Behaviour of the monotone single index model under repeated measurements.
Cecile DUROT, Fadoua BALABDAOUI, Hanna JANKOWSKI
2021
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.
On the population least-squares criterion in the monotone single index model.
Cecile DUROT, Christopher FRAGNEAU, Fadoua BALABDAOUI
2021
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.
On the population least‐squares criterion in the monotone single index model.
Fadoua BALABDAOUI, Cecile DUROT, Christopher FRAGNEAU
Statistica Neerlandica | 2021
No summary available.
Least squares estimation in the monotone single index model.
Fadoua BALABDAOUI, Cecile DUROT, Hanna JANKOWSKI
Bernoulli | 2019
No summary available.
Testing convexity of a discrete distribution.
Fadoua BALABDAOUI, Cecile DUROT, Babagnide francois KOLADJO, Francois KOLADJO
Statistics & Probability Letters | 2018
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.
Score estimation in the monotone single‐index model.
Fadoua BALABDAOUI, Piet GROENEBOOM, Kim HENDRICKX
Scandinavian Journal of Statistics | 2018
No summary available.
Least squares estimation in the monotone single index model.
Cecile DUROT, Fadoua BALABDAOUI, Hanna JANKOWSKI
2018
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.
Testing convexity of a discrete distribution.
Fadoua BALABDAOUI, Cecile DUROT, Francois KOLADJO
2018
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.
Least squares estimation of a discrete density under k-monotonicity constraints and risk bounds. Application to the estimation of the number of species in a population.
Jade GIGUELAY, Christophe GIRAUD, Pascal MASSART, Christophe GIRAUD, Pascal MASSART, Fadoua BALABDAOUI, Sylvie HUET, Cecile DUROT, Beatrice LAURENT, Fadoua BALABDAOUI, John BUNGE
2017
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.
On asymptotics of the discrete convex LSE of a pmf.
Cecile DUROT, Francois KOLADJO, Fadoua BALABDAOUI
revue bernoulli | 2017
No summary available.
Revisiting the Hodges-Lehmann estimator in a location mixture model: is asymptotic normality good enough?
Fadoua BALABDAOUI
Electronic journal of statistics | 2017
No summary available.
Inference for a mixture of symmetric distributions under log-concavity.
Fadoua BALABDAOUI, Charles DOSS
revue bernoulli | 2017
No summary available.
Calibration tests for multivariate Gaussian forecasts.
Fadoua BALABDAOUI, Wei WEI, Held LEO
Journal of Multivariate Analysis | 2017
No summary available.
Letter to the Editor. Comments on Groparu-Cojocaru and Doray (2013).
Fadoua BALABDAOUI, Saonli BASU
Communications in Statistics - Simulation and Computation | 2017
No summary available.
Maximum Likelihood estimation of a unimodal probability mass function.
Fadoua BALABDAOUI, Hanna JANKOWSKI
Statistica Sinica | 2016
No summary available.
On asymptotics of the discrete convex LSE of a pmf.
Fadoua BALABDAOUI, Cecile DUROT, Francois KOLADJO
Bernoulli journal | 2015
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.
Marshall lemma in discrete convex estimation.
Fadoua BALABDAOUI, Cecile DUROT
Statistics and Probability Letters | 2015
No summary available.
Global convergence of the log-concave MLE when the truth is geometric.
Fadoua BALABDAOUI
Journal of Nonparametric Statistics | 2014
No summary available.
Chernoff's density is log-concave.
Fadoua BALABDAOUI, Jon WELLNER
revue bernoulli | 2014
No summary available.
Testing monotonicity via local least concave majorants.
Nathalie AKAKPO, Fadoua BALABDAOUI, Cecile DUROT
Bernoulli | 2014
No summary available.
Testing monotonicity via local least concave majorants.
Fadoua BALABDAOUI, Nathalie AKAKPO, Cecile DUROT
revue bernoulli | 2014
No summary available.
Chernoff’s density is log-concave.
Fadoua BALABDAOUI, Jon a WELLNER
Bernoulli | 2014
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.
On location mixtures with Polya frequency components.
Fadoua BALABDAOUI, Cristina BUTUCEA
Statistics and Probability Letters | 2014
No summary available.
Asymptotic distribution of the discrete log-concave MLE and related applications.
Fadoua BALABDAOUI, Kaspar RUFIBACH, Hanna JANKOWSKI, Marios PAVLIDES
Journal of the Royal Statistical Society: Series B | 2013
No summary available.
Discussion on "How to find an appropriate clustering for mixed type variables with application to socio-economic stratification" by Christian Henning and Tim Liao.
Fadoua BALABDAOUI
2013
No summary available.
