Conference Agenda

Nonparametric maximum likelihood estimation in monotone binary regression models is studied when the impact of the features on the labels is weak. To define a notion of weak feature impact and to investigate the statistical behaviour of the nonparametric maximum likelihood estimator (NPMLE) in this context, we introduce a novel mathematical model. We prove consistency of the NPMLE in Hellinger distance, as well as its pointwise, $L^{1}$- and uniform consistency in the introduced weak feature impact model. Moreover, we derive consistency rates and limiting distributions respectivley. While consistency is shown to be independent of the level of feature impact, we observe a phase transition affecting both, convergence rates and limiting distributions, as a result of the level of feature impact.

Maximum likelihood estimation of a totally positive ($TP_2$) density is ill-defined due to the unboundedness of the likelihood function. An alternative is the consideration of a nonparametric maximum empirical likelihood estimator (NPMeLE) of $TP_2$ distributions. In terms of arbitrary distributions this refers to approximating a distribution with finite support optimally by a $TP_2$ distribution with respect to the Kullback-Leibler divergence. We study d-dimensional $TP_2$ distributions without assuming the existence of (Lebesgue-) densities and show for the 2-dimensional case that it is possible to characterise the NPMeLE solely by the bivariate distribution function. This leads naturally to some conjectures about projections of arbitrary bivariate distributions onto the space of $TP_2$ distributions. To further explore this, we investigate the continuity of the projection with respect to the Kolmogorov or Kuiper distance and the preservation of the $TP_2$ dependence structure for certain probability integral transforms. Finally, we connect the approach to nonparametric estimation of the conditional distributions of a real response given a real covariate under the assumption that these conditional distributions are non-decreasing with respect to the likelihood ratio order (isotonic distributional regression under likelihood ratio order).