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

Wojciech Niemiro

Statistics & Probability Letters, 2019, vol. 153, issue C, 151-156

Abstract: 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.

Keywords: Lévy processes; Relative error; Sequential estimation; Worst case efficiency; ε-α-approximation; Confidence estimation (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1016/j.spl.2019.06.009

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