A note on Hall's triple sampling procedure: A multiple sample second order sequential analogue of the Behrens–Fisher problem

In many statistical problems, the variances of the populations cannot be assumed to be equal. These inhomogeneity problems are often more difficult to handle than the corresponding homogeneity problems. In sequential estimation, this often means that only first order sequential procedures are available in the statistics literature for inhomogeneity problems. The purpose of this paper is to illustrate by using the classical Behrens–Fisher problem how to construct a second order sequential procedure using the batch sampling idea of Hall [Ann. Statist. 9 (1981) 1229–1238]; the cost of assuming variance inhomogeneity even when the two variances are equal turns out to be very limited. The approach of this paper can readily be applied to many other inhomogeneous problems.