Abstract
In this work, we are concerned with nonparametric estimation of the distribution function when the data are possibly censored and satisfy the \(\alpha \)-mixing condition, also called strong mixing. Among various mixing conditions used in the literature, \(\alpha \)-mixing is reasonably weak and has many practical applications as it is fulfilled by many stochastic processes including some time series models. In practice the observed data can be complete or subject to censorship, so we deal with these different cases. More precisely, the rate of the almost complete convergence is established, under the \(\alpha \)-mixing condition, for complete, singly censored and twice censored data. To lend further support to our theoretical results, a simulation study is carried out to illustrate the good accuracy of the studied method, for relatively small sample sizes. Finally, an application to censored dependent data is provided via the analysis of Chromium concentrations collected from two stations of the Niagara River in Canada.
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Notes
Let \((Z_n)\) be a sequence of real random variables and let Z be a real random variable. The sequence \((Z_n)\) converges almost completely to Z means that the probability that the distance between \(Z_n\) and Z exceeds a strictly positive number (as small as it is) tends to zero quickly enough to be the term of a convergent series.
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Acknowledgements
The authors are very pleased to thank the editor and the reviewers. Funding was provided by Ministère de l’Enseignement Supérieur et de la Recherche scientifique, Algeria.
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Rouabah, N.E.H., Nemouchi, N. & Messaci, F. A rate of consistency for nonparametric estimators of the distribution function based on censored dependent data. Stat Methods Appl 28, 259–280 (2019). https://doi.org/10.1007/s10260-018-00445-7
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DOI: https://doi.org/10.1007/s10260-018-00445-7