Nonparametric inference and Bayesian computing

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The nonparametrics that I developed (during the period of my previous discovery grant), so far in the context of inference on constrained functions, has shown to work well. The functions of interest here are constrained essentially because they characterize a copula, being for one a distribution function, with uniform margins. Copulas are of fundamental importance because they hold the entire dependency structure between the variates of a random vector, and so they appear naturally in many hypothesis tests. When they are combined with marginals, they become a powerful tool for modelling joint distributions. Much of the inferential methodology for the dependence structure has found concrete applications in many applied sciences, such as genetics and econometrics. The inferential approach is usually frequentist, and in many cases nonparametric. In the latter setup, processes linked to the copula (e.g the Kendall process, or empirical copula process) are the main working tools. These do not usually lead to intrinsic estimators for finite samples, and one often needs to modify them in order to satisfy the constraints, an inconvenience here. Another approach is to construct models by further exploiting the geometry of the problem and providing infinite dimensional models, still remaining tractable. Continuing in this direction, I have two main objectives for the next years. ***My first objective is to work on extensions of sieves models that I have studied in dimension 2. I want to explore Bayesian possibilities, the advantage being essentially prediction, where the uncertainty of the future observations is naturally combined to that of the parameters via the predictive density.***In my previous work, I have always made the assumption that the marginal distributions are known or that they belong to some family of sieves. My second objective consists in removing all assumptions on the marginal distributions using ranks only. It is natural to work with ranks when inferring on the dependence structure for two main reasons discussed in the proposal. Essentially, the effect of the nuisance parameters here (the margins) is eliminated, and in particular, the prior on the margins has no effect on the posterior. A main (challenging) goal is the calculation of the rank-likelihood. While asymptotics are currently being developed here, I will focus on the construction of good stochastic approximations via rapidly mixing Markov chains, for finite samples.****A frequentist competitor that uses ranks only is the empirical copula (which is not a genuine copula). Moreover, the covariance structure of the limiting Gaussian field process is complicated and bootstrap approximations become essential. I propose a model based estimator, that has the same limiting process, but hope to show that it has better finite samples behaviour than the empirical copula. In particular, the model should give better bootstrap approximations.**