Opuscula Math. 36, no. 1 (2016), 25-48
http://dx.doi.org/10.7494/OpMath.2016.36.1.25

Opuscula Mathematica

Kernel conditional quantile estimator under left truncation for functional regressors

Nacéra Helal
Elias Ould Saïd

Abstract. Let $$Y$$ be a random real response which is subject to left-truncation by another random variable $$T$$. In this paper, we study the kernel conditional quantile estimation when the covariable $$X$$ takes values in an infinite-dimensional space. A kernel conditional quantile estimator is given under some regularity conditions, among which in the small-ball probability, its strong uniform almost sure convergence rate is established. Some special cases have been studied to show how our work extends some results given in the literature. Simulations are drawn to lend further support to our theoretical results and assess the behavior of the estimator for finite samples with different rates of truncation and sizes.

Keywords: almost sure convergence, functional variables, kernel estimator, Lynden-Bell estimator, small-ball probability, truncated data.

Mathematics Subject Classification: 62G05, 62G07, 62G20, 62M09.

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• Nacéra Helal
• Département de Mathématiques, Université Djillali Liabès, BP 89, 22000, Sidi Bel Abbès, Algeria
• Elias Ould Saïd
• Université Lille Nord de France, F-59000 Lille, France
• ULCO, LMPA, CS: 80699 Calais, France
• Communicated by Zbigniew Szkutnik.
• Revised: 2015-02-26.
• Accepted: 2015-03-21.
• Published online: 2015-09-19.