Bruno Rémillard

Emprical Processes Based on Pseudo-Observations*

Publisher Summary Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions θ (X) of the observable random variable X. Many researchers are using empirical distribution functions constructed from residuals, which are estimations of a nonobservable error terms in linear models. This falls under a class of more general problems, in which one is interested in the estimation of the distribution function of a nonobservable random variable depending on an observable random variable X, together with its unknown law Q. Some work is done recently when the pseudo-observations are the so-called residuals of linear models. The aim of this paper is to provide some tools to study the asymptotic behavior of empirical processes constructed from general pseudo-observations. The aim of this paper is to provide some tools to study the asymptotic behavior of empirical processes constructed from general pseudo-observations.

Relating quantiles and expectiles under weighted-symmetry

Recently, quantiles and expectiles of a regression function have been investigated by several authors. In this work, we give a sufficient condition under which a quantile and an expectile coincide. We extend some classical results known for mean, median and symmetry to expectiles, quantiles and weighted-symmetry. We also study split-models and sample estimators of expectiles.

Rank-Based Extensions of the Brock, Dechert, and Scheinkman Test

This article proposes new tests of randomness for innovations in a large class of time series models. These tests are based on functionals of empirical processes constructed from either the model residuals or their associated ranks. The asymptotic behavior of these processes is determined under the null hypothesis of randomness. The limiting distributions are seen to be independent of estimation errors under appropriate regularity conditions. Several test statistics are derived from these processes; the classical Brock, Dechert, and Scheinkman statistic and a rank-based analog are included as special cases. Because the limiting distributions of the rank-based test statistics are margin-free, their finite-sample p values can be easily calculated by simulation. Monte Carlo experiments show that these statistics are quite powerful against several classes of alternatives.

A limit theorem for Brownian motion in a random scenery

We find the limiting distribution ofl/an Jjf V(BU)du,t G [0,1 ], where {Bu}u>o is the standard Brownian motion on K , V is a particular random potential and {an}n>i is a normalizing sequence.

A note on tightness

This note describes an extension of Billingsleys classical tightness criterion for sequences of cadlag processes on [0, 1]. Applications of the new criterion to the convergence of Gaussian and other processes in D[0, 1] are provided.

Laws of the iterated logarithm and large deviations for a class of diffusionl processes

We use the martingale approach to study large deviations and laws of the iterated logarithm for certain multidimensional diffusion processes. The criteria for the validity of these properties are expressed in terms of averaging properties of the coefficients of the infinitesimal generator. In particular we apply our results to diffusion processes with random coefficients. Utilisant la methode des martingales, nous etudions les proprietes de grandes deviations ainsi que les lois du logarithme itere pour une classe de processus de diffusion multidimensionels. Les criteres de validite de ces proprietes sont exprimes en termes de propriete de moyenne des coefficients du generateur infinitesimal. En particulier, nous appliquons nos resultats aux processus de diffusion avec coefficients aleatoires.

Occupation times in systems of null recurrent Markov processes

SummaryWe study asymptotic properties of differences of occupation times for infinite systems of noninteracting Markovian particles. Under a suitable normalisation we prove convergence in law to a nondegerate Gaussian field. We also obtain large deviations properties. These results generalise previous results obtained separately by both authors.

Estimation Based on the Empirical Characteristic Function

There exist distributions for which standard estimation techniques based on the probability density function are not applicable. As an alternative, the characteristic function is used. Certain distributions whose characteristic functions can be expressed in terms of |t|α are such examples. Tailweight properties are first examined; it is shown that these laws are Paretian, their tail index a being one of the parameters defining these laws. Estimators similar to those proposed by Press (1972) for stable laws are then used for the estimation of the parameters of such laws and asymptotic properties are proved. As an illustration, the Linnik distribution is examined.