2014/05/31 by Pierre C. Bellec · 6 citations
Decision Sciences · Mathematics · #Adaptive estimator #Advanced Causal Inference Techniques #Context (archaeology) #Convex hull #Empirical risk minimization #Estimator #Exponential family #Exponential function #Large deviations theory #Minification #Minimax #Risk and Portfolio Optimization #Statistical Methods and Inference #math.ST #stat.TH
paper · pdf · doi:10.3150/15-bej742
published in Bernoulli 23(1) (Chapman and Hall London) · Published at http://dx.doi.org/10.3150/15-BEJ742 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)
openalex created_date 2016/06/24 · openalex publication_date 2016/09/27 · arxiv created 2016/09/28 · arxiv updated 2016/09/29 · openalex updated_date 2026/08/06
We consider the problem of model selection type aggregation in the context of density estimation. We first show that empirical risk minimization is sub-optimal for this problem and it shares this property with the exponential weights aggregate, empirical risk minimization over the convex hull of the dictionary functions, and all selectors. Using a penalty inspired by recent works on the Q-aggregation procedure, we derive a sharp oracle inequality in deviation under a simple boundedness assumption and we show that the rate is optimal in a minimax sense. Unlike the procedures based on exponential weights, this estimator is fully adaptive under the uniform prior. In particular, its construction does not rely on the sup-norm of the unknown density. By providing lower bounds with exponential tails, we show that the deviation term appearing in the sharp oracle inequalities cannot be improved.