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On the Bahadur representation of sample quantiles for sequences of [phi]-mixing random variables


  • Sen, Pranab Kumar


The object of the present investigation is to show that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of [phi]-mixing random variables. Two different orders of the remainder term, under different [phi]-mixing conditions, are obtained and used for proving two functional central limit theorems for sample quantiles. It is also shown that the law of iterated logarithm holds for quantiles in stationary [phi]-mixing processes.

Suggested Citation

  • Sen, Pranab Kumar, 1972. "On the Bahadur representation of sample quantiles for sequences of [phi]-mixing random variables," Journal of Multivariate Analysis, Elsevier, vol. 2(1), pages 77-95, March.
  • Handle: RePEc:eee:jmvana:v:2:y:1972:i:1:p:77-95

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    Cited by:

    1. Ghalibaf, M. Bolbolian & Fakoor, V. & Azarnoosh, H.A., 2010. "Strong Gaussian approximations of product-limit and quantile processes for truncated data under strong mixing," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 581-586, April.
    2. Coeurjolly, Jean-François, 2008. "Bahadur representation of sample quantiles for functional of Gaussian dependent sequences under a minimal assumption," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2485-2489, October.
    3. Ho, Hwai-Chung, 2015. "Sample quantile analysis for long-memory stochastic volatility models," Journal of Econometrics, Elsevier, vol. 189(2), pages 360-370.
    4. Ajami, M. & Fakoor, V. & Jomhoori, S., 2011. "The Bahadur representation for kernel-type estimator of the quantile function under strong mixing and censored data," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1306-1310, August.
    5. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    6. Dominicy, Yves & Hörmann, Siegfried & Ogata, Hiroaki & Veredas, David, 2013. "On sample marginal quantiles for stationary processes," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 28-36.
    7. Chen, E. Jack & Kelton, W. David, 2006. "Quantile and tolerance-interval estimation in simulation," European Journal of Operational Research, Elsevier, vol. 168(2), pages 520-540, January.
    8. Wendler, Martin, 2011. "Bahadur representation for U-quantiles of dependent data," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1064-1079, July.
    9. Bernd Heidergott & Warren Volk-Makarewicz, 2013. "A Measure-Valued Differentiation Approach to Sensitivity Analysis of Quantiles," Tinbergen Institute Discussion Papers 13-082/III, Tinbergen Institute.
    10. Lin Fan & Peter W. Glynn & Markus Pelger, 2018. "Change-Point Testing and Estimation for Risk Measures in Time Series," Papers 1809.02303,
    11. Qinchi Zhang & Wenzhi Yang & Shuhe Hu, 2014. "On Bahadur representation for sample quantiles under α-mixing sequence," Statistical Papers, Springer, vol. 55(2), pages 285-299, May.
    12. Bernd Heidergott & Warren Volk-Makarewicz, 2016. "A Measure-Valued Differentiation Approach to Sensitivities of Quantiles," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 293-317, February.
    13. Dembińska, Anna, 2014. "Asymptotic behavior of central order statistics from stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 348-372.
    14. Yoshihara, Ken-ichi, 1995. "The Bahadur representation of sample quantiles for sequences of strongly mixing random variables," Statistics & Probability Letters, Elsevier, vol. 24(4), pages 299-304, September.


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