Abstract
The estimation of the asymptotic variance of sample median based on a random sample of univariate observations has been extensively studied in the literature. The appearance of a “local object” like the density function of the observations in this asymptotic variance makes its estimation a difficult task, and there are several complex technical problems associated with it. This paper explores the problem of estimating the dispersion matrix of the multivariateL 1 median. Though it is absolutely against common intuition, this problem turns out to be technically much simpler. We exhibit a simple estimate for the large sample dispersion matrix of the multivariateL 1 median with excellent asymptotic properties, and to construct this estimate, we do not use any of the computationally intensive resampling techniques (e.g. the generalized jackknife, the bootstrap, etc. that have been used and thoroughly investigated by leading statisticians in their attempts to estimate the asymptotic variance of univariate median). However surprising may it sound, our analysis exposes that most of the technical complicacies associated with the estimation of the sampling variation in the median are only characteristics of univariate data, and they disappear as soon as we enter into the realm of multivariate analysis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubuchon, J. C. and Hettmansperger, T. P. (1984). A note on the estimation of the integral off 2(x),J. Statist. Plann. Inference,9, 321–331.
Babu, G. J. (1986). A note on bootstrapping the variance of sample quantile,Ann. Inst. Statist. Math.,38, 439–443.
Bahadur, R. R. (1966). A note on quantiles in large samples,Ann. Math. Statist.,37, 577–580.
Barnett, V. (1976). The ordering of multivariate data (with discussion),J. Roy. Statist. Soc. Ser. A,139, 318–354.
Bloch, D. A. and Gastwirth, J. L. (1968). On a simple estimate of the reciprocal of the density function,Ann. Math. Statist.,39, 1083–1085.
Bose, A. and Chandra, T. K. (1993). Cesaro uniform integrability andL p convergence,Sankhyā Ser. A,55 (to appear).
Brown, B. M. (1983). Statistical use of the spatial median,J. Roy. Statist. Soc. Ser. B,45, 25–30.
Chaudhuri, P. (1992). Multivariate location estimation using extension ofR-estimates throughU-statistics type approach,Ann. Statist.,20, 897–916.
Choudhury, J. and Serfling, R. J. (1988). Generalized order statistics, Bahadur representation and sequential nonparametric fixed width confidence intervals,J. Statist. Plann. Inference,19, 269–282.
Ducharme, G. R. and Milasevick, P. (1987). Spatial median and directional data,Biometrika,74, 212–215.
Efron, B. (1979). Bootstrap methods: another look at the jackknife,Ann. Statist.,7, 1–26.
Efron, B. (1982).The Jackknife, the Bootstrap and Other Resampling Plans, SIAM, Philadelphia.
Falk, M. (1986). On the estimation of the quantile density function,Statist. Probab. Lett.,4, 69–73.
Ghosh, M., Parr, W. C., Singh, K. and Babu, G. J. (1984). A note on bootstrapping the sample median,Ann. Statist.,12, 1130–1135.
Gini, C. and Galvani, L. (1929). Di talune estensioni dei concetti di media ai caratteri qualitativi,Metron,8 (Partial English translation inJ. Amer. Statist. Assoc.,25, 448–450).
Gower, J. C. (1974). The mediancenter,Appl. Statist.,23, 466–470.
Haldane, J. B. S. (1948). Note on the median of a multivariate distribution,Biometrika,35, 414–415.
Hall, P. and Martin, M. A. (1988). Exact convergence rate of bootstrap quantile variance estimator,Probab. Theory Related Fields,80, 261–268.
Hall, P. and Martin, M. A. (1991). On the error incurred in using the bootstrap variance estimate when constructing confidence intervals for quantiles,J. Multivariate Anal.,38, 70–81.
Hall, P., DiCiccio, T. J. and Romano, J. P. (1989). On smoothing and the bootstrap,Ann. Statist.,17, 692–704.
Hettmansperger, T. P. (1984).Statistical Inference Based on Ranks, Wiley, New York.
Hodges, J. L. and Lehmann, E. L. (1963). Estimates of location based on rank tests,Ann. Math. Statist.,34, 598–611.
Isogai, T. (1985). Some extension of Haldane's multivariate median and its application,Ann. Inst. Statist. Math.,37, 289–301.
Kemperman, J. H. B. (1987). The median of a finite measure on a Banach space,Statistical Data Analysis Based on the L 1 Norm and Related Methods (ed. Y. Dodge), 217–230, North-Holland, Amsterdam.
Kendall, M. and Stuart, A. (1958).The Advanced Theory of Statistics, Griffin, London.
Kiefer, J. (1967). On Bahadur's representation of sample quantiles,Ann. Math. Statist.,38, 1323–1342.
Lehmann, E. L. (1963). Nonparametric confidence intervals for a shift parameter,Ann. Math. Statist.,34, 1507–1512.
Lehmann, E. L. (1975).Nonparametrics: Statistical Methods Based on Ranks, Holden-Day, San Francisco.
Maritz, J. S. and Jarrett, R. G. (1978). A note on estimating the variance of the sample median,J. Amer. Statist. Assoc.,73, 194–196.
Milasevick, P. and Ducharme, G. R. (1987). Uniqueness of the spatial median,Ann. Statist.,15, 1332–1333.
Pollard, D. (1984).Convergence of Stochastic Processes, Springer, New York.
Pyke, R. (1965). Spacings,J. Roy. Statist. Soc. Ser. B,27, 395–449.
Rao, C. R. (1988). Methodology based onL 1-norm in statistical inference,Sankhyā Ser. A,50, 289–313.
Schuster, E. (1974). On the rate of convergence of an estimate of a functional of a probability density,Scand. Actuar. J.,1, 103–107.
Schweder, T. (1975). Window estimation of the asymptotic variance of rank estimators of location,Scand. J. Statist.,2, 113–126.
Sen, P. K. (1960). On some convergence properties ofU-statistics,Calcutta Statist. Assoc. Bull.,10, 1–18.
Sen, P. K. (1981).Sequential Nonparametrics, Wiley, New York.
Serfling, R. J. (1980).Approximation Theorems of Mathematical Statistics, Wiley, New York.
Shao, J. (1990). Bootstrap estimation of the asymptotic variances of statistical functionals,Ann. Inst. Statist. Math.,42, 737–752.
Shao, J. and Wu, C. F. J. (1989). A general theory for jackknife variance estimation,Ann. Statist.,17, 1176–1197.
Small, C. G. (1990). A survey of multidimensional medians,Internat. Statist. Rev.,58, 263–277.
Welsh, A. H. (1987). Kernel estimates of the sparsity function,Statistical Data Analysis Based on the L 1-norm and Related Methods (ed. Y. Dodge), 369–377, North-Holland, Amsterdam.
Wilks, S. S. (1932). Certain generalizations in the analysis of variance,Biometrika,24, 471–494.
Author information
Authors and Affiliations
Additional information
The research of the second author was partially supported by a Wisconsin Alumni Research Foundation Grant from University of Wisconsin, Madison.
About this article
Cite this article
Bose, A., Chaudhuri, P. On the dispersion of multivariate median. Ann Inst Stat Math 45, 541–550 (1993). https://doi.org/10.1007/BF00773354
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773354