Multivariate mixture representations for (weakly) likelihood ratio ordered distributions
In many applications, domain knowledge often suggests that two distributions should satisfy a stochastic order. Jauch et al. (2023) introduced univariate mixture representations for a distribution function G in terms of F when F <= G with respect to the likelihood ratio order. The ratio of two probability density functions is monotone if and only if one can be expressed as a mixture of one-sided truncations of the other. They proposed a Bayesian nonparametric density estimation model using Dirichlet Process Mixtures to enforce this constraint and applied it to medical data. We extend this framework using the weak likelihood ratio order, a multivariate generalization, first in the bivariate case and then to a full multivariate mixture representation. Additionally, we consider the case of sequences of K+1 stochastically ordered distributions F_0 <= F_1 <=, ..., <= F_K. These extensions enable multivariate Bayesian nonparametric logistic and ordinal regression models.
Keywords: Mixture representations Bayesian methods Monotone likelihood ratio order Stochastic order Regression models