Sergio Alvarez-Andrade

Some Nonparametric Tests for Change-Point Detection Based on the ℙ-ℙ and ℚ-ℚ Plot Processes

Abstract We propose nonparametric procedures for testing change-point by using the ℙ-ℙ and ℚ-ℚ plots processes. The limiting distributions of the proposed statistics are characterized under the null hypothesis of no change and also under contiguous alternatives. We give an estimator of the change-point coefficient and obtain its strong consistency. We introduce the bootstrapped version of ℙ-ℙ and ℚ-ℚ processes, requiring the estimation of quantile density, and obtain their limiting laws. Finally, we propose and investigate the exchangeable bootstrap of the empirical ℙ-ℙ plot and ℚ-ℚ plot processes which avoids the problem of the estimation of quantile density, which is of its own interest. These results are used for calculating p-values of the proposed test statistics. Emphasis is placed on the explanation of the strong approximation methodology.

Some asymptotic results for the integrated empirical process with applications to statistical tests

ABSTRACT The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial-sum process representation of the integrated empirical process.

On the Local Time of the Weighted Bootstrap and Compound Empirical Processes

This article is mainly concerned with the local times of the weighted bootstrap process. We prove a strong approximation theorem for the local time of the weighted bootstrap process by the local time of a Brownian bridge. We consider also the local time of the compound empirical processes that can be seen, asymptotically, as the local time of the convolution of two independent Gaussian processes.

On Complete Convergence for the Hybrid Process

We consider the hybrids of empirical and partial-sum processes given by (1) under suitable conditions on the sequence of random variables {Xi}i ⩾ 1 and {εi}i ⩾ 1 given below. Our aim here is to establish some asymptotic results of some weighted series of , for some function gn(ε) of n and ε → 0 with (i.e., the sup-norm) and ‖f‖ = ‖f‖2 = (∫10(f(t))2dt)1/2. This work is in the same spirit of Zhang and Yang (2008) where the authors have considered the uniform empirical process under the sup-norm ‖f‖ = ‖f‖[0, 1].

Almost sure central limit theorem for the hybrid process

ABSTRACT In this paper, we investigate some problems related to the almost sure central limit theorem for the hybrids of empirical and partial sum processes. More precisely, under mild technical conditions, we study the almost sure (a.s.) convergence of where denote the hybrids of empirical and partial sum processes, is a bounded Lipschitz function, denotes a sequence of positive weights, and

Strong approximations for the p -fold integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the p-fold integrated empirical process, p being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (Z Wahrscheinlichkeitstheorie und Verw Gebiete 32:111–131, 1975)’s results. Applications include the two-sample testing procedures together with the change-point problems. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.