Constrained Polynomial Likelihood

38 Pages Posted: 24 May 2021 Last revised: 29 Sep 2022

See all articles by Caio Almeida

Caio Almeida

Princeton University

Paul Schneider

University of Lugano - Institute of Finance; Swiss Finance Institute

Date Written: September 25, 2022


We develop a non-negative polynomial minimum-norm likelihood ratio (PLR) of two distributions of which only moments are known under shape restrictions. The PLR converges to the true, unknown, likelihood ratio under mild conditions. We establish asymptotic theory for the PLR coefficients and present two empirical applications. The first develops a PLR for the unknown transition density of a jump-diffusion process. The second modifies the Hansen-Jagannathan pricing kernel framework to accommodate non-negative polynomial return models consistent with no-arbitrage while simultaneously nesting the linear return model. In both cases, we show the value of implementing the non-negative restriction.

Keywords: Likelihood ratio, positive polynomial, Reproducing Kernel Hilbert Space (RKHS)

JEL Classification: C13, C51, C61

Suggested Citation

Almeida, Caio and Schneider, Paul Georg, Constrained Polynomial Likelihood (September 25, 2022). Swiss Finance Institute Research Paper No. 21-45, Available at SSRN: or

Caio Almeida (Contact Author)

Princeton University ( email )

26 Prospect Avenue
Princeton, NJ 08540
United States

Paul Georg Schneider

University of Lugano - Institute of Finance ( email )

Via Buffi 13
CH-6900 Lugano

Swiss Finance Institute ( email )

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4

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