Set estimation under convexity type assumptions
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Alberto Rodríguez Casal (2007)
Annales de l'I.H.P. Probabilités et statistiques
Vladimir Koltchinskii (2009)
Annales de l'I.H.P. Probabilités et statistiques
Let (X, Y) be a random couple in S×T with unknown distribution P. Let (X1, Y1), …, (Xn, Yn) be i.i.d. copies of (X, Y), Pn being their empirical distribution. Let h1, …, hN:S↦[−1, 1] be a dictionary consisting of N functions. For λ∈ℝN, denote fλ:=∑j=1Nλjhj. Let ℓ:T×ℝ↦ℝ be a given loss function, which is convex with respect to the second variable. Denote (ℓ•f)(x, y):=ℓ(y; f(x)). We study the following penalized empirical risk minimization problem which is an empirical version of the problem (hereɛ≥0...
Savitsky, Terrance, Vannucci, Marina (2010)
Journal of Probability and Statistics
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