A Course on Point Processes

Reiss, Rolf–Dieter

doi:10.1007/978-1-4613-9308-5

Cited by 200 publications

(139 citation statements)

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“…To be more precise, define Q i,n,f,f 0 , i = 1, 2, 3 to be the distributions of (y(t), t ∈ [0, 1]) in (27), (28), (29). Consider a "compound experiment" given by joint observations y 1 , .…”

Section: From Local To Global Resultsmentioning

confidence: 99%

Asymptotic equivalence of density estimation and Gaussian white noise

Nussbaum¹

1996

Ann. Statist.

215

221

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Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance ∆ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression (Brown and Low, 1993). We consider the analogous problem for the experiment given by n i. i. d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Hölder ball with exponent α > 1 2 and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift f 1/2 and variance 1 4 n −1 . This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i. i. d. model from white noise. As first applications we discuss exact constants for L 2 and Hellinger loss.

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Section: From Local To Global Resultsmentioning

confidence: 99%

Asymptotic equivalence of density estimation and Gaussian white noise

Nussbaum¹

1996

Ann. Statist.

215

221

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“…This and other definitions of the spatial Poisson processes as well as their properties and examples can be found in many books devoted to point processes (see, e.g., Daley and Vere-Jones [2], Krickeberg [5], Reiss [10], Ripley [11], Snyder and Miller [13]). The spatial Poisson processes are widely used in many fields.…”

Section: Preliminariesmentioning

confidence: 99%

Power Loss for Inhomogeneous Poisson Processes

Fazli

2011

Communications in Statistics - Theory and Methods

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In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on a real unknown parameter, we consider a simple hypothesis against a sequence of close (contiguous) alternatives. Under certain regularity conditions we obtain the power loss of the score test with respect to the Neyman-Pearson test. The power loss measures the performance of a second order efficient test by the help of third order asymptotic properties of the problem under consideration. AMS 1991 Classification: 62M05.

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“…The set Γ ω is the Poisson configuration (the support of the Poisson point process) with intensity measure ρdx dy, where ρ is a positive constant (for the definition of the Poisson point process, see e.g. Reiss [31] or Ando-Iwatsuka-KaminagaNakano [3]). The random variables {α γ } γ∈Γω are i.i.d.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%