A Frequentistic Approach to Sequential Estimation in the General Linear Model

Abstract

Independent observations Y 1 , Y 2 , … are generated via the general linear model Y i = X i β + ∈ i , X i being either a random or a sequentially designed matrix. The parameter β is estimated sequentially using the least squares estimator subject to a loss structure that is the sum of a cost due to sampling and a loss due to estimation error. If ∈ 1 , ∈ 2 , … are iid N (0, σ 2 I ) with σ unknown, no fixed-sample procedure will minimize the risk for all 0 < σ < ∞. The sequential procedure presented here performs nearly as well as the constant risk best fixed-sample procedure when σ 2 is known. Its risk and the regret in not knowing σ 2 are calculated up to terms that are o ( c ), as the cost of one observation c → 0 (or as σ → ∞). The asymptotic moments of the stopping time are given and the normalized stopping time's limiting distribution is shown to be normal. Its first two moments are calculated up to second-order terms.