Nonparametrics without tuning parameters: shape-constraints and mixture models

The talk is a glimpse back at the line of work that led from the research on shape-constrained statistical inference to opening new perspectives in mixture models, with outcomes in deconvolution and empirical Bayes prediction. Particular personal topics include s-unimodality in density estimation, shape-constrained aspects of empirical  Bayes prediction, and Kiefer-Wolfowitz nonparametric estimators of mixing distributions. Those unfamiliar with the subject may discover some data-analytic stimuli, new methods illustrated on the presented examples; those well in touch may recognize some recent developments. The unifying theme, and thus of an interest by itself, is the role played by the modern convex optimization methodology, both from the algorithmic and theoretical point of view.