Charles El-Nouty

The Lower Classes of the Sub-Fractional Brownian Motion

Let \(\{{B}_{H}(t),t \in{\mathbb{R}}^{\}}\) be a fractional Brownian motion with Hurst index 0 < H < 1. Consider the sub-fractional Brownian motion X H defined as follows :

On the smoothed bootstrap

{X}_{H}(t) = \dfrac{{B}_{H}(t) + {B}_{H}(-t)} {\sqrt{2}},t \geq0.

On the large increments of fractional Brownian motion

We characterize the lower classes of the sup-norm statistic of X H by an integral test.

A Hanson-Russo-type law of the iterated logarithm for fractional Brownian motion

Abstract. A niajor question in extreme value theory is to obtain workable finite sample confidence intervcds for the Pareto index. The first answer was given by Caers et al. [1]. They suggested a non peirametric bootstrap Solution and validated it by providing Monte Carlo simulations. The problem addressed in the present paper is the validity (from a theoretical point of view) of the bootstrap method in extreme value theory. To this aim, we consider the naive estimator of the tail index, introduced by Csörgö et al. [3], and use Los representation method [12] for analyzing bootstrap accuracy. Denote by ö a Parameter of interest, d„ an estimator of 6, based on the original sample, and ö* its bootstrapped version. Los approach [12] consists of expressing (Ö* — &„) as the sum of two terms, one of which has the same distribution as {

The increments of a sub-fractional Brownian motion

„ — 9), and the other is a relatively negligible remainder. We illustrate how this method may be used to determine the bootstrap accuracy of the Pareto index.

On a convergent stochastic estimation algorithm for frailty models

We give necessary conditions for the smoothed boostrapped mean to be consistent. Then, we apply Edgeworth expansions to show that better confidence intervals in terms of coverage probability can be obtained using the smoothed bootstrap than via the standard Efrons bootstrap.

Parametric estimation of a prey-predator model with Allee effect

Abstract A new estimator (the presmoothed one) of the hazard cumulative function in a competing risk model is introduced. Its asymptotic properties are proved. As soon as the censorship is not too heavy, numerical studies show its efficiency in terms of mean squarred errors, compared to the classic Nelson–Aalen estimator.

Some properties of the integrated fractional Brownian motion

Abstract Let { B H ( t ), t ⩾ 0} be a fractional Brownian motion with index 0 H V T = sup 0⩽s⩽T−a T β T |B H (s + a T ) − B H (s)| , where β T and α T are suitably chosen functions of T ⩾ 0. We establish some laws of the iterated logarithm for V T .