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STOCHASTIC COMPARISONS OF PARALLEL SYSTEMS WHEN COMPONENTS HAVE PROPORTIONAL HAZARD RATES

Published online by Cambridge University Press:  22 October 2007

Subhash Kochar  and
Maochao Xu
Subhash Kochar
Affiliation:
Department of Mathematics and StatisticsPortland State UniversityPortland, OR 97201 E-mail: kochar@pdx.edu
Maochao Xu
Affiliation:
Department of Mathematics and StatisticsPortland State UniversityPortland, OR 97201 E-mail: kochar@pdx.edu
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Abstract

Let X1, … , Xn be independent random variables with Xi having survival function λi, i = 1, … , n, and let Y1, … ,Yn be a random sample with common population survival distribution , where = ∑i=1nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n − X1:n is larger than Yn:n − Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 21 , Issue 4 , October 2007 , pp. 597 - 609
Copyright
Copyright © Cambridge University Press 2007

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References

1.Balakrishnan, N. & Rao, C.R. (1998). Handbook of statistics. Vol. 16: Order statistics: Theory and methods. New York: Elsevier.Google Scholar
2.Balakrishnan, N. & Rao, C.R. (1998). Handbook of statistics. Vol. 17: Order statistics: Applications. New York: Elsevier.Google Scholar
3.Boland, P.J., El-Neweihi, E. & Proschan, F. (1994). Applications of the hazard rate ordering in reliability and order statistics. Journal of Applied Probability 31: 180–192.CrossRefGoogle Scholar
4.Bon, J.L. & Paltanea, E. (1999). Ordering properties of convolutions of exponential random variables. Lifetime Data Analysis 5: 185–192.CrossRefGoogle ScholarPubMed
5.David, H.A. & Nagaraja, H.N. (2003). Order statistics, 3rd ed.New York: Wiley.CrossRefGoogle Scholar
6.Dykstra, R., Kochar, S.C. & Rojo, J. (1997). Stochastic comparisons of parallel systems of heterogeneous exponential components. Journal of Statistical Planning and Inference 65: 203–211.CrossRefGoogle Scholar
7.Khaledi, B. & Kochar, S.C. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37: 1123–1128.CrossRefGoogle Scholar
8.Khaledi, B. & Kochar, S.C. (2000). Sample range-some stochastic comparisons results. Calcutta Statistical Association Bulletin 50: 283–291.CrossRefGoogle Scholar
9.Khaledi, B. & Kochar, S.C. (2002). Dispersive ordering among linear combinatons of uniform random variables. Journal of Statistical Planning and Inference 100: 13–21.CrossRefGoogle Scholar
10.Khaledi, B. & Kochar, S.C. (2002). Stochastic orderings among order statistics and sample spacings. In Mara, J.C. (ed.), Uncertainty and optimality: Probability, statistics and operations research. Singapore: World Scientific, pp. 167–203.CrossRefGoogle Scholar
11.Kochar, S.C. & Rojo, J. (1996). Some new results on stochastic comparisons of spacings from heterogeneous exponential distributions. Journal of Multivariate Analysis 59: 272–281.CrossRefGoogle Scholar
12.Marshall, A.W. & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic Press.Google Scholar
13.Mitrinović, D.S. (1970). Analytic inequalities. Berlin: Springer-Verlag.CrossRefGoogle Scholar
14.Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. New York: Wiley.Google Scholar
15.Pledger, P. & Proschan, F. (1971). Comparisons of order statistics and of spacings from heterogeneous distributions. In Rustagi, J.S. (ed.), Optimizing methods in statistics. New York: Academic Press, pp. 89–113.Google Scholar
16.Proschan, F. & Sethuraman, J. (1976). Stochastic comparisons of order statistics from heterogeneous populations, with applications in reliability. Journal of Multivariate Analysis 6: 608–616.CrossRefGoogle Scholar
17.Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. San Diego: Academic Press.Google Scholar