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Some Results on Interarrival Times of Nonhomogeneous Poisson Processes

Published online by Cambridge University Press:  27 July 2009

Subhash C. Kochar
Subhash C. Kochar
Affiliation:
Indian Statistical Institute, 7, SJS Sansanwal Marg, New Delhi 110016, India
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Abstract

It is well known that in the case of a Poisson process with constant intensity function the interarrival times are independent and identically distributed, each having exponential distribution. We study this problem when the intensity function is monotone. In particular, we show that in the case of a nonhomogeneous Poisson process with decreasing (increasing) intensity the interarrival times are increasing (decreasing) in the hazard rate ordering sense and they are also jointly likelihood ratio ordered (cf. Shanthikumar and Yao, 1991, Bivariate characterization of some stochastic order relations, Advances in Applied Probability 23: 642–659). This result is stronger than the usual stochastic ordering between the successive interarrival times. Also in this case, the interarrival times are conditionally increasing in sequence and, as a consequence, they are associated. We also consider the problem of comparing two nonhomogeneous Poisson processes in terms of the ratio of their intensity functions and establish some results on the successive number of events from one process occurring between two consecutive occurrences from the second process.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 1 , January 1996 , pp. 75 - 85
Copyright
Copyright © Cambridge University Press 1996

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References

1.Bagai, I. & Kochar, S.C. (1986). On tail ordering and comparison of failure rates. Communications in Statistical Theories and Methods 15: 1377–1388.CrossRefGoogle Scholar
2.Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, MD: To Begin With.Google Scholar
3.Baxter, I.A. (1982). Reliability applications of the relevation transform. Naval Research Logistics Quarterly 29: 323–330.CrossRefGoogle Scholar
4.Bickel, P. & Lehmann, E. (1975). Descriptive statistics for nonparametric models II. Annals of Statistics 3: 1045–1069.Google Scholar
5.Boyett, J. & Saw, J. (1980). On comparing two Poisson intensity functions. Communications in Statistical Theories and Methods 9: 943–948.Google Scholar
6.Gupta, R.C. & Kirmani, S.N.U.A. (1988). Closure and monotonicity properties of nonhomogeneous Poisson processes and record values. Probability in the Engineering and Informational Sciences 2: 475–484.CrossRefGoogle Scholar
7.Hollander, M., Proschan, R., & Sethuraman, J. (1977). Functions decreasing in transposition and their applications in ranking problems. Annals of Statistics 4: 722–733.Google Scholar
8.Joag-dev, K., Kochar, S., & Proschan, F. (1995). A general composition theorem and its applications to certain partial orderings of distributions. Statistics & Probability Letters 22: 111–119.CrossRefGoogle Scholar
9.Kirmani, S.N.U.A. & Gupta, R.C. (1992). Some moment inequalities for the minimal repair process. Probability in the Engineering and Informational Sciences 6: 245–255.CrossRefGoogle Scholar
10.Kochar, S.C. (1994). Monotonicity properties of the ordered ranks in the two-sample problem. Statistics & Probability Letters 20: 63–67.CrossRefGoogle Scholar
11.Lee, L. (1982). Distribution free tests for comparing trends in Poisson series. Stochastic Processes and Their Applications 12: 107–113.CrossRefGoogle Scholar
12.Lee, L. & Pirie, W. (1981). A graphical method for comparing trends in series of events.Communications in Statistical Theories and Methods 9: 827–848.CrossRefGoogle Scholar
13.Liu, W. (1995). On the existence of tests uniformly more powerful than the likelihood ratio tests. Journal of Statistical Planning and Information 44: 19–35.CrossRefGoogle Scholar
14.Marshall, A.W. & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic Press.Google Scholar
15.Parzen, E. (1962). Stochastic processes. San Francisco: Holden-Day.Google Scholar
16.Robertson, T. & Wright, F.T. (1982). On measuring the conformity of a parameter set to a trend, with applications. Annals of Statistics 4: 1234–1245.Google Scholar
17.Ross, S.M. (1983). Stochastic processes. New York: Wiley.Google Scholar
18.Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego: Academic Press.Google Scholar
19.Shanthikumar, J.G. & Yao, D.D. (1991). Bivariate characterization of some stochastic order relations. Advances in Applied Probability 23: 642–659.CrossRefGoogle Scholar