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Asymptotic behavior of central order statistics from stationary processes

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  • Dembińska, Anna

Abstract

In this paper, we show that central order statistics from strictly stationary and ergodic sequences are strongly consistent estimators of population quantiles provided that the quantiles are unique. We generalize this result to strictly stationary but not necessarily ergodic sequences. We also describe three types of possible asymptotic behavior of central order statistics in the case when the corresponding population quantile is not unique. We give applications of the presented results to linear processes with both absolutely continuous and discrete innovations.

Suggested Citation

  • Dembińska, Anna, 2014. "Asymptotic behavior of central order statistics from stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 348-372.
    • Handle: RePEc:eee:spapps:v:124:y:2014:i:1:p:348-372
    DOI: 10.1016/j.spa.2013.08.001
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    1. 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.
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    3. Miao, Yu & Yang, Guangyu, 2008. "The law of the iterated logarithm for additive functionals of Markov chains," Statistics & Probability Letters, Elsevier, vol. 78(3), pages 265-270, February.
    4. Pham, Tuan D. & Tran, Lanh T., 1985. "Some mixing properties of time series models," Stochastic Processes and their Applications, Elsevier, vol. 19(2), pages 297-303, April.
    5. Dutta, Kalyan & Sen, Pranab Kumar, 1971. "On the Bahadur representation of sample quantiles in some stationary multivariate autoregressive processes," Journal of Multivariate Analysis, Elsevier, vol. 1(2), pages 186-198, June.
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