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Dependency measures under bivariate homogeneous shock models

The bivariate exponential (BVE) model, defined and explained by Marshall and Olkin (Marshall, A.W. and Olkin, I., 1967, J. Am. Stat. Assoc., 62, 30–44.), is a well-motivated model derived from practical failure process. Edwardes (Edwardes, M.D., 1993, Stat. Prob. Lett., 17, 415–419.) further studied BVE model and proved that under BVE model, Kendall's τ and correlation coefficient are equal. In this article, a more general bivariate model, referred to as bivariate homogeneous shock (BHS) model, is introduced and some of its dependency measures are investigated. Despite the singularity feature of BHS model, these dependency measures are found to be of simple forms with mathematical elegance. In virtue of these simple forms, we prove that BVE is also a necessary condition for the equality between Kendall's τ and correlation coefficient.