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 β$$ \beta $$ of coefficients. The estimation of a Pearson density is challenging, as small variations in β$$ \beta $$ can induce wild changes in the shape of the corresponding density fβ$$ {f}_{\beta } $$. The authors show how to estimate β$$ \beta $$ and fβ$$ {f}_{\beta } $$ 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. [ABSTRACT FROM AUTHOR]