Nonparametric estimation of the conditional distributions of a real response Y, given a real covariate X, is possible if one imposes that these conditional distributions are non-decreasing with respect to the usual stochastic order or the stronger likelihood ratio order. We discuss briefly these both paradigms. Then we present various results connecting the likelihood ratio ordering of conditional distributions with total positivity of order 2 (TP2) of bivariate distributions. These considerations lead to a maximum empirical likelihood estimator. We also discuss some conjectures about projections of arbitrary bivariate distributions onto the space of TP2 distributions.

We consider survival data with a cure fraction, meaning that some subjects never experience the event of interest. For example, in oncology the event of interest is cancer relapse/death and the cured patients after treatment are immune to such event. It is common in this context to use a mixture cure model, consisting of two sub-models: one for the probability of being uncured (incidence) and one for the survival of the uncured subjects (latency). Various approaches, ranging from parametric to nonparametric, have been proposed to model the incidence component, with the logistic model being the standard choice. We consider a monotone single-index model for the incidence, which relaxes the parametric logistic assumption, while maintaining interpretability of the regression coefficients and avoiding the curse-of-dimensionality. A new estimation method is introduced that relies on the profile maximum likelihood principle, techniques from isotonic regression and kernel smoothing. We discuss some unique and challenging issues that arise when incorporating the monotone single-index model within the mixture cure model. The consistency of the proposed estimator is established, and its practical performance is investigated through a simulation study and an application to melanoma cancer data.

We consider a linear regression model in which the regression coefficients are estimated by minimising the empirical risk based on a convex loss function. The accuracy of the estimator depends on the choice of loss function; for instance, when the errors are non-Gaussian, ordinary least squares can be outperformed by estimators based on alternative loss functions. A natural question then is how to select a data-driven convex loss function that leads to optimal downstream estimation of the regression coefficients. We propose a nonparametric approach that approximates the derivative of the log-density of the noise distribution by a decreasing function, and explicitly identifies the convex loss function for which the asymptotic variance of the resulting M-estimator is minimal. We show that this optimisation problem is equivalent to a version of score matching, which corresponds to a log-concave projection of the noise distribution not in the usual Kullback–Leibler sense, but instead with respect to the so-called Fisher divergence.

Consider these two problems. The first one: astronomers are interested in the study of the way stars are distributed in the universe. In this setting, globular clusters - spherical aggregations of stars held together by gravity - are a topic of particular focus. But how to determine the distribution of stars within these three-dimensional structures when given only a 2D photo of the clusters? The second one: having material that presents a globular microstructure and that cannot be scanned but only sectioned, how can we determine the distribution of particles' size inside the material when given only a limited number of 2D cross sections?

Despite their seemingly distant nature, both of these problems share a common thread: the inherent structure of a nonlinear inverse problem, originally conceptualized by Wicksell.

In this talk, we explore the properties of nonparametric Bayesian estimators based on the Dirichlet process prior for F in this problem, which is the first time the Dirichlet process is studied in an inverse setting. In particular, we illustrate contraction rates results and uncertainty quantification (Bernstein-von Mises type of results) for our methodology and we compare it to the state-of-the-art nonparametric shape-constrained estimator.