✍ Scribed by Peter Mathé
No coin nor oath required. For personal study only.
For many typical instances where Monte Carlo methods are applied attempts were made to find unbiased estimators, since for them the Monte Carlo error reduces to the statistical error. These problems usually take values in the scalar field. If we study vector valued Monte Carlo methods, then we are confronted with the question of 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.
1996 Academic Press, Inc.
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.
The ``crude Monte Carlo integration'', cf. [HH64, Chapter 5.2], is certainly the most prominent example. Suppose we want to compute the integral I( f ) := 0 f(|) d+(|) for some (square integrable) function f: 0 Ä R and probability +. We may regard the mapping f as a real valued random variable. Then the expectation of this random variable is just I( f ). This may be rephrased by stating that | Ä f (|) is an unbiased Monte Carlo method for the functional I( f ). The basic result within the classical theory claims, that the sample mean of n independent copies of | Ä f (|) article no. 0025
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