On non-negative unbiased estimators

We study the existence of algorithms generating almost surely non-negative unbiased estimators of some quantities of practical interest. Our work is motivated by recent "exact approximation" methods to compute integrals with respect to any target distribution up to any arbitrary precision. These techniques can be implemented as long as non-negative unbiased estimators of the target density evaluations are available. In particular we show that given a non-constant function f from \R to \R^+ and real-valued unbiased estimators {X_n} of lambda in \R, there is no algorithm yielding almost surely non-negative unbiased estimators of f(lambda). Even if {X_n} is itself a.s. non-negative, then there is no algorithm yielding a.s. non-negative unbiased estimators of f(lambda) if f is not increasing. However if the support of {X_n} is a known interval of \R, then such algorithms exist as long as the function f satisfies some polynomial bound away from zero. The results are discussed in the light of several typical situations corresponding to different functions f.