(Received November 6, 1995; revised April 15, 1996)
Abstract. The problem of selecting one model from a family of linear models to describe a normally distributed observed data vector is considered. The notion of the model of given dimension nearest to the observation vector is introduced and methods of estimating the risk associated with such a nearest model are discussed. This leads to new model selection criteria one of which, called the ``partial bootstrap'', seems particularly promising. The methods are illustrated by specializing to the problem of estimating the non-zero components of a parameter vector on which noisy observations are available.
Key words and phrases: Linear model selection, risk estimation, little bootstrap estimator, Stein estimation, selection criteria.