5.1 The Robbins-Monro Procedure

Let Y(x) denote the response to a stimulus or dose level x and assume that Y(x) takes the value 0 or 1 with E[ Y( x)]= P{ Y( x)=1}= M( x) where M(x) is unknown. We wish to estimate ? such M( ?)= ? where ? is specified (0< ?<1). Next the Robbins-Monro (1951) procedure is as follows:

Guess an initial value x ₁ and let y _r( x _r) denote the response at x _r . Then choose x _{n +1} by the recursion formula:

(5.1.1)

where a _n, n=1, 2, is a decreasing sequence of positive constants and a _n tends to 0 as n tends to infinity. If we stop after n iterations, x _{n +1} will be the estimate of ?. Without loss of generality we can set ?=0. Then (5.1.1) becomes

(5.1.2)

A suitable choice for a _n is c/n where c is chosen optimally in some sense. Further it is not unreasonable to assume that M(x)>0 for all x>0. With a _n= c/n, Sacks (1958) has shown that is approximately normal with mean 0 and variance ? ² c ²/ (2c ? ₁