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Tail Probability via Tube Formula and Euler Characteristic Method when Critical Radius is Zero

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Author Info
Akimichi Takemura (Faculty of Economics, University of Tokyo)
Satoshi Kuriki (The Institute of Statistical Mathematics)
Abstract

In Takemura and Kuriki(1999b) we have established that the tube formula and the Euler characteristic method give identical and valid asymptotic expansion of tail probability of the maximum of Gaussian random field when the random field has finite Karhunen-Loeve expansion and the index set has positive critical radius. The purpose of this paper is to show that the positiveness of the critical radius is an essential condition. Namely, we prove that if the critical radius is zero, only the main term is valid and other higher order terms are generally not valid in the formal asymptotic expansion based on the tube formula or the Euler characteristic method. Our examples show that index sets with zero critical radius are commonly used in statistics.

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File URL: http://www.e.u-tokyo.ac.jp/cirje/research/dp/99/cf59/contents.htm
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Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-59.

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Length: 21 pages
Date of creation: Sep 1999
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Handle: RePEc:tky:fseres:99cf59

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  1. Satoshi, Kuriki & Akimichi Takemura, 1997. "James-Stein Type Estimator by Shrinkage to Closed Convex Set with Smooth Boundary," CIRJE F-Series 97-F-22, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
  2. Akimichi Takemura & Satoshi Kuriki, 1998. "Tail Probabilities of the Maxima of Multilinear Forms and Their Applications," CIRJE F-Series CIRJE-F-4, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
  3. Akimichi Takemura & Satoshi Kuriki, 1999. "Maximum of Gaussian Field on Piecewise Smooth Domain: Equivalence of Tube Method and Euler Characteristic Method," CIRJE F-Series CIRJE-F-54, CIRJE, Faculty of Economics, University of Tokyo. [Downloadable!]
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