×

Moderate and large deviations for the smoothed estimate of sample quantiles. (English) Zbl 1426.60033

Summary: We derive the moderate and large deviations principle for the smoothed sample quantile from a sequence of independent and identically distributed samples of size \(n\).

MSC:

60F10 Large deviations
62G05 Nonparametric estimation
62G30 Order statistics; empirical distribution functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lahiri, S. N.; Sun, S., A Berry-Esseen theorem for sample quantiles under weak dependence, The Annals of Applied Probability, 19, 1, 108-126, (2009) · Zbl 1158.60007 · doi:10.1214/08-AAP533
[2] Wu, W. B., On the Bahadur representation of sample quantiles for dependent sequences, The Annals of Statistics, 33, 4, 1934-1963, (2005) · Zbl 1080.62024 · doi:10.1214/009053605000000291
[3] Miao, Y.; Chen, Y.-X.; Xu, S.-F., Asymptotic properties of the deviation between order statistics and p-Quantile, Communications in Statistics—Theory and Methods, 40, 1, 8-14, (2011) · Zbl 1208.62082 · doi:10.1080/03610920903350523
[4] Xu, S. F.; Ge, L.; Miao, Y., On the Bahadur representation of sample quantiles and order statistics for NA sequences, Journal of the Korean Statistical Society, 42, 1, 1-7, (2013) · Zbl 1294.62109 · doi:10.1016/j.jkss.2012.04.003
[5] Ma, Y.; Genton, M. G.; Parzen, E., Asymptotic properties of sample quantiles of discrete distributions, Annals of the Institute of Statistical Mathematics, 63, 2, 227-243, (2011) · Zbl 1432.62035 · doi:10.1007/s10463-008-0215-z
[6] Nadaraya, E. A., Some new estimates for distribution function, Theory of Probability and Its Applications, 9, 497-500, (1964) · Zbl 0152.17605
[7] Parzen, E., Nonparametric statistical data modeling, Journal of the American Statistical Association, 74, 365, 105-131, (1979) · Zbl 0407.62001 · doi:10.1080/01621459.1979.10481621
[8] Reiss, R.-D., Estimation of quantiles in certain nonparametric models, The Annals of Statistics, 8, 1, 87-105, (1980) · Zbl 0424.62023 · doi:10.1214/aos/1176344893
[9] Falk, M., Relative deficiency of kernel type estimators of quantiles, The Annals of Statistics, 12, 1, 261-268, (1984) · Zbl 0533.62040 · doi:10.1214/aos/1176346405
[10] Yang, S.-S., A smooth nonparametric estimator of a quantile function, Journal of the American Statistical Association, 80, 392, 1004-1011, (1985) · Zbl 0593.62037 · doi:10.1080/01621459.1985.10478217
[11] Padgett, W. J., A kernel-type estimator of a quantile function from right-censored data, Journal of the American Statistical Association, 81, 393, 215-222, (1986) · Zbl 0596.62043 · doi:10.1080/01621459.1986.10478263
[12] Cai, Z.; Roussas, G. G., Smooth estimate of quantiles under association, Statistics & Probability Letters, 36, 3, 275-287, (1997) · Zbl 0946.62039 · doi:10.1016/s0167-7152(97)00074-6
[13] Dembo, A.; Zeitouni, O., Large Deviations Techniques and Applications. Large Deviations Techniques and Applications, Applications of Mathematics (New York), 38, (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0896.60013 · doi:10.1007/978-1-4612-5320-4
[14] Louani, D., Large deviations limit theorems for the kernel density estimator, Scandinavian Journal of Statistics. Theory and Applications, 25, 1, 243-253, (1998) · Zbl 0904.62060 · doi:10.1111/1467-9469.00101
[15] Gao, F., Moderate deviations and large deviations for kernel density estimators, Journal of Theoretical Probability, 16, 2, 401-418, (2003) · Zbl 1041.62025 · doi:10.1023/A:1023574711733
[16] He, X.; Gao, F., Moderate deviations and large deviations for a test of symmetry based on kernel density estimator, Acta Mathematica Scientia. Series B. English Edition, 28, 3, 665-674, (2008) · Zbl 1174.60006 · doi:10.1016/S0252-9602(08)60068-5
[17] Korbe Diallo, A. O.; Louani, D., Moderate and large deviation principles for the hazard rate function kernel estimator under censoring, Statistics & Probability Letters, 83, 3, 735-743, (2013) · Zbl 1463.62112 · doi:10.1016/j.spl.2012.11.010
[18] Xu, S. F.; Miao, Y., Limit behaviors of the deviation between the sample quantiles and the quantile, Filomat, 25, 2, 197-206, (2011) · Zbl 1299.62042 · doi:10.2298/fil1102197x
[19] Serfling, R. J., Approximation Theorems of Mathematical Statistics, (1980), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0423.60030
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.