### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 201-208 |

Number of pages | 8 |

Journal | Journal of Multivariate Analysis |

Volume | 114 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2013 |

Externally published | Yes |

### Fingerprint

### Keywords

- Baker's bivariate distribution
- Fréchet-hoeffding bounds
- Hoeffding's representation for covariance
- Negative quadrant dependent
- Pearson's correlation
- Positive quadrant dependent

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Numerical Analysis
- Statistics and Probability

### Cite this

*Journal of Multivariate Analysis*,

*114*(1), 201-208. https://doi.org/10.1016/j.jmva.2012.07.009

**Dependence structure of bivariate order statistics with applications to bayramoglu's distributions.** / Huang, J. S.; Dou, Xiaoling; Kuriki, Satoshi; Lin, G. D.

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 114, no. 1, pp. 201-208. https://doi.org/10.1016/j.jmva.2012.07.009

}

TY - JOUR

T1 - Dependence structure of bivariate order statistics with applications to bayramoglu's distributions

AU - Huang, J. S.

AU - Dou, Xiaoling

AU - Kuriki, Satoshi

AU - Lin, G. D.

PY - 2013

Y1 - 2013

N2 - 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.

AB - 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.

KW - Baker's bivariate distribution

KW - Fréchet-hoeffding bounds

KW - Hoeffding's representation for covariance

KW - Negative quadrant dependent

KW - Pearson's correlation

KW - Positive quadrant dependent

UR - http://www.scopus.com/inward/record.url?scp=84867740607&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867740607&partnerID=8YFLogxK

U2 - 10.1016/j.jmva.2012.07.009

DO - 10.1016/j.jmva.2012.07.009

M3 - Article

VL - 114

SP - 201

EP - 208

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -