A symptotic robustness of sequential confidence intervals and the connection with influence functions

Abstract

The robustness of sequential confidence intervals is studied by considering contamination with probability ε of the basic underlying distribution in a so-called gross errors model. Asymptotic theory is considered when d → 0, where the prescribed length of the interval is 2d, and simultaneously ε ≏ ε(d) → 0. A general theorem, in a distribution free setting, is given which provides expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable. The results depend on the rate of ε(d)/d as d → 0 and on the contaminating distribution. If the latter distribution is degenerate, it turns out that the influence functions of the above mentioned two estimators used in the construction of the procedure, appear in the expressions for the asymptotic coverage probability and the asymptotic distribution of the stopping variable respectively. This shows how the sequential procedure inherits the robustness properties of the estimators concerned and how this is quantified. The general theorem is specialized to two procedures for the estimation of the mean of a symmetric distribution. Results of Monte Carlo studies indicate agreement between the asymptotic theory and the actual behavior of the procedures.