文章基本信息

标题：Statistical inference across time scales
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作者：Céline Duval ; Marc Hoffmann
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2011
卷号：5
页码：2004-2030
DOI：10.1214/11-EJS660
语种：English
出版社：Institute of Mathematical Statistics
摘要：We consider a compound Poisson process with symmetric Bernoulli jumps, observed at times iΔ for i=0,1,… over [0,T], for different sizes of Δ=ΔT relative to T in the limit T→∞. We quantify the smooth statistical transition from a microscopic Poissonian regime (when ΔT→0) to a macroscopic Gaussian regime (when ΔT→∞). The classical quadratic variation estimator is efficient for estimating the intensity of the Poisson process in both microscopic and macroscopic scales but surprisingly, it shows a substantial loss of information in the intermediate scale ΔT→Δ∞∈(0,∞). This loss can be explicitly related to Δ∞. We provide an estimator that is efficient simultaneously in microscopic, intermediate and macroscopic regimes. We discuss the implications of these findings beyond this idealised framework.