Study of some measures of dependence between order statistics and systems

  • Authors:

  • Jorge Navarro;N. Balakrishnan

  • Affiliations:

  • Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain;Department of Math. and Stat., McMaster University, Hamilton, Ontario L8S 4K1, Canada

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2010

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Abstract

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.