A Frequentistic Approach to Sequential Estimation in the General Linear Model
针对一般线性模型中的参数估计，提出一种序贯最小二乘估计方法，在观测成本趋近于零时，其风险与已知方差的最优固定样本方法几乎相同，并给出了停止时间的渐近分布和矩。
Abstract Independent observations Y 1, Y 2, … are generated via the general linear model Yi = Xi 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, σ2 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 σ2 is known. Its risk and the regret in not knowing σ2 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.