Study of some measures of dependence between order statistics and systems

Let X=(X"1,X"2,...,X"n) be a random vector, and denote by X"1":"n,X"2":"n,...,X"n":"n the corresponding order statistics. When X"1,X"2,...,X"n represent the lifetimes of n components in a system, the order statistic X"n"-"k"+"1":"n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson's correlation coefficient between X"i":"n and X"j":"n. We pay special attention to the case n=2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman's rho and Kendall's tau coefficients when the variables X"1,X"2,...,X"n are independent and identically distributed or when they jointly have an exchangeable distribution.

[1]  Some characterizations of distributions based on order statistics , 1990 .

[2]  N. Loperfido,et al.  Some Relationships Between Skew-Normal Distributions and Order Statistics from Exchangeable Normal Random Vectors , 2007 .

[3]  J. Navarro,et al.  Properties of Coherent Systems with Dependent Components , 2007 .

[4]  Yung-Pin Chen.  A note on the relationship between Spearman's ρ and Kendall's τ for extreme order statistics , 2007 .

[5]  D. Lindley,et al.  MULTIVARIATE DISTRIBUTIONS FOR THE LIFE LENGTHS OF COMPONENTS OF A SYSTEM SHARING A COMMON ENVIRONMENT , 1986 .

[6]  K. Joag-dev,et al.  Bivariate Dependence Properties of Order Statistics , 1996 .

[7]  Tomasz Rychlik.  Projecting Statistical Functionals , 2001 .

[8]  C. Genest,et al.  On the dependence structure of order statistics , 2005 .

[9]  J. Shanthikumar,et al.  Multivariate Stochastic Orders , 2007 .

[10]  M. C. Jones,et al.  How are moments and moments of spacings related to distribution functions , 2002 .

[11]  D. Walkup,et al.  Association of Random Variables, with Applications , 1967 .

[12]  Tomasz Rychlik.  Distributions and expectations of order statistics for possibly dependent random variables , 1994 .

[13]  J. Navarro,et al.  Multivariate weighted distributions: a review and some extensions , 2006 .

[14]  G. Terrell,et al.  A Characterization of Rectangular Distributions , 1983 .

[15]  R. Nelsen,et al.  On the relationship between Spearman's rho and Kendall's tau for pairs of continuous random variables , 2007 .

[16]  Herbert A. David,et al.  Order Statistics, Third Edition , 2003, Wiley Series in Probability and Statistics.

[17]  Marco Burkschat.  Multivariate dependence of spacings of generalized order statistics , 2009, J. Multivar. Anal..

[18]  A. W. Kemp,et al.  Continuous Bivariate Distributions, Emphasising Applications , 1991 .

[19]  B. Arnold,et al.  A first course in order statistics , 2008 .

[20]  J. Navarro,et al.  Properties of systems with two exchangeable Pareto components , 2007 .

[21]  Christian Genest,et al.  Spearman's ρ is larger than kendall's τ for positively dependent random variables , 1993 .

[22]  Xiaohu Li,et al.  Proof of a conjecture on Spearman's p and Kendall's tau for sample minimum and maximum , 2007 .

[23]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[24]  Equivalence of Hartley--David--Gumbel and Papathanasiou bounds and some further remarks , 1993 .

[25]  Udo Kamps,et al.  Ch. 12. Sequential k-out-of-n systems , 2001 .

[26]  Xiaohu Li,et al.  Letter to the Editor Proof of a conjecture on Spearman's and Kendall's for sample minimum and maximum , 2007 .

[27]  H. Joe.  Multivariate models and dependence concepts , 1998 .

[28]  Ordering Properties of Systems with Two Dependent Components , 2007 .

[29]  N. L. Johnson,et al.  Continuous Multivariate Distributions, Volume 1: Models and Applications , 2019 .

[30]  Adriana Hornikova,et al.  Stochastic Ageing and Dependence for Reliability , 2007, Technometrics.

[31]  Narayanaswamy Balakrishnan,et al.  On the application and extension of system signatures in engineering reliability , 2008 .

[32]  N. Balakrishnan,et al.  Conditional correlation analysis of order statistics from bivariate normal distribution with an application to evaluating inventory effects in futures market , 2003 .

[33]  Subhash C. Kochar,et al.  Dependence orderings for generalized order statistics , 2005 .

[34]  P. Bickel.  Some contributions to the theory of order statistics , 1967 .

[35]  N. Balakrishnan,et al.  Continuous Bivariate Distributions , 2009 .

[36]  Marco Scarsini,et al.  On measures of concordance , 1984 .

[37]  E. Lehmann.  Some Concepts of Dependence , 1966 .

[38]  F. Samaniego.  On Closure of the IFR Class Under Formation of Coherent Systems , 1985, IEEE Transactions on Reliability.

[39]  Christian Genest,et al.  On the dependence between the extreme order statistics in the proportional hazards model , 2008 .

[40]  V. Schmitz,et al.  Revealing the dependence structure between X(1) and X(n) , 2004 .

[41]  R. Nelsen.  An Introduction to Copulas , 1998 .

[42]  Taizhong Hu,et al.  Dependence properties of order statistics , 2008 .