BALABDAOUI Fadoua

The generalized linear model is an important method in the statistical toolkit. The isotonic single index model can be thought of as a further generalization whereby the link function is assumed to be monotone non-decreasing as opposed to known and fixed. Such a shape constraint is quite natural in many statistical problems, and is fulfilled by the usual generalized linear models. In this paper we consider inference in this model in the setting where repeated measurements of predictor values and associated responses are observed. This setting is encountered in medical studies and is very different from the one considered in the classical monotone single index model studied in the literature. Here, we use nonparametric maximum likelihood estimation to infer the unknown regression vector and link function. We present a detailed study of finite and asymptotic properties of this estimator and propose goodness-of-fit tests for the model. Through an extended simulation study, we show that the model has competitive predictive performance. We illustrate our estimation approach using a Leukemia data set.

Monotone single index models have gained over the past decades increasing popularity due to their flexibility and versatile use in diverse areas. Semi-parametric estimators such as the least squares and maximum likelihood estimators of the unknown index and monotone ridge function were considered to make inference in such models without having to choose some tuning parameter. Description of the asymptotic behavior of those estimators crucially depends on acquiring a good understanding of the optimization problems associated with the corresponding population criteria. In this paper, we give several insights into these criteria by proving existence of minimizers thereof over general classes of parameters. In order to describe these minimizers, we prove different results which give the direction of variation of the population criteria in general and in the special case where the common distribution of the covariates is Gaussian. A complementary simulation study was performed and whose results give support to our main theorems.

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Based on the convex least-squares estimator, we propose two different procedures for testing convexity of a probability mass function supported on N with an unknown finite support. The procedures are shown to be asymptotically calibrated.

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We study the monotone single index model where a real response variable $Y $ is linked to a $d$-dimensional covariate $X$ through the relationship $E[Y | X] = \Psi_0(\alpha^T_0 X)$ almost surely. Both the ridge function, $\Psi_0$, and the index parameter, $\alpha_0$, are unknown and the ridge function is assumed to be monotone on its interval of support. Under some regularity conditions, without imposing a particular distribution on the regression error, we show the $n^{-1/3}$ rate of convergence in the $\ell_2$-norm for the least squares estimator of the bundled function $\psi_0({\alpha}^T_0 \cdot),$ and also that of the ridge function and the index separately. Furthermore, we show that the least squares estimator is nearly parametrically rate-adaptive to piecewise constant ridge functions.

Based on the convex least-squares estimator, we propose two different procedures for testing convexity of a probability mass function supported on N with an unknown finite support. The procedures are shown to be asymptotically calibrated.

This thesis is a contribution to the field of non-parametric estimation under shape constraints. The functions are discrete and the form considered, called k-monotonicity, where k denotes an integer greater than 2, is a generalization of convexity. The integer k is an indicator of the degree of hollowness of a convex function. The manuscript is structured in three parts in addition to the introduction, the conclusion and an appendix.Introduction:The introduction includes three chapters. The first one presents a state of the art of density estimation under shape constraints. The second one is a synthesis of the results obtained during the thesis, available in French and in English. Finally, Chapter 3 gathers some notations and mathematical results used during the manuscript.Part I: Estimation of a discrete density under k-monotonicity constraintTwo least squares estimators of a discrete distribution p* under k-monotonicity constraint are proposed. Their characterization is based on the spline decomposition of k-monotonic sequences, and on the properties of their primitives. The statistical properties of these estimators are studied. Their estimation quality, in particular, is assessed. It is measured in terms of squared error, the two estimators converge at the parametric speed. An algorithm derived from the Support Reduction Algorithm is implemented and available in the R-package pkmon. A study on simulated data sets illustrates the properties of these estimators. This work has been published in Electronic Journal of Statistics (Giguelay, 2017).Part II: Risk Bound ComputationsIn the first chapter of Part II, the quadratic risk of the previously introduced least squares estimator is bounded. This bound is adaptive in the sense that it depends on a trade-off between the distance of p* from the boundary of the set of finitely supported k-monotone densities, and the complexity (in terms of decomposition in the spline basis) of the densities belonging to this set that are sufficiently close to p*. The method is based on a variational risk formulation proposed by Chatterjee (2014) andgeneralized to the density estimation framework. Subsequently, the bracketed entropies of the corresponding function spaces are computed to control the supremum of empirical processes involved in the squared error. The optimality of the risk bound is then discussed with respect to the results obtained in the continuous case and in the regression framework.In the second chapter of Part II, additional results on the bracketed entropies for k-monotone function spaces are given.Part III: Estimation of the number of species in a population and k-monotonicity testsThe last part deals with the problem of the estimation of the number of species in a population. The chosen model is that of an abundance distribution common to all species and defined as a mixture. The proposed method is based on the assumption of k-monotonicity of abundance. This hypothesis makes the problem of estimating the number of species identifiable. Two approaches are proposed. The first one is based on the least squares estimator under k-monotonicity constraint, while the second one is based on the empirical estimator. The two estimators are compared on a study on simulated data. Since the estimate of the number of species is strongly dependent on the degree of k-monotonicity chosen in the model, three multiple testing procedures are then proposed to infer the degree k directly on the basis of the observations. The level and power of these procedures are calculated and then evaluated by means of a study on simulated data sets and the method is applied on real data sets from the literature.

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In this article, we derive the weak limiting distribution of the least squares estimator (LSE) of a convex probability mass function (pmf) with a finite support. We show that it can be defined via a certain convex projection of a Gaussian vector. Furthermore, samples of any given size from this limit distribution can be generated using an efficient Dykstra-like algorithm.

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We show that the density of Z = argmaxfW (t) t 2 g, sometimes known as Cherno's density, is log-concave. We conjecture that Cherno's density is strongly log-concave or \super-Gaussian", and provide evidence in support of the conjecture. We also show that the standard normal den- sity can be written in the same structural form as Cherno's density, make connections with L. Bondesson's class of hyperbolically completely mono- tone densities, and identify a large sub-class thereof having log-transforms to R which are strongly log-concave.

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