Sequential Fixed-Width Confidence Interval Procedures for the Mean Under Multiple Boundary Crossings

The Anscombe-Ray-Chow-Robbins purely sequential procedure provided a breakthrough for constructing a fixed-width (=2d) confidence interval for a normal population mean mu with a confidence coefficient 1 - alpha when the standard deviation sigma was unknown. They proposed a purely sequential procedure of the following kind: Q(1) = inf {q(1) >= m(>= 2) : n(1) >= z(alpha/2)(2)S(q1)(2)/d(2)}, with the associated confidence interval I-Q1 equivalent to [(X) over bar (Q1) +/- d] for mu. But, a purely sequential procedure does not meet the exact consistency property. While Simons (1968) proved the existence of a universal fixed non negative integer r such that P-mu,P-sigma{mu is an element of [(X) over bar (Q1+r) +/- d]} >= 1 - alpha for all fixed mu, sigma, alpha and d, the magnitude of r has remained unknown. We introduce a new methodology along with appropriate truncation in which the number of additional observations required Q(1) is determined by the sequential sampling process itself. Interesting properties and performances of this methodology under successive crossings are explored theoretically and compared with those of the existing procedures by large-scale simulations.