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Statistical inference across time scales

Abstract : We consider a compound Poisson process with symmetric Bernoulli jumps, observed at time i Delta for i = 0, 1, ... over [0, T], for different sizes of Delta = Delta(T) relative to T in the limit T -> infinity. We quantify the smooth statistical transition from a microscopic Poissonian regime (when Delta(T) -> 0) to a macroscopic Gaussian regime (when Delta(T) -> infinity). The classical quadratic variation estimator is efficient for estimating the intensity of the Poisson process in both microscopic and macroscopoic scales but surprisingly, it shows a substantial loss of information in the intermediate scale Delta(T) -> Delta(infinity) is an element of (0, infinity). This loss can be explicitly related to Delta(infinity). 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.
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Submitted on : Tuesday, May 1, 2012 - 7:21:26 PM
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Celine Duval, Marc Hoffmann. Statistical inference across time scales. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2011, 5 (?), pp.2004--2030. ⟨10.1214/11-EJS660⟩. ⟨hal-00692996⟩



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