## Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables.(English)Zbl 0934.62052

For each $$n=1,2,\dots$$, suppose that $$X_{n1}$$, $$X_{n2},\dots, X_{nn}$$ are independent random variables with distribution functions $$F_{n1}$$, $$F_{n2},\dots,F_{nn}$$. Let $$\mathbb{F}_n$$ denote the empirical distribution function of the $$X_{ni}$$’s: $\mathbb{F}_n(x)\equiv n^{-1}\sum^n_{i=1}1_{(-\infty, x]}(X_{ni}),$ and let $$\overline F_n$$ be the average distribution function $\overline F_n\equiv n^{-1}\sum^n_{i=1}F_{ni}.$ For any distribution function $$G$$, let $$G^{-1}$$ be the left-continuous inverse of $$G$$ defined by $$G^{-1}(u) \equiv\inf\{x:G(x)\geq u\}$$, $$0<u<1$$. Throughout this paper, unless otherwise noted, we call $$G^{-1}(1/2)$$ the median of $$G$$, even when there is a nondegenerate interval $$[m_0,m_1]$$ of median points in the sense that $$P_G(Y\leq m)\geq 1/2$$ and $$P_G(Y\geq m)\geq 1/2$$ for $$m\in [m_0,m_1]$$. The problem in this paper is to give necessary and sufficient conditions for weak consistency of the sample median $$\mathbb{F}_n^{-1}(1/2)$$: under what conditions on the $$F_{ni}$$’s does it hold that $$\mathbb{F}_n^{-1} (1/2)-\overline F_n^{-1}(1/2)\to_P0$$?

### MSC:

 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics 62F12 Asymptotic properties of parametric estimators

### Keywords:

majorization; empirical distributions; weak consistency; median
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### References:

 [1] Aubin, J.-P. and Frankowska, H. (1990). Set-Valued Analy sis. Birkhäuser, Boston. [2] Dufour, J.-M., Hallin, M. and Mizera, I. (1995). Generalized runs tests for heteroscedastic time series. J. Nonparametr. Statist. · Zbl 0899.62058 [3] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Wiley, New York. · Zbl 0219.60003 [4] Gleser, L. J. (1975). On the distribution of the number of successes in independent trials. Ann. Probab. 3 182-188. · Zbl 0301.60010 [5] Hallin, M. and Mizera, I. (1996). Sample heterogeneity and the asy mptotics of M-estimators. Preprint IS-P 1996-15 (n. 49), Inst. Statistique, Univ. Libre de Bruxelles. (Available via http://www.dcs.fmph.uniba.sk/ mizera/writings.html.) URL: · Zbl 0933.62017 [6] Hallin, M. and Mizera, I. (1997). Unimodality and the asy mptotics of M-estimators. In L1Statistical Procedures and Related Topics (Y. Dodge, ed.) 47-56. IMS, Hay ward, CA. · Zbl 0933.62017 [7] Hoeffding, W. (1956). On the distribution of the number of successes in independent trials. Ann. Math. Statist. 27 713-721. · Zbl 0073.13902 [8] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 713-721. JSTOR: · Zbl 0127.10602 [9] Huber, P. J. (1981). Robust Statistics. Wiley, New York. · Zbl 0536.62025 [10] Kiefer, J. (1970). Old and new methods for studying order statistics and sample quantiles. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 349-357. Cambridge Univ. Press, Cambridge. [11] Kolmogorov, A. N. (1931). Method of median in the theory of errors. Mat. Sb. 38 47-50. [Reprinted in English in Selected Works of A. N. Kolmogorov (A. N. Shiry ayev, ed.) Kluwer, Dordrecht, 1991.] [12] Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. · Zbl 0437.26007 [13] Mizera, I. (1993). Weak continuity and identifiability of M-functionals. Ph.D. dissertation, Charles Univ., Prague. (In Slovak.) · Zbl 0797.62042 [14] Mizera, I. and Wellner, J. A. (1996). Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables. Technical Report 306, Dept. Statistics, Univ. Washington, Seattle. (Available via http://www.stat.washington.edu/jaw/jaw.research.available.html.) URL: · Zbl 0934.62052 [15] Moore, A. W., Jr. and Jorgenson, J. W. (1993). Median filtering for removal of low-frequency background drift. Analy tical Chemistry 65 188-191. [16] Petrov, V. V. (1995). Limit Theorems of Probability Theory. Clarendon, Oxford. · Zbl 0826.60001 [17] Portnoy, S. (1991). Asy mptotic behavior of regression quantiles in nonstationary, dependent cases. J. Multivariate Anal. 38 100-113. · Zbl 0737.62078 [18] Sen, P. K. (1968). Asy mptotic normality of sample quantiles for m-dependent processes. Ann. Math. Statist. 39 1724-1730. · Zbl 0197.16102 [19] Sen, P. K. (1970). A note on order statistics for heterogeneous populations. Ann. Math. Statist. 41 2137-2139. · Zbl 0216.22003 [20] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York. · Zbl 0538.62002 [21] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365 [22] Stigler, S. M. (1976). The effect of sample heterogeneity on linear functions of order statistics with applications to robust estimation. J. Amer. Statist. Assoc. 71 956-960. JSTOR: · Zbl 0359.62041 [23] Weiss, L. (1969). The asy mptotic distribution of quantiles from mixed samples. Sankhy\?a Ser. A 31 313-318. · Zbl 0183.48504
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