CASE WESTERN RESERVE UNIVERSITY

STATISTICS COLLOQUIUM

Consider a line& model Yk = 9'Xk + u~, k = ~ where 9 = ....... , 9~)' is a vector of unknown parameters, the errors are i.i.d. standard normal random variables, and each design variable Xk may be a function of previous responses, say Xk = xk(y1,... , yk-i), k = 1, ~ The adaptive design does not affect the likelihood function, and the formula for the maxi mum likelihood estimator is the same in this case as in the non-adaptive one in which X1, X2, are constants. However, the adaptive design may affect the sampling distribution of the maximum likelihood estimator, and may do so dramatically. Denote the design matrices by Xfl = (x1,... , x~)', write Xfl'Xfl = B~B~', where B~ is a p x p matrix, and let Zfl = B~(9 - &), where 9~ denotes the maximum likelihood estimator of 9. Then Zfl is approxi mately normal with a mean vector and covariance matrix that are slightly different from 0 and the identity to order o(1/n). Using estimates of the mean vector and covariance matrix to restandardize Zfl leads to confidence sets of high likelihood with confidence levels that differ from their nominal values by o(1/~). The justification for these assertions is in the very weak sense of Woodroofe (1986, Ann. Statist, 14,1049-1067), and the proofs use Bayesian mathematics to arrive at frequentist conclusions.