A Stochastic Newton-Raphson Method

A stochastic approximation procedure of the Robbins-Monro type is considered. The original idea behind the Newton-Raphson method is used as follows. Given n approximations X(1),...,X(n) with observation Y(1),...,Y(n), a least squares line is fitted to the points (X(m),Y(m)),...,(X(n),Y(n)) where m n...

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Zusammenfassung:A stochastic approximation procedure of the Robbins-Monro type is considered. The original idea behind the Newton-Raphson method is used as follows. Given n approximations X(1),...,X(n) with observation Y(1),...,Y(n), a least squares line is fitted to the points (X(m),Y(m)),...,(X(n),Y(n)) where m n may depend on n. The (n+1)st approximation is taken to be the intersection of the least squares line with y = 0. A variation of the resulting process is studied. It is shown that this process yields a strongly consistent sequence of estimates which is asymptotically normal with minimal asymptotic variance.