Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems (Q1117623)

Many statistical problems, such as the problem of finding statistical estimators, are reduced to the following general class of problems: Find \(x^*\in R^ n\) that minimizes \(\int f(x,\xi)P(d\xi)\) where (\(\Xi\),A,P) is a probability space, P is known and \(f: R^ n\times \Xi \to R\cup \{+\infty \}\) is an extended real-valued function and \(\xi\) is a random variable with values in \(\Xi\). But, if we only have limited information available on P, then to estimate \(x^*\) we usually have to rely on the solution of an optimization problem of the following type: Find \(x^{\nu}\in R^ n\) that minimizes \(\int f(x,\xi)P^{\nu}(d\xi)\) where the measure \(P^{\nu}\) is not necessarily the empirical measure, but more generally the ``best'' approximate to P on the basis of the available information. The authors consider a sequence \(\{P^{\nu}(\cdot,\zeta)\), \(\nu =1,2,...\}\) of probability measures on (\(\Xi\),A) which are constructed by another sample \(\zeta\), independent of \(\xi\). Then, assuming that \(\{P^{\nu}(\cdot,\xi)\), \(\nu =1,2,...\}\) converges in distribution to P a. s., the authors obtain a sufficient condition which assures the existence of \(x^*\). This consistency result generalizes those of \textit{A. Wald} [Ann. Math. Statist. 20, 595-601 (1949; Zbl 0034.229)] and \textit{P. J. Huber} [Proc. 5th Berkeley Sympos. Math. Statist. Probab., Univ. Calif., 1965/1966, 1, 221-233 (1967)]. They also consider convergence rates in probabilistic terms and the asymptotic behavior of estimators \(\{x^{\nu}\}\).