Jean-Marie Rolin

Consistency of the beta kernel density function estimator

The authors give the exact asymptotic behaviour of the expected average absolute error of a beta kernel density estimator proposed by Chen (1999). They also prove the uniform weak consistency of this estimator for the class of continuous densities.

Noncausality and Marginalization of Markov-processes

In this paper it is shown that a subprocess of a Markov process is markovian if a suitable condition of noncausality is satisfied. Furthermore, a markovian condition is shown to be a natural condition when analyzing the role of the horizon (finite or infinite) in the property of noncausality. We also give further conditions implying that a process is both jointly and marginally markovian only if there is both finite and infinite noncausality and that a process verifies both finite and infinite noncausality only if it is markovian. Counterexamples are also given to illustrate the cases where these further conditions are not satisfied.

Identification of the 1PL Model with Guessing Parameter: Parametric and Semi-parametric Results

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available.

Product-limit estimators of the survival function with twice censored data

A model for competing (resp. complementary) risks survival data where the failure time can be left (resp. right) censored is proposed. Product-limit estimators for the survival functions of the individual risks are derived. We deduce the strong convergence of our estimators on the whole real half-line without any additional assumptions and their asymptotic normality under conditions concerning only the observed distribution. When the observations are generated according to the double censoring model introduced by Turnbull, the product-limit estimators represent upper and lower bounds for Turnbulls estimator.

Non Parametric Bayesian Statistics: A Stochastic Process Approach

The specification of a prior distribution on the set of all distribution functions permits to consider a non parametric bayesian experiment as an abstract probability space on which are defined the sampling process and a stochastic distribution process.

On two definitions of identification

Summary. The paper analyzes the connection between the identification of a statistical experiment defined by the injectivity of the sampling probability indexation and by the minimal sufficiency of the c-field characterizing the parametrization. If the parameter space is a BLACKWELL space and if the observation generates a separable 7-field, these two concepts are proved to be equivalent.

Competing Risks Models : Problems of Modelling and of Identification

Competing risks models are presented in the framework of single transition-multiple causes models. Particular attention is paid to general distributions of latent durations and to model identification problems, which are different from parameter identification problems. Implications for modelling and interpreting empirical findings are considered. The exposition is fairly elementary and the techniques in use, although not the most usual ones, are deemed to be more powerful and, hopefully, more illuminating than the more traditional ones. In particular, continuous and discrete distributions are treated in a unified framework.