Stochastic Approximation and NonLinear Regression [Paperback]

This monograph addresses the problem of real-time curve fitting in the presence of noise, from the computational and statistical viewpoints. It examines the problem of nonlinear regression, where observations are made on a time series whose mean-value function is known except for a vector parameter. In contrast to the traditional formulation, data are imagined to arrive in temporal succession. The estimation is carried out in real time so that, at each instant, the parameter estimate fully reflects all available data. Specifically, the monograph focuses on estimator sequences of the so-called differential correction type. The term differential correction refers to the fact that the difference between the components of the updated and previous estimators is proportional to the difference between the current observation and the value that would be predicted by the regression function if the previous estimate were in fact the true value of the unknown vector parameter. The vector of proportionality factors (which is generally time varying and can depend upon previous estimates) is called the gain or smoothing vector. The main purpose of this research is to relate the large-sample statistical behavior of such estimates (consistency, rate of convergence, large-sample distribution theory, asymptotic efficiency) to the properties of the regression function and the choice of smoothing vectors. Furthermore, consideration is given to the tradeoff that can be effected between computational simplicity and statistical efficiency through the choice of gains.

Part I deals with the special cases of an unknown scalar parameter-discussing probability-one and mean-square convergence, rates of mean-square convergence, and asymptotic distribution theory of the estimators for various choices of the smoothing sequence. Part II examines the probability-one and mean-square convergence of the estimators in the vector case for various choices of smoothing velu