Model Selection for Estimating the Non Zero Components of a Gaussian Vector

We propose a method based on a penalised likelihood criterion, for estimating the number on non-zero components of the mean of a Gaussian vector. Following the work of Birgé and Massart in Gaussian model selection, we choose the penalty function such that the resulting estimator minimises the Kullback risk. Mathematics Subject Classification. 62G05, 62G09. Received January 13, 2004. Revised September 28, 2005. Introduction The following regression model is considered: X = m + τε, ε ∼ Nn(0, In), where X = (X1, . . . Xn) is the vector of observations. The expectation of X, say m = (m1, . . . ,mn) , and the variance τ are unknown. Assuming that some of the components of m are equal to zero, our objective is to estimate the number of zero components as well as their positions. We propose an estimation method based on a model choice procedure. We denote by J a subset of Jn = {1, 2, . . . , n} with dimension kJ , and we consider the collection J of all subsets of Jn with dimension less than kn for some kn less than n: J = {J ⊂ Jn, kJ ≤ kn} . Let x = (x1, . . . , xn) , then for each subset J ∈ J we denote by xJ the vector in R whose component i equals xi if i belongs to J and 0 if not. We denote by ‖x‖2 the Euclidean distance in R and we set ‖x‖n = ‖x‖2/n. For each subset J in the collection J , assuming that m = mJ , the maximum likelihood estimators of the parameters (m, τ) are (XJ , ‖XJc‖n), where J denotes the complement of J in Jn. We thus define a collection of estimators, (XJ , ‖XJc‖n) and the problem is now to choose an estimator of (m, τ) in this collection, or equivalently, to choose the best J in J , say Ĵ , and to take (m̂, τ̂) = (XĴ , ‖XĴc‖n). We associate to each estimator in the collection a risk defined as R(J) = E {K(m,τ2) (XJ , ‖XJc‖2n)} ,