it leads to estimates of Value-at-Risk and Expected Shortfall that can substantially improve market risk assessment by outperforming the estimates currently used by financial institutions and regulators. This talk is based on joint work with Michelle Carey (Dublin) and Jim Ramsay.