A Note on Extremes of Concomitants of Order Statistics

Abstract

Let $$\{ (X_n ,Y_n )\} _{n \geqslant 1} $$ be a sequence of identically distributed variables. We study the asymptotic distribution of $$M_{k:n} = \max (Y_{[n - k + 1:n]} , \ldots ,Y_{[n:n]} )$$ , where Y [r:n] denotes the concomitant of the rth order statistic X r:n , corresponding to $$X_1 , \ldots ,X_n $$ , and $$k \geqslant 1$$ is held fixed while $$n \to \infty $$ . Conditions are given for the $$Y_{[r:n]} $$ and $$M_{k:n} $$ to have the same asymptotic behavior as that we would apply if $$\{ (X_n ,Y_n )\} _{n \geqslant 1} $$ were i.i.d. The result is illustrated with a simple linear regression model $$Y_n = \beta X_n + \varepsilon _n ,n \geqslant 1$$ , where $$\{ (X_n \} _{n \geqslant 1} $$ is a stationary sequence with extremal index $$\theta < 1$$ .