Limit theorems for functions of marginal quantiles. (English) Zbl 1253.60024

Babu, G. Jogesh; Bai, Zhidong; Choi, Kwok Pui; Mangalam, Vasudevan

Bernoulli 17, No. 2, 671-686 (2011).

Summary: Multivariate distributions are explored using the joint distributions of marginal sample quantiles. Limit theory for the mean of a function of order statistics is presented. The results include a multivariate central limit theorem and a strong law of large numbers. A result similar to Bahadur’s representation of quantiles is established for the mean of a function of the marginal quantiles. In particular, it is shown that \[ \sqrt{n} \Biggl({1\over n} \sum^n_{i=1} \phi(X^{(1)}_{n:i},\dots, X^{(d)}_{n:i}- \overline\gamma\Biggr)= {1\over\sqrt{n}}\,\sum^n_{i= 1} Z_{n,i}+ o_P(1) \] as \(n\to\infty\), where \(\overline\gamma\) is a constant and \(Z_{n,i}\) are i.i.d. random variables for each \(n\). This leads to the central limit theorem. Weak convergence to a Gaussian process using equicontinuity of functions is indicated. The results are established under very general conditions. These conditions are shown to be satisfied in many commonly occurring situations.

MSC:

60F05	Central limit and other weak theorems
60F15	Strong limit theorems

Keywords:

central limit theorem; Cramér-Wold device; lost association; quantiles; strong law of large numbers; weak convergence of a process

References:

[1]	Babu, G.J. and Rao, C.R. (1988). Joint asymptotic distribution of marginal quantiles and quantile functions in samples from a multivariate population. J. Multivariate Anal. 27 15-23. · Zbl 0649.62050 · doi:10.1016/0047-259X(88)90112-1
[2]	Bai, Z.D. and Hsing, T. (2005). The broken sample problem. Probab. Theory Related Fields 131 528-552. · Zbl 1062.62041 · doi:10.1007/s00440-004-0384-5
[3]	Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics . New York: Wiley. · Zbl 0944.60003
[4]	Copas, J.B. and Hilton, F.J. (1990). Record linkage: Statistical models for matching computer records. J. R. Statist. Soc. A 153 287-320.
[5]	Chan, H.P. and Loh, W.L. (2001). A file linkage problem of DeGroot and Goel revisited. Statist. Sinica 11 1031-1045. · Zbl 0984.62039
[6]	David, H.A. (1981). Order Statistics . New York: Wiley. · Zbl 0553.62046
[7]	DeGroot, M.H. and Goel, P.K. (1980). Estimation of the correlation coefficient from a broken sample. Ann. Statist. 8 264-278. · Zbl 0446.62049 · doi:10.1214/aos/1176344952
[8]	Hardy, G.H., Littlewood, J.E. and Pólya, G. (1952). Inequalities . Cambridge: Cambridge Univ. Press. · Zbl 0047.05302
[9]	Kiefer, J. (1970). Deviations between the sample quantile process and the sample d.f. In Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969) 299-319. London: Cambridge Univ. Press.
[10]	Mangalam, V. (2010). Regression under lost association.