Sparse estimation within Pearson's system, with an application to financial market risk

Pearson's system is a rich class of models that includes many classical univariate distributions. It comprises all continuous densities whose logarithmic derivative can be expressed as a ratio of quadratic polynomials governed by a vector $\beta$ of coefficients. The estimation of a Pearson density is challenging, as small variations in $\beta$ can induce wild changes in the shape of the corresponding density $f_{\beta }$. The authors show how to estimate $\beta$ and $f_{\beta }$ effectively through a penalized likelihood procedure involving differential regularization. The approach combines a penalized regression method and a profiled estimation technique. Simulations and an illustration with S&P 500 data suggest that the proposed method can improve market risk assessment substantially through value-at-risk and expected shortfall estimates that outperform those currently used by financial institutions and regulators.