Corrected confidence intervals for parameters in adaptive linear models

Consider an adaptive linear model yt=xt′θ+σet, where xt=(xt1,…,xtp)′ may depend on previous responses. Woodroofe and Coad [1999. Corrected confidence sets for sequentially designed experiments: examples. In: Ghosh, S. (Ed.), Multivariate Analysis, Design of Experiments, and Survey Sampling. Marcel Dekker, Inc., New York, pp. 135-161] derived very weak asymptotic expansions for the distributions of an appropriate pivotal quantity and constructed corrected confidence sets for θ, where the correction terms involve the limit of ∑t=1nxtxt′/n (as n approaches infinity) and its derivatives with respect to θ. However, the analytic form of this limit and its derivatives may not be tractable in some models. This paper proposes a numerical method to approximate the correction terms. For the resulting approximate pivot, we show that under mild conditions the error induced by numerical approximation is op(1/n). Then, we assess the accuracy of the proposed method by an autoregressive model and a threshold autoregressive model.