Constrained Polynomial Likelihood

TL;DR

This paper introduces a non-negative polynomial likelihood ratio method for comparing distributions using only moments, with applications to jump-diffusion processes and option pricing, emphasizing the importance of non-negativity constraints.

Contribution

It develops a novel polynomial likelihood ratio approach that incorporates shape restrictions and demonstrates its effectiveness in complex financial and stochastic models.

Findings

Sample PLR converges to the true population PLR under mild conditions.

Incorporating non-negativity improves estimation accuracy.

Applications show the method's utility in finance and stochastic processes.

Abstract

We develop a non-negative polynomial minimum-norm likelihood ratio (PLR) of two distributions of which only moments are known. The sample PLR converges to the unknown population PLR under mild conditions. The methodology allows for additional shape restrictions, as we illustrate with two empirical applications. The first develops a PLR for the unknown transition density of a jump-diffusion process, while the second extracts a positive density directly from option prices. In both cases, we show the importance of implementing the non-negativity restriction.