Joint distribution of new sample rank of bivariate order statistics

Abstract

Let $(X_k,Y_k),k=1,2,\dots ,n$, be independent copies of bivariate random vector $(X,Y)$ with joint cumulative distribution function $F(x,y)$ and probability density function $f(x,y)$. For $1\le r,s\le n$, the vector of order statistics of $X_{1:n}\le X_{2:n}\le \cdots \le X_{n:n}$ and $Y_{1:n}\le Y_{2:n}\le \cdots \le Y_{n:n}$, respectively, is denoted by $(X_{r:n},Y_{s:n})$. Let $(X_{n+i},Y_{n+i})$, $i=1,2,\dots ,m$, be a new sample from $F(x,y)$, which is independent from $(X_k,Y_k),k=1,2,\dots ,n$. Let ${\xi }_1$ be the rank of order statistics $X_{r:n}$ in a new sample $X_{n+1},X_{n+2},\dots ,X_{n+m}$ and ${\xi }_2$ be the rank of order statistics $Y_{s:n}$ in a new sample $Y_{n+1},Y_{n+2},\dots ,Y_{n+m}$. We derive the joint distribution of discrete random vector $({\xi }_1,{\xi }_2)$ and a general scheme wherein the distributions of new and old samples are different is considered. Numerical examples for given well-known distribution are also provided.

Keywords:

• bivariate order statistics
• probability density function
• rank
• bivariate binomial distribution
• discrete random vector

Acknowledgements

The authors are grateful to the Editor and anonymous reviewers for their valuable comments and suggestions which resulted in improvement of the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data and research materials

Supplemental data for this article can be accessed at http://dx.doi.org/10.1080/02664763.2015.102370510.1080/02664763.2015.1023705.