Abstract: Pearson’s family of distributions consists of all continuous densities f which are solutions to the differential equation:

f' = −g_βf, where g_β(x) = (x − β1) / (β2 + β3x + β4x^2) for all x in a connected subset of the real line and β = (β1, β2, β3, β4) is a given vector.

It is a rich class of models which includes many classical distributions and which can accommodate both skewness and flexible tail behavior. However, estimation of a Pearson density is challenging because a small variation in β can induce a wild change in the shape of the solution fβ. In this talk, I will show how β and fβ can be estimated effectively through a penalized likelihood procedure incorporating Pearson’s differential equation. The approach relies on a parameter cascading method from the functional data analysis literature. 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 M. Carey (Dublin) and my colleague J.O. Ramsay.