## Approximation theory of Monte Carlo methods

- P. Math e
- Habilitation thesis,
- 1994

1 Excerpt

- Published 1995

For many typical instances where Monte Carlo methods are applied attempts were made to nd unbiased estimators, since for them the Monte Carlo error reduces to the statistical error. These problems usually take values in the scalar eld. If we study vector valued Monte Carlo methods, then we are confronted with the question whether there can exist unbiased estimators. This problem is apparently new. Below it is settled precisely. Partial answers are given, indicating relations to several classes of linear operators in Banach spaces. 1. Introduction, Notation In many practical applications the program designer is confronted with the "curse of dimensionality", an exponential dependence on the dimension, which is inherent in most error estimates provided by classical numerical analysis, see e.g. TWW88] for a sample of typical numerical problems and the respective error estimates. Often this can be overcome by choosing Monte Carlo methods, i.e., numerical methods involving random parameters in the computational process, see HH64] for an excellent, by now classical treatment on the applicability of Monte Carlo methods. Within the classical theory one prefers unbiased Monte Carlo estimators, since they are self-focusing if the numerical simulation is repeated. Such unbiased estimators are often hard to nd, see KW86, ENS89]. In statistics, the problem whether there are unbiased estimates (in statistical sense) has also been attractive, recently. While it has been known since 1956 that there cannot exist such unbiased estimates for density estimation, see Ros56], advantage has been made by LB93]. They developed a machinery which enables to prove that there are no unbiased l-informative estimates for singular problems. Within our framework, the notion of l-informativity corresponds to the requirement of nite errors. On the other hand, all problems studied below are not singular in their sense. This may express that our arguments are completely diierent, indicating geometric properties of the target space, where the error is measured. Below we are concerned with vector valued Monte Carlo methods for the randomized approximation of linear mappings. The classical results extend easily from real valued

@inproceedings{MATH1995UnbiasedMC,
title={Unbiased Monte Carlo Estimators 3},
author={PETER MATH},
year={1995}
}