Concomitants of order statistics.

*(English)*Zbl 0905.62055
Balakrishnan, N. (ed.) et al., Order statistics: theory & methods. Amsterdam: North Holland/ Elsevier. Handb. Stat. 16, 487-513 (1998).

Let \((X_{i},Y_{i})\), \(i=1,\dots,n\), be a sample from a bivariate distribution. If the sample \(X_{i}\) is ordered, then the variate \(Y\) associated with the \(r\)-th order statistic \(X_{r:n}\) is denoted by \(Y_{[r:n]}\) and termed the concomitant of the \(r\)-th order statistic \(X_{r:n}\). Another term is ‘induced order statistics’. Concomitants can also be associated with record values and with generalized order statistics which include order statistics and record values as special cases. The authors provide a unified account of the basic finite dimensional and asymptotic theory of concomitants. Multivariate generalizations of the theory are provided, too. Estimation and hypothesis testing problems are considered.

Applications of the theory of concominants to: (i) estimation of regression and correlation coefficients, (ii) analysis of censored bivariate data, (ii) ranked-set sampling, (iv) double sampling, are described. Selection procedures based on concomitants are presented. The asymptotic theory of various functions of concomitants is examined with application to: (i) inference on regression functions, (ii) the induced selection differential, (iii) bootstrapping and (iv) file-matching procedures.

This review includes an almost complete set of references to the literature on concomitants. Earlier reviews of concomitants were given by P. K. Bhattacharya [Nonparametric Methods, Handb. Stat. 4, 383-403 (1984; Zbl 0596.62056)] and by H. A. David [in F. M. Hoppe (ed.), Multiple comparisons, selection, and applications in biometry. Stat., Textb. Monogr. 134, 507-518 (1993; Zbl 0827.62043)].

For the entire collection see [Zbl 0894.00024].

Applications of the theory of concominants to: (i) estimation of regression and correlation coefficients, (ii) analysis of censored bivariate data, (ii) ranked-set sampling, (iv) double sampling, are described. Selection procedures based on concomitants are presented. The asymptotic theory of various functions of concomitants is examined with application to: (i) inference on regression functions, (ii) the induced selection differential, (iii) bootstrapping and (iv) file-matching procedures.

This review includes an almost complete set of references to the literature on concomitants. Earlier reviews of concomitants were given by P. K. Bhattacharya [Nonparametric Methods, Handb. Stat. 4, 383-403 (1984; Zbl 0596.62056)] and by H. A. David [in F. M. Hoppe (ed.), Multiple comparisons, selection, and applications in biometry. Stat., Textb. Monogr. 134, 507-518 (1993; Zbl 0827.62043)].

For the entire collection see [Zbl 0894.00024].

Reviewer: M.P.Moklyachuk (Kyïv)

##### MSC:

62G30 | Order statistics; empirical distribution functions |