Abstract
A necessary condition for the asymptotic normality of the sample quantile estimator isf(Q(p))=F′(Q(p))>0, whereQ(p) is thep-th quantile of the distribution functionF(x). In this paper, we estimate a quantile by a kernel quantile estimator when this condition is violated. We have shown that the kernel quantile estimator is asymptotically normal in some nonstandard cases. The optimal convergence rate of the mean squared error for the kernel estimator is obtained with respect to the asymptotically optimal bandwidth. A law of the iterated logarithm is also established.
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References
Billingsley, P. (1986).Probability and Measure, Wiley, New York.
Csörgő, S. and Hall, P. (1982). Upper and lower classes for triangular arrays,Z. Wahrsch. Verw. Gebiete,61, 207–222.
Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1956). Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator,Ann. Math. Statist.,27, 642–669.
Falk, M. (1984). Relative deficiency of kernel type estimators of quantiles,Ann. Statist.,12, 261–268.
Falk, M. (1985). Asymptotic normality of the kernel quantile estimators,Ann. Statist.,13, 428–433.
Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent r.v.'s and the sample d.f. I,Z. Wahrsch. Verw. Gebiete,32, 111–132.
Parzen, E. (1979). Nonparametric statistical data modeling,J. Amer. Statist. Assoc.,74, 105–131.
Sheather, S. J. and Marron, J. S. (1990). Kernel quantile estimators,J. Amer. Statist. Assoc.,85, 410–416.
Shorack, G. R. and Wellner, J. A. (1986).Empirical Processes with Applications to Statistics, Wiley, New York.
Xiang, X. (1993). Deficiency of sample quantile estimator with respect to kernel estimator from randomly right-censored data, Manuscript (submitted for publication).
Yang, S. S. (1985). A smooth nonparametric estimator of a quantile function,J. Amer. Statist. Assoc.,80, 1004–1011.
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This research was partially supported by the new faculty award from the University of Oregon.
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Xiang, X. Estimation of a quantile in some nonstandard cases. Ann Inst Stat Math 47, 105–117 (1995). https://doi.org/10.1007/BF00773415
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DOI: https://doi.org/10.1007/BF00773415