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.

翻译：我们针对仅知矩信息的两个分布，提出一种非负多项式最小范数似然比（PLR）。在温和条件下，样本PLR收敛于未知总体PLR。该方法允许施加额外形状约束，我们通过两个实证应用予以说明：其一是为跳扩散过程的未知转移密度构建PLR，其二是直接从期权价格中提取正密度函数。在两种情形中，我们均展示了实施非负性约束的重要性。