Christophe Ley, ULB : Maximum likelihood characterizations of distributions: a generalization of Gauss's principle
Abstract:It is a well-known fact that the maximum likelihood estimate (MLE) of the location parameter in a normal population is given by the sample mean. It is less well-known that the converse holds true as well: if the MLE of a location parameter of a population is given by the sample mean, then the distribution is necessarily of normal type. This result is due to Gauss and commonly referred to as Gauss' principle. Several extensions of Gauss' result have been proposed in the literature. For example, Teicher~(1961) studies a similar MLE characterization of the normal and the exponential distribution, but with respect to the scale parameter (the result for the exponential has been later on extended by Marshall and Olkin~1993 to the case of Gamma distributions), while Ferguson~(1962) characterizes a one-parameter generalized normal distribution via the MLE of its location parameter. MLE characterizations have also been examined for spherical distributions (i.e., distributions taking their values only on the unit sphere) in dimensions $k>1$ (e.g., by Bingham and Mardia 1975). In this talk, we shall propose a unified framework for (i) the univariate and (ii) the spherical MLE characterizations, which allows us to retrieve all existing results and to construct quite easily many new ones.
Catherine Legrand, UCL : A general class of time-varying coefficients models for right censored data.
Abstract: While the Cox proportional hazards model is probably the most popular one to model survival data, there exist several alternatives, such as the additive risk model and the proportional odds model. These models assume that the regression coefficients are constant over time. However, in practice, this is not always the case, e.g., the impact of a prognostic value (measured at baseline) may decline over time. Following the ideas of McCullagh and Nelder (1989) on generalized linear models, all these models can be studied in an unified framework using a link function :[0,1] to model the impact of the covariates on the transformed conditional survival function (S(y|x)). Extensions of these various models to time-dependent coefficients can also be studied in this unified framework. Teodorescu and Van Keilegom (2010) proposed a least squares based estimation methods for the model (S(y|x))=O(y) + 1(y)x, with X a continuous covariate and Y possibly right-censored event-times. Building further on their work, we will discuss the practical implementation of this method, including the choice of the bandwidth necessary to estimate S(y|x) non parametrically . We will also discuss for various choice of the link function, what are possible choices for introducing the time-dependency of the coefficients. This will reveal concrete families of conditional survival functions, resulting in an easy way to generate data from models with time-dependent regression coefficients. Results on a real bladder cancer database will be used to illustrate how the proposed procedure works in practice and results of preliminary simulations will also be discussed.