• Bernoulli
  • Volume 17, Number 2 (2011), 671-686.

Limit theorems for functions of marginal quantiles

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

Full-text: Open access


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 $$\sqrt{n}\Biggl(\frac{1}{n}\sum_{i=1}^{n}\phi\bigl(X_{n:i}^{(1)},\ldots,X_{n:i}^{(d)}\bigr)-\bar{\gamma}\Biggr)=\frac{1}{\sqrt{n}}\sum _{i=1}^{n}Z_{n,i}+\mathrm{o}_{P}(1)$$ as $n → ∞$, where $γ̄>$ 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.

Article information

Bernoulli, Volume 17, Number 2 (2011), 671-686.

First available in Project Euclid: 5 April 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Babu, G. Jogesh; Bai, Zhidong; Choi, Kwok Pui; Mangalam, Vasudevan. Limit theorems for functions of marginal quantiles. Bernoulli 17 (2011), no. 2, 671--686. doi:10.3150/10-BEJ287.

Export citation


  • [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.
  • [2] Bai, Z.D. and Hsing, T. (2005). The broken sample problem. Probab. Theory Related Fields 131 528–552.
  • [3] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley Series in Probability and Statistics: Probability and Statistics. New York: Wiley.
  • [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.
  • [6] David, H.A. (1981). Order Statistics. New York: Wiley.
  • [7] DeGroot, M.H. and Goel, P.K. (1980). Estimation of the correlation coefficient from a broken sample. Ann. Statist. 8 264–278.
  • [8] Hardy, G.H., Littlewood, J.E. and Pólya, G. (1952). Inequalities. Cambridge: Cambridge Univ. Press.
  • [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. To appear.