Local linear conditional cumulative distribution function with mixing data
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Oussama Bouanani
Arabian Journal of Mathematics
· Saâdia Rahmani · Larbi Ait-Hennani
Local linear conditional cumulative distribution function with mixing data
Received: 31 May 2018 / Accepted: 14 February 2019 © The Author(s) 2019
Abstract This paper investigates a conditional cumulative distribution of a scalar response given by a functional random variable with an α-mixing stationary sample using a local polynomial technique. The main purpose of this study is to establish asymptotic normality results under selected mixing conditions satisfied by many time-series analysis models in addition to the other appropriate conditions to confirm the planned prospects. Mathematics Subject Classification
62G05 · 62G08 · 62G20 · 62G07 · 62G30 · 62H12
1 Introduction and motivations With the evolution of the measuring instruments and the growth of research studies mainly since the publication of the pioneer paper of Ferraty and Vieu [19], functional data analysis (FDA) has attracted the attention of many works as in the recent monograph of Horváth and Kokoszka [23]. On the other hand, alternative conditional predictions of the classical regression have also gained a considerable interest in basically all the fields of statistics, especially for estimating conditional models using the kernel approach (or local constant) as investigated in the papers of Ferraty et al. [18], Dabo-Niang and Laksaci [8], or Ezzahrioui and Ould-Saïd [15]. In numerous nonparametric statistic problems, the estimation of a conditional distribution function (CDF) constitutes a key aspect of inference. Accordingly, the present study employs a specific CDF model for constructing prediction intervals that can be involved in many applications such as the survival analysis and reliability. Interestingly, it is well known that the CDF has the advantage to completely characterize the conditional law of the considered random variables. The determination of the CDF allows, in fact, to obtain the conditional density and conditional hazard functions. Moreover, several prediction tools can also be implemented for the nonparametric statistics modeling, taking the example of conditional mode, median, or quantile. In addition, an extensive literature including various nonparametric approaches has taken place in the conditional estimation of independent samples and dependent observations in finite- and infinite-dimensional spaces (see, for instance, Berlinet et al. [3], Honda [22], and Ferraty and Vieu [17]). In many situations, the O. Bouanani (B) · S. Rahmani Laboratoire de Modèles Stochastiques, Statistique et Applications, Université Docteur Moulay Taher, 20000 Saida, Algeria E-mail: [email protected] S. Rahmani E-mail: [email protected] L. Ait-Hennani Université Lille 2, Droit et santé, IUT C, Roubaix, France E-mail: [email protected]
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kernel approach cannot adequately estimate the conditional models for the reason that this technique suffers from a large bias particularly at the boundary region. However, the
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