J. Appl. Prob. 43, 391–408 (2006) Printed in Israel  Applied Probability Trust 2006 HAZARD RATE ORDERING OF ORDER STATISTICS AND SYSTEMS JORGE NAVARRO, ∗ Universidad de Murcia MOSHE SHAKED, ∗∗ University of Arizona Abstract Let X = (X 1 ,X 2 ,...,X n ) be an exchangeable random vector, and write X (1:i) = min{X 1 ,X 2 ,...,X i }, 1 ≤ i ≤ n. In this paper we obtain conditions under which X (1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples. Keywords: Exchangeable distribution; signature; coherent system; hazard rate order; stochastic order; Freund distribution; hazard gradient; cause specificity; Farlie–Gumbel– Morgenstern distribution 2000 Mathematics Subject Classification: Primary 60E15; 60K10 1. Introduction Let X = (X 1 ,X 2 ,...,X n ) be an exchangeable random vector, and write X (1:i) = min{X 1 , X 2 ,...,X i }, 1 ≤ i ≤ n. Intuitively we expect X (1:i) to get smaller as i gets larger. Indeed, it is not hard to verify that X (1:i) decreases in i in the usual stochastic order (see Section 2 for more details). In this paper we study the (possible) monotonicity of X (1:i) in the hazard rate order (see Section 2 for the exact definition of this order). The hazard rate order is stronger than the usual stochastic order, and it turns out that it is not always true that X (1:i) decreases in i in the hazard rate order; this is shown in Section 2. The purpose of this paper is to obtain conditions under which X (1:i) indeed decreases in i in the hazard rate order. This is also done in Section 2, where we actually obtain a result involving more general (that is, not necessarily exchangeable) random vectors. Throughout the paper, the notions ‘increasing’and ‘decreasing’ are used in the weak sense. In Section 3 we apply the results of Section 2 to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in Received 29 April 2005; revision received 17 January 2006. ∗ Postal address: Facultad de Matematicas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es ∗∗ Postal address: Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. Email address: shaked@math.arizona.edu Partially supported by the Ministerio de Ciencia y Tecnologia under grant BFM2003-02947 and Fundacion Seneca under grant 00698/PI/04. 391 at https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1152413730 Downloaded from https://www.cambridge.org/core. IP address: 54.163.42.124, on 25 May 2020 at 00:21:34, subject to the Cambridge Core terms of use, available