Dependence structure of bivariate order statistics with applications to bayramoglu's distributions

We study the dependence structure of bivariate order statistics from bivariate distributions, and prove that if the underlying bivariate distribution H is positive quadrant dependent (PQD) then so is each pair of bivariate order statistics. As an application, we show that if H is PQD, the bivariate distribution K⁽ⁿ⁾₊, recently proposed by Bairamov and Bayramoglu (2012) [1], is greater than or equal to Baker's (2008) [2] distribution H⁽ⁿ⁾₊, and hence K⁽ⁿ⁾_' attains a correlation higher than that of H⁽ⁿ⁾₊. We give two explicit forms of the intractable K⁽ⁿ⁾₊ and prove that for all n ≥ 2, K⁽ⁿ⁾₊ is PQD regardless of H. We also show that if H is PQD, K⁽ⁿ⁾₊ converges weakly to the Fréchet-Hoeffding upper bound as n tends to infinity.