On risk concentration for convex combinations of linear estimators

International audienceWe consider the estimation problem for an unknown vector beta epsilon R-p in a linear model Y = X beta + sigma xi, where xi epsilon R-n is a standard discrete white Gaussian noise and X is a known n x p matrix with n >= p. It is assumed that p is large and X is an ill-conditioned matrix. To estimate beta in this situation, we use a family of spectral regularizations of the maximum likelihood method (beta) over tilde (alpha)(Y) = H (alpha)(X (T) X) (beta) over cap (o)(Y), alpha epsilon R+, where (beta) over cap (o)(Y) is the maximum likelihood estimate for beta and {H (alpha)(center dot): R+ -> [0, 1], alpha a R+} is a given ordered family of functions indexed by a regularization parameter alpha. The final estimate for beta is constructed as a convex combination (in alpha) of the estimates (beta) over tilde (alpha)(Y) with weights chosen based on the observations Y. We present inequalities for large deviations of the norm of the prediction error of this method