×

On the \(u\)th geometric conditional quantile. (English) Zbl 1118.62051

Summary: Motivated by P. Chaudhuri’s [On a geometric notion of quantiles for multivariate data. J. Am. Stat. Assoc. 91, No. 434, 862–872 (1996; Zbl 0869.62040)] work on unconditional geometric quantiles, we explore the asymptotic properties of sample geometric conditional quantiles, defined through kernel functions, in high-dimensional spaces. We establish a Bahadur-type linear representation for the geometric conditional quantile estimator and obtain the convergence rate for the corresponding remainder term. From this, asymptotic normality including bias on the estimated geometric conditional quantile is derived. Based on these results, we propose confidence ellipsoids for multivariate conditional quantiles. The methodology is illustrated via data analysis and a Monte Carlo study.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation
62H12 Estimation in multivariate analysis
62G15 Nonparametric tolerance and confidence regions
65C05 Monte Carlo methods

Citations:

Zbl 0869.62040
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abdous, B.; Theodorescu, R., Note on the spatial quantile of a random vector, Statist. Probab. Lett., 13, 333-336 (1992) · Zbl 0743.62040
[2] Bosq, D.; Lecoutre, J. P., Théorie de l’estimation fonctionnelle (1987), Economica: Economica Paris
[3] Brown, B. M.; Hettmansperger, T. P., Affine invariant rank methods in the bivariate location model, J. Roy. Statist. Soc. Ser. B, 49, 301-310 (1987) · Zbl 0653.62039
[4] Brown, B. M.; Hettmansperger, T. P., An affine invariant bivariate version of the sign test, J. Roy. Statist. Soc. B, 51, 117-125 (1989) · Zbl 0675.62036
[5] Cadre, B.; Gannoun, A., Asymptotic normality of consistent estimate of the conditional \(L_1\)-median, Ann. I.S.U.P., 44, 13-35 (2000) · Zbl 0957.62024
[6] Chakraborty, B., On affine equivariant multivariate quantiles, Ann. Inst. Statist. Math., 53, 380-403 (2001) · Zbl 1027.62035
[7] Chakraborty, B., On multivariate quantile regression, J. Statist. Plann. Inference, 110, 109-132 (2003) · Zbl 1030.62046
[8] Chaudhuri, P., Multivariate location estimation using extension of \(R\)-estimates through U-statistics type approach, Ann. Statist., 20, 897-916 (1992) · Zbl 0762.62013
[9] Chaudhuri, P., On a geometric notion of quantiles for multivariate data, J. Amer. Statist. Assoc., 91, 862-872 (1996) · Zbl 0869.62040
[10] De Gooijer, J. G.; Gannoun, A.; Zerom, D., A multivariate quantile predictor, Commun. Statist. Theory Methods, 35, 133-147 (2006) · Zbl 1084.62097
[11] Devroye, L., On the almost everywhere convergence of nonparametric regression function estimates, Ann. Statist., 9, 1310-1319 (1981) · Zbl 0477.62025
[12] Kemperman, J. H.B., The median of a finite measure on a Banach space, (Dodge, Y., Statistical Data Analysis Based on the \(L_1\) Norm and Related Methods (1987), North-Holland: North-Holland Amsterdam), 217-230
[13] Schuster, E. F., Joint asymptotic distribution of the estimated regression function at a finite number of distinct points, Ann. Math. Statist., 43, 84-88 (1972) · Zbl 0248.62027
[14] Serfling, R. J., Approximation Theorems of Mathematical Statistics (1980), Wiley: Wiley New York · Zbl 0456.60027
[15] Serfling, R. J., Quantile functions for multivariate analysis: approaches and applications, Statist. Neerlandica, 56, 214-232 (2002) · Zbl 1076.62054
[16] Serfling, R. J., Nonparametric multivariate descriptive measures based on spatial quantiles, J. Statist. Plann. Inference, 123, 259-278 (2004) · Zbl 1045.62048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.