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Uniform strong estimation under [alpha]-mixing, with rates

Author

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  • Cai, Zongwu
  • Roussas, George G.

Abstract

Let s{;Xns};, n [greater-or-equal, slanted] 1, be a stationary [alpha]-mixing sequence of real-valued r.v.'s with distribution function (d.f.) F, probability density function (p.d.f.) f and mixing coefficient [alpha](n). The d.f. F is estimated by the empirical d.f. Fn, based on the segment X1,..., Xn. By means of a mixingale argument, it is shown that Fn(x) converges almost surely to F(x) uniformly in x[set membership, variant]. An alternative approach, utilizing a Kiefer process approximation, establishes the law of the iterated logarithm for sups{;vb;Fn(x)-F(xvb;; x[set membership, variant]. The d.f. F is also estimated by a smooth estimate n, which is shown to converge almost surely (a.s.) to F, and the rate of convergence of sups{;vb;n(x) - F(x)vb;;; x[set membership, variant]s}; is of the order of O((log log n/n)). The p.d.f. f is estimated by the usual kernel estimate fn, which is shown to converge a.s. to f uniformly in x[set membership, variant], and the rate of this convergence is of the order of O((log log n/nh2n)), where hn is the bandwidth used in fn. As an application, the hazard rate r is estimated either by rn or n, depending on whether Fn or n is employed, and it is shown that rn(x) and n(x) converge a.s. to r(x), uniformly over certain compact subsets of , and the rate of convergence is again of the order of O((log log n/nh2n)). Finally, the rth order derivative of f, f(r), is estimated by f(r)n, and is shown that f(r)n(x) converges a.s. to f(r)(x) uniformly in x[set membership, variant].The rate of this convergence is of the order of O((log log n/nh2(r+1)n)).

Suggested Citation

  • Cai, Zongwu & Roussas, George G., 1992. "Uniform strong estimation under [alpha]-mixing, with rates," Statistics & Probability Letters, Elsevier, vol. 15(1), pages 47-55, September.
  • Handle: RePEc:eee:stapro:v:15:y:1992:i:1:p:47-55
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    Cited by:

    1. Gao, Min & Yang, Wenzhi & Wu, Shipeng & Yu, Wei, 2022. "Asymptotic normality of residual density estimator in stationary and explosive autoregressive models," Computational Statistics & Data Analysis, Elsevier, vol. 175(C).
    2. R. Maya & E. Abdul-Sathar & G. Rajesh & K. Muraleedharan Nair, 2014. "Estimation of the Renyi’s residual entropy of order $$\alpha $$ with dependent data," Statistical Papers, Springer, vol. 55(3), pages 585-602, August.
    3. Cai, Zongwu, 1998. "Asymptotic properties of Kaplan-Meier estimator for censored dependent data," Statistics & Probability Letters, Elsevier, vol. 37(4), pages 381-389, March.
    4. Xin Chen & Jieli Ding & Liuquan Sun, 2018. "A semiparametric additive rate model for a modulated renewal process," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 24(4), pages 675-698, October.
    5. Nour El Houda Rouabah & Nahima Nemouchi & Fatiha Messaci, 2019. "A rate of consistency for nonparametric estimators of the distribution function based on censored dependent data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 28(2), pages 259-280, June.

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