Asymptotic Behavior of Optimal Weighting in Generalized Self-Normalization for Time Series

Self-normalization has been celebrated for its ability to avoid direct estimation of the nuisance long-run variance and its versatility in handling the mean and other quantities. The self-normalizer in its original form uses only recursive estimators of one direction, and generalizations involving both forward and backward estimators were recently given. Unlike existing results that weigh the forward and backward estimators in a deterministic manner, the current article focuses on a data-driven weight that corresponds to confidence intervals with minimal lengths. We study the asymptotic behavior of such a data-driven weight choice, and find an interesting dichotomy between linear and nonlinear quantities. Another interesting phenomenon is that, for nonlinear quantities, the data-driven weight typically distributes over an uncountable set in finite-sample problems but in the limit it converges to a discrete distribution with a finite support.