Aimé Lachal
Research Subjects
Aimé Lachal has a total of 13 co-authors. Here are the top co-authors by number of shared publications:
- Salim Bouzebda
Université de Technologie de Compiègne
4 shared publications - Sergio Alvarez-Andrade
Université de Technologie de Compiègne
3 shared publications E. Orsingher
2 shared publicationsValentina Cammarota
2 shared publications- Valentina Cammarota
Department of Statistics, Sapienza University of Rome, Rome, Italy
2 shared publications - Enzo Orsingher
Dipartimento di Scienze Statistiche, Sapienza Universitá di Roma, Rome, Italy
1 shared publication
Latest Publications
- 13 December 2017
- Published by Springer Nature in TEST
- Vol. 27 (4) , 826-849
Strong approximations for the p-fold integrated empirical process with applications to statistical tests
- 13 December 2017
- Published by Springer Nature in TEST
- Vol. 27 (4) , 826-849
Strong approximations for the p-fold integrated empirical process with applications to statistical tests
- Open Access
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 results of Koml\'os, Major and Tusn\'ady (1975). We also obtain an exponential bound for the tail probability of the weighted approximation to the $p$-fold integrated empirical process. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. We study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes.Open AccessStrong 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 results of Koml\'os, Major and Tusn\'ady (1975). We also obtain an exponential bound for the tail probability of the weighted approximation to the $p$-fold integrated empirical process. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. We study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes. Finally, simulation results are provided to illustrate the finite sample performance of the proposed statistical tests based on the integrated empirical processes. - 3 April 2016
- Published by Taylor & Francis in Communications in Statistics - Theory and Methods
- Vol. 46 (7) , 3365-3392
Some asymptotic results for the integrated empirical process with applications to statistical tests
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 Komlós, J. , Major, P. , Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32:111–131. [Crossref], [Web of Science ®] [Google Scholar] )'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.
- 3 April 2016
- Published by Taylor & Francis in Communications in Statistics - Theory and Methods
- Vol. 46 (7) , 3365-3392
Some asymptotic results for the integrated empirical process with applications to statistical tests
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 Komlós, J. , Major, P. , Tusnády, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32:111–131. [Crossref], [Web of Science ®] [Google Scholar] )'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.
- Open Access
Some asymptotic results for the integrated empirical process with applications to statistical tests
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 Koml\'os, Major and Tusn\'ady'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. Reference: Koml\'os, J., Major, P. and Tusn\'ady, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32, 111-131.
Open AccessSome asymptotic results for the integrated empirical process with applications to statistical tests
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 Koml\'os, Major and Tusn\'ady'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. Reference: Koml\'os, J., Major, P. and Tusn\'ady, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32, 111-131.
- Open Access
Entrance and sojourn times for Markov chains. Application to $(L,R)$-random walks
In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.
Open AccessEntrance and sojourn times for Markov chains. Application to $(L,R)$-random walks
In this paper, we provide a methodology for computing the probability distribution of sojourn times for a wide class of Markov chains. Our methodology consists in writing out linear systems and matrix equations for generating functions involving relations with entrance times. We apply the developed methodology to some classes of random walks with bounded integer-valued jumps.
- Open Access
- 26 March 2014
- Published by Hindawi Limited in International Journal of Stochastic Analysis
- Vol. 2014, 1-49
From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval
Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the (positive or negative) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.- 26 March 2014
- Published by Hindawi Limited in International Journal of Stochastic Analysis
- Vol. 2014, 1-49
Open AccessFrom Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval
Let be a positive integer, a positive constant and be a sequence of independent identically distributed pseudorandom variables. We assume that the ’s take their values in the discrete set and that their common pseudodistribution is characterized by the (positive or negative) real numbers for any . Let us finally introduce the associated pseudorandom walk defined on by and for . In this paper, we exhibit some properties of . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess. - Open AccessFirst exit time from a bounded interval for pseudo-processes driven by the equation $\partial/\partial t=(-1)^{N-1}\partial^{2N}/\partial x^{2N}$Let $N$ be a positive integer. We consider pseudo-Brownian motion $X=(X(t))_{t\ge 0}$ driven by the high-order heat-type equation $\partial/\partial t=(-1)^{N-1}\partial^{2N}/\partial x^{2N}$. Let us introduce the first exit time {\tau}ab from a bounded interval $(a,b)$ by $X$ ($a,b\in\mathbb{R}$). In this paper, we provide a representation of the joint pseudo-distribution of the vector $(\tau_{ab},X(\tau_{ab}))$ by means of Vandermonde-like determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of the location of $X$ at time $\tau_{ab}$, namely $X(\tau_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.Open AccessFirst exit time from a bounded interval for pseudo-processes driven by the equation $\partial/\partial t=(-1)^{N-1}\partial^{2N}/\partial x^{2N}$Let $N$ be a positive integer. We consider pseudo-Brownian motion $X=(X(t))_{t\ge 0}$ driven by the high-order heat-type equation $\partial/\partial t=(-1)^{N-1}\partial^{2N}/\partial x^{2N}$. Let us introduce the first exit time {\tau}ab from a bounded interval $(a,b)$ by $X$ ($a,b\in\mathbb{R}$). In this paper, we provide a representation of the joint pseudo-distribution of the vector $(\tau_{ab},X(\tau_{ab}))$ by means of Vandermonde-like determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of the location of $X$ at time $\tau_{ab}$, namely $X(\tau_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.
- 1 February 2014
- Published by Elsevier in Stochastic Processes and their Applications
- Vol. 124 (2) , 1084-1111
First exit time from a bounded interval for pseudo-processes driven by the equation- 1 February 2014
- Published by Elsevier in Stochastic Processes and their Applications
- Vol. 124 (2) , 1084-1111
First exit time from a bounded interval for pseudo-processes driven by the equation- Open AccessA survey on the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial x^N$ concerning the first hitting times and sojourn timesFix an integer n>2 and let $(X(t))_{t\ge 0}$ be the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial x^N$. The denomination "pseudo-process" means that $(X(t))_{t\ge 0}$ is related to a signed measure (which is not a probability measure) with total mass equal to 1. In this note, we present some results and discuss some problems concerning the pseudo-distributions of the first overshooting times of a single barrier $\{a\}$ or a double barrier $\{a,b\}$ by $(X(t))_{t\ge 0}$, as well as those of the sojourn times of $(X(t))_{t\ge 0}$ in the intervals $[a,+\infty)$ and $[a,b]$ up to a fixed time.Open AccessA survey on the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial x^N$ concerning the first hitting times and sojourn timesFix an integer n>2 and let $(X(t))_{t\ge 0}$ be the pseudo-process driven by the high-order heat-type equation $\partial/\partial t=\pm\partial^N/\partial x^N$. The denomination "pseudo-process" means that $(X(t))_{t\ge 0}$ is related to a signed measure (which is not a probability measure) with total mass equal to 1. In this note, we present some results and discuss some problems concerning the pseudo-distributions of the first overshooting times of a single barrier $\{a\}$ or a double barrier $\{a,b\}$ by $(X(t))_{t\ge 0}$, as well as those of the sojourn times of $(X(t))_{t\ge 0}$ in the intervals $[a,+\infty)$ and $[a,b]$ up to a fixed time.
- Open Access
A trick around Fibonacci, Lucas and Chebyshev
In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of this trick to more general sequences. This study leads us to interesting connections between Fibonacci, Lucas sequences and Chebyshev polynomials.
Open AccessA trick around Fibonacci, Lucas and Chebyshev
In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of this trick to more general sequences. This study leads us to interesting connections between Fibonacci, Lucas sequences and Chebyshev polynomials.