Runs in superpositions of renewal processes with applications to discrimination

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:186

Article

Alsmeyer, G ; Irle, A

关键词: run test;    discrimination;    superposition;    Poisson process;    renewal process;    Markov renewal process;    Harris chain;   

DOI  :  10.1016/j.cam.2005.02.017

来源: Elsevier

【 摘 要 】

Wald and Wolfowitz [Ann. Math. Statist. 11 (1940) 147-162] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let (S-n)(n >= 1) and (T-n)(n >= 1) be the arrival processes for channel 1 and channel 2, respectively, and (W-n)(n >= 1) their be superposition with counting process N(t) (def) double under bar sup{n >= 1 : W-n <= t). Let further R-n(*) be the number of runs in W-1,..., W-n and R-t = R-N(t)(*) the number of runs observed up to time t. We study the asymptotic behavior of R-n* and R-t first for the case where (S-n)(n >= 1) and (T-n)(n >= 1) have exponentially distributed increments with parameters lambda(1) and lambda(2), and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level alpha tests for testing lambda(1) = lambda(2) against lambda(1) not equal lambda(2) in the first case, and for testing for equal scale parameters in the second. (c) 2005 Elsevier B.V. All rights reserved.

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