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In this paper we present a sequential scheme for allocating doses so that asymptotically we achieve full efficiency for estimating LD,. This scheme is based on an algorithm due to Fedorov (1969) and Wynn (1970) and studied in the context of design by White (unpublished Ph.D. thesis, Imperial College, 1975), Wu and Wynn (1978), and Wu (1978a, 1978b). Bayesian optimal designs for the logistic regression problem treated here are discussed in Chaloner and Larntz (1989). An excellent exposition of the basic sequential algorithms as well as a general discussion of optimal design is found in Silvey (1980). In principle, the method we employ is applicable in a wide variety of problems in which design parameters cannot be optimally allocated without the knowledge of the underlying unknown parameters of the distribution. The versatility of this scheme and its adaptivity are its primary virtues. Suppose that observations are made sequentially at levels xl, x2, ..., with ni observations at dose xi. Suppose that the probability that a response is observed at dose x takes the form P{l(x O)j, where P(x) is some known distribution function and d and 0 are unknown parameters. Let pi be the proportion of responses observed at dose xi. Then the likelihood function based on pi, P2, n, P and on xi, x2, ..., x,Z is, with zi= (xi 6),
McLeish et al. (Thu,) studied this question.