Convex optimization in the theory and practice of statistical estimation, prediction, and inference

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The proposed research elaborates on the experience the applicant gained in about past ten years of his research, concentrating on methods where convex, especially constrained and non-differentiable optimization plays a pivotal role. The impact of convex optimization is not limited only to the algorithmic aspects of the considered methods, but transcends also into their theory - the mathematics of convex optimization, in particular the study of the dual versions of the investigated formulations, is employed to obtain additional theoretical insights. These include viable strategies for the inference, the probabilistic evaluation of their results. ***From the statistical point of view, the unifying aspect is the study of methods involving shrinkage/regularization, a prominent place occupied by compound decision (empirical Bayes) problems and "sparsity-promoting" regularization methods (like various versions of basis pursuit/lasso, adaptive lasso, Dantzig selector, nonnegative garrote). The proposed topics include nonparametric estimation of mixtures (improved algorithms for compound decision problems capable of dealing with the curse of dimensionality); additional shape constrained approaches in empirical Bayes prediction (mixtures with unimodality restrictions); regularized periodogram analysis (including also l1 and quantile periodograms and l1 and l2 penalties) with applications to period hunting in unequally spaced time series data; the simultaneous use of l2 and l1 penalties motivated by statistical considerations in linear models (hierarchically organized sparse models in the analysis of designed experiments); nonparametric regression (additive models in the same variable arising from change-point considerations); and topics in quantile and composite quantile regression.**