The distribution of a linear predictor after model selection: conditional finite-sample distributions and asymptotic approximations

This paper studies the distribution of a linear predictor that is constructed after a data-driven model selection step in a linear regression model. The finite-sample cumulative distribution function (cdf) of the linear predictor is derived and a detailed analysis of the effects of the model selection step is given. Moreover, a simple approximation to the (complicated) finite-sample cdf is proposed. This approximation facilitates the study of the large-sample limit behavior of the linear predictor and its cdf, in the fixed-parameter case and under local alternatives. The focus of this paper is on the conditional distribution of a linear predictor, conditional on the event that a fixed (possibly incorrect) model has been selected. The unconditional distribution of a linear predictor is studied in the companion paper Leeb (The distribution of a linear predictor after model selection: unconditional finite-sample distributions and asymptotic approximations, Technical Report, Department of Statistics, University of Vienna, 2002).