Proceedings Volume 2010 (format .pdf) - SimpBTH

arbitrarily chosen and succeding levels for each set of experiments are defined by:∞∞2⎡ ⎤n+ 1 n n n n,n0,n,n1 1X = X + a Y − Y a > a = ∞ a < ∞⎣ ⎦ ∑ ∑ , (7)where:1Xn= [ X1n,X2n] , Yn = [ Y1 n,Y2n ] and Yn = ⎡Yn , Y ⎤⎣ n ⎦, Yn = ( Y1 n+ Y2n ) . (8)2We assume that both experiments in each set are performed independently and wesuppose for all x andi = 1,2 , there exist V, A and B, positive real numbers, suchthat:2σi ( x)< V < ∞ , (9)i( ) .M x < A x + B < ∞ (10)this means that the averages and the variances are real numbers.There is a theorem which ensures that under the above-mentioned conditions:2P ⎡lim E Xn− θs= 0⎤= 1. (11)⎣n→∞⎦We remind the most important aspects in a procedure of stochasticapproximation. First of all, one will be interested in the convergence and mode ofconvergence of the sequence generated by the method to the desired solution of theequation. Next, one would like to know the asymptotic distribution of thesequence. Finally, one will be interested to know an optimum stopping rule for agiven situation.As a conclusion of what was said above, we mention that when a charactervariable depends on two factors, each of whom enforcement response is a randomvariable with mean, dispersion, distribution function and the correspondingregression and checking certain conditions, the approximation method ensures thatan iterative sequence of values assigned to variables which depends on theexperimental nature is convergent and the limit is that value of the variable forwhich we obtain optimal response.An alternative to stochastic approximation is Newton-Raphson method. Thisis a iterative technique of numerical analysis, which is frequently used in problems.M : a, b → a,b . We are interested in solving the followingLet M be a function [ ] [ ]equation:( )M x= α , (12)where the functional form of M is not known, but it is known that M is increasingX by recursive relation:and continuous. We construct a sequence { } n n 1≥330