Fixed relative precision estimators of growth rate for compound Poisson and Lévy processes

We consider compound Poisson processes or, more generally, Lévy processes X(t) with positive bounded jumps. The problem is to estimate the “growth rate” μ=EX(t)∕t with fixed relative precision, i.e. to construct an estimator μˆ such that P(|μˆ−μ|<με)≥1−α, for a given precision parameter ε and confidence parameter α, given a trajectory X(t) for 0≤t≤T. Such an estimator must be sequential, i.e. the length T of the observed trajectory must be random and chosen adaptively. Assume that the upper bound on jumps is known (w.l.o.g. equal to 1). We consider the estimator μˆr=r∕Tr, where Tr=min{t:X(t)≥r}, with a suitably chosen r=r(ε,α). We show that this estimator is “nearly worst case optimal” in a certain asymptotic sense, for ε→0 and α→0. The “worst case” turns out to be the process with jumps 1, i.e. the Poisson process with intensity μ.