## Bernoulli

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

### Limit theorems for functions of marginal quantiles

#### Abstract

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

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

Dates
First available in Project Euclid: 5 April 2011

https://projecteuclid.org/euclid.bj/1302009242

Digital Object Identifier
doi:10.3150/10-BEJ287

Mathematical Reviews number (MathSciNet)
MR2787610

Zentralblatt MATH identifier
1253.60024

#### Citation

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. https://projecteuclid.org/euclid.bj/1302009242

#### 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.
• [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.