ROBUST-TESTS IN GROUP SEQUENTIAL-ANALYSIS - ONE-SIDED AND 2-SIDED HYPOTHESES IN THE LINEAR-MODEL

Consider the linear model Y = Xtheta + E in the usual matrix notation where the errors are independent and identically distributed. We devel op robust tests for a large class of one- and two-sided hypotheses abo ut theta when the data are obtained and tests are carried out accordin g to a group sequential design. To illustrate the nature of the main r esults, let theta and theta be an M- and the least squares estimator o f theta respectively which are asymptotically normal about theta with covariance matrices sigma2(X(t)X)-1 and tau2(X(t)X)-1 respectively. Le t the Wald-type statistics based on theta and theta be denoted by RW a nd W respectively. It is shown that RW and W have the same asymptotic null distributions; here the limit is taken with the number of groups fixed but the numbers of observations in the groups increase proportio nately. Our main result is that the asymptotic Pitman efficiency of RW relative to W is (sigma2/tau2). Thus, the asymptotic efficiency-robus tness properties of theta relative to theta translate to asymptotic po wer-robustness of RW relative to W. Clearly, this is an attractive res ult since we already have a large literature which shows that theta is efficiency-robust compared to theta. The results of a simulation stud y show that with realistic sample sizes, RW is likely to have almost a s much power as W for normal errors, and substantially more power if t he errors have long tails. The simulation results also illustrate the advantages of group sequential designs compared to a fixed sample desi gn, in terms of sample size requirements to achieve a specified power.