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.

翻译：我们开发了一种非负多边最低-北位概率比(PLR),其中两种分布方式在形状限制下只知道瞬间。PLR在温和条件下会与真实的、未知的、可能的比例相融合。我们为PLR系数建立了无症状理论,并提出了两种经验应用。第一种为跳跃扩散过程的未知过渡密度开发了一个PLR。第二种修改汉森-贾甘纳汉定价内核框架,以适应非负多边回报模式,使之与无套利模式一致,同时嵌入线性返回模式。在这两种情况下,我们展示了执行非负性限制的价值。