Consider bivariate observations $(X_1,Y_1), \ldots, (X_n,Y_n) \in \mathbb{R}\times \mathbb{R}$ with unknown conditional distributions $Q_x$ of $Y$, given that $X = x$. The goal is to estimate these distributions under the sole assumption that $Q_x$ is isotonic in $x$ with respect to likelihood ratio order. If the observations are identically distributed, a related goal is to estimate the joint distribution $\mathcal{L}(X,Y)$ under the sole assumption that it is totally positive of order two in a certain sense. An algorithm is developed which estimates the unknown family of distributions $(Q_x)_x$ via empirical likelihood. The benefit of the stronger regularization imposed by likelihood ratio order over the usual stochastic order is evaluated in terms of estimation and predictive performances on simulated as well as real data.

翻译：考虑双变量观测值$(X_1,Y_1), \ldots, (X_n,Y_n) \in \mathbb{R}\times \mathbb{R}$，其中$Y$的条件分布$Q_x$未知，给定$X = x$。目标是在$Q_x$关于$x$满足似然比序单调性的唯一假设下估计这些分布。若观测值独立同分布，则相关目标是在联合分布$\mathcal{L}(X,Y)$在某种意义下满足二阶完全正性的唯一假设下估计该分布。本文开发了一种算法，通过经验似然方法估计未知分布族$(Q_x)_x$。相较于通常的随机序，由似然比序施加的更强正则化在模拟数据和真实数据上从估计和预测性能两方面进行了评估。