Stochastic Comparisons and Dependence among Concomitants of Order Statistics

✍ Scribed by Baha-Eldin Khaledi; Subhash Kochar

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 214 KB
• Volume: 73
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1999.1882

✦ Synopsis

Let (X i , Y i ) i=1, 2, ..., n be n independent and identically distributed random variables from some continuous bivariate distribution. If X (r) denotes the r th ordered X-variate then the Y-variate, Y [r] , paired with X (r) is called the concomitant of the r th order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP 2 dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering.