Asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems.

*(English)*Zbl 0667.62018Many 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 A. Wald [Ann. Math. Statist. 20, 595-601 (1949; Zbl 0034.229)] and 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 \(\{x^{\nu}\}\).

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 A. Wald [Ann. Math. Statist. 20, 595-601 (1949; Zbl 0034.229)] and 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 \(\{x^{\nu}\}\).

Reviewer: K.I.Yoshihara

##### MSC:

62F12 | Asymptotic properties of parametric estimators |

90C15 | Stochastic programming |

62A01 | Foundations and philosophical topics in statistics |