Revealing the dependence structure between X(1) and X(n) |
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| Institution: | 1. Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China;2. Department of Probability and Statistics, School of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China;3. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR 97006, USA |
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| Abstract: | In this paper we address the dependence structure of the minimum and maximum of n iid random variables X1,…,Xn by determining their copula. It is then easy to give an alternative proof for their asymptotic independence and to calculate Kendall's τ and Spearman's ρ for (X(1),X(n)). This will show that the dependence between the variables is already small for small sample sizes. Finally, it can be shown that 3τn⩾ρn⩾τn>0. Although closed-form expressions are available for τn and ρn, we cannot compare them directly but have to use the concept of positive likelihood ratio dependence to establish this result. |
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