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On Some Smooth Estimators of the Quantile Function for a Stationary Associated Process

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  • Yogendra P. Chaubey

    (Concordia University)

  • Isha Dewan

    (Indian Statistical Institute, Delhi Centre)

  • Jun Li

    (Nanjing Audit University
    Hainan Normal University)

Abstract

Let {Xn, n ≥ 1} be a sequence of stationary non-negative associated random variables with common marginal distribution function F(x) and quantile function Q(u), where Q(u) is defined as F(Q(u)) = u. Here we consider the smooth estimation of Q(u), adapted from generalized kernel smoothing (Cheng and Parzen J. Stat. Plann. Infer. 59, 291–307, 1997) of the empirical quantile function. Some asymptotic properties of the kernel quantile estimator, for associated sequences, are also established parallel to those in the i.i.d. case. Various estimators in this class of estimators are contrasted, through a simulation study, among themselves and with an indirect smooth quantile estimator obtained by inverting the Poisson weights based estimator of the distribution function studied in Chaubey et al. (Statist. Probab. Lett. 81, 267–276, 2011). The indirect smoothing estimator seems to be the best estimator on account of smaller MSE, however, a quantile estimator based on the Bernstein polynomials and that using the corrected Poisson weights turn out to be almost as good as the inverse distribution function estimator using Poisson weights.

Suggested Citation

  • Yogendra P. Chaubey & Isha Dewan & Jun Li, 2021. "On Some Smooth Estimators of the Quantile Function for a Stationary Associated Process," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 114-139, May.
    • Handle: RePEc:spr:sankhb:v:83:y:2021:i:1:d:10.1007_s13571-020-00242-x
    DOI: 10.1007/s13571-020-00242-x
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    References listed on IDEAS

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    1. Bagai, Isha & Prakasa Rao, B. L. S., 1991. "Estimation of the survival function for stationary associated processes," Statistics & Probability Letters, Elsevier, vol. 12(5), pages 385-391, November.
    2. Balakrishnapillai Vineshkumar & Narayanan Unnikrishnan Nair, 2019. "Bivariate Quantile Functions and their Applications to Reliability Modelling," Statistica, Department of Statistics, University of Bologna, vol. 79(1), pages 3-21.
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    4. Mahboubeh Akbari & Majid Rezaei & Sarah Jomhoori & Vahid Fakoor, 2019. "Nonparametric estimators for quantile density function under length-biased sampling," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(19), pages 4918-4935, October.
    5. Chen, Song Xi, 1999. "Beta kernel estimators for density functions," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 131-145, August.
    6. M. Jones, 1992. "Estimating densities, quantiles, quantile densities and density quantiles," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 44(4), pages 721-727, December.
    7. Lai Wei & Dongliang Wang & Alan D. Hutson, 2015. "An Investigation of Quantile Function Estimators Relative to Quantile Confidence Interval Coverage," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 44(10), pages 2107-2135, May.
    8. Chaubey, Yogendra P. & Dewan, Isha & Li, Jun, 2011. "Smooth estimation of survival and density functions for a stationary associated process using Poisson weights," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 267-276, February.
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    Cited by:

    1. Nour-Eddine Berrahou & Salim Bouzebda & Lahcen Douge, 2024. "The Bahadur Representation for Empirical and Smooth Quantile Estimators Under Association," Methodology and Computing in Applied Probability, Springer, vol. 26(2), pages 1-37, June.

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