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Local linear approach: conditional density estimate for functional and censored data

Benkhaled Abdelkader, Madani Fethi

Demonstratio Mathematica

2022

Vol. 55, nr 1

315--327

Let Y be a random real response, which is subject to right censoring by another random variable C . In this paper, we study the nonparametric local linear estimation of the conditional density of a scalar response variable and when the covariable takes values in a semi-metric space. Our main aim is to prove under some regularity conditions both the pointwise and the uniform almost-sure consistencies with convergence rates of the conditional density estimator related by this estimation procedure.

Kernel conditional quantile estimator under left truncation for functional regressors

Helal N., Said E. O.

Opuscula Mathematica

2016

Vol. 36, no. 1

25--48

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.

Path regularity of Gaussian processes via small deviations

Aurzada F.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 1

61--78

We study the a.s. sample path regularity of Gaussian processes. To this end we relate the path regularity directly to the theory of small deviations. In particular, we show that if the process is n-times differentiable, then the exponential rate of decay of its small deviations is at most ε-1/n. We also show a similar result if n is not an integer. Further generalizations are given, which parallel the entropy method to determine the small deviations. In particular, the present approach seems to be a probabilistic interpretation of the multiplicativity property of the entropy numbers.