Stochastic approximation for functionals
D. L. Hanson, Ralph P. Russo · Lecture notes-monograph series · 1986
Suppose F_ is a class of distributions containing the discrete distributions and the distribution F for each real x Suppose φ is a real valued functional on _F and define θ(x) = φ(F ) so that θ(.) is a parameter of the family {F }.Fix α A stochastic approximation procedure for finding the x for which θ(x) = α is presented.When φ(F) is the mean of F, a form of this procedure is just the Robbins-Monro process.When φ(F) is the p-th quantile of F, a form of this procedure is just the quantile process introduced by the authors in an earlier paper.Some convergence theorems, examples, and generalizations are presented.1. Introduction.Suppose that for each real x (or for each x in some interval) there is a distribution F χ from which we can sample at will Suppose F^is a collection of distribution functions containing all empirical distribution functions (i.e., 1 n all distribution functions of the form F(t) =-Σ I, x (t)) and all of the \=i [ V"> distribution functions F χ .Let φ be a real valued functional on F_ and define θ(x) » φ(F ) so that θ(.) is some parameter of the family {F }.Our objective