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Corrected discrete approximations for the conditional and unconditional distributions of the continuous scan statistic

Part of: Markov processes Geometric probability and stochastic geometry Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  04 April 2017

Yi-Ching Yao ,
Daniel Wei-Chung Miao  and
Xenos Chang-Shuo Lin
Yi-Ching Yao*
Affiliation:
Academia Sinica
Daniel Wei-Chung Miao*
Affiliation:
National Taiwan University of Science and Technology
Xenos Chang-Shuo Lin*
Affiliation:
Aletheia University
*
* Postal address: Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan, ROC. Email address: [email protected]
** Postal address: Graduate Institute of Finance, National Taiwan University of Science and Technology, Taipei 106, Taiwan, ROC. Email address: [email protected]
*** Postal address: Accounting Information Department, Aletheia University, New Taipei City, 25103, Taiwan, ROC. Email address: [email protected]
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Abstract

The (conditional or unconditional) distribution of the continuous scan statistic in a one-dimensional Poisson process may be approximated by that of a discrete analogue via time discretization (to be referred to as the discrete approximation). Using a change of measure argument, we derive the first-order term of the discrete approximation which involves some functionals of the Poisson process. Richardson's extrapolation is then applied to yield a corrected (second-order) approximation. Numerical results are presented to compare various approximations.

Keywords

Poisson processRichardson’s extrapolationMarkov chain embeddingchange of measuresecond-order approximation

MSC classification

Primary: 60E05: Distributions 62E20: Asymptotic distribution theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60D05: Geometric probability and stochastic geometry
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 304 - 319
Copyright
Copyright © Applied Probability Trust 2017 

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