arXiv subjects

François Portier

Publications and source records attributed to François Portier.

46 records · Page 3Linked to original sources

arXiv2018

Parametric versus nonparametric: the fitness coefficient

The fitness coefficient, introduced in this paper, results from a competition between parametric and nonparametric density estimators within the likelihood of the data. As illustrated on several real datasets, the fitness coefficient generally agrees with p-values but is easier to compute and interpret. Namely, the fitness coefficient can be interpreted as the proportion of data coming from the parametric model. Moreover, the fitness coefficient can be used to build a semiparamteric compromise which improves inference over the parametric and nonparametric approaches. From a theoretical perspective, the fitness coefficient is shown to converge in probability to one if the model is true and to zero if the model is false. From a practical perspective, the utility of the fitness coefficient is illustrated on real and simulated datasets.

math.ST↗

arXiv2018

On an extension of the promotion time cure model

We consider the problem of estimating the distribution of time-to-event data that are subject to censoring and for which the event of interest might never occur, i.e., some subjects are cured. To model this kind of data in the presence of covariates, one of the leading semiparametric models is the promotion time cure model \citep{yakovlev1996}, which adapts the Cox model to the presence of cured subjects. Estimating the conditional distribution results in a complicated constrained optimization problem, and inference is difficult as no closed-formula for the variance is available. We propose a new model, inspired by the Cox model, that leads to a simple estimation procedure and that presents a closed formula for the variance. We derive some asymptotic properties of the estimators and we show the practical behaviour of our procedure by means of simulations. We also apply our model and estimation method to a breast cancer data set.

math.ST↗

arXiv2017

Integral estimation based on Markovian design

Suppose that a mobile sensor describes a Markovian trajectory in the ambient space. At each time the sensor measures an attribute of interest, e.g., the temperature. Using only the location history of the sensor and the associated measurements, the aim is to estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g., Monte Carlo, the proposed approach does not require any knowledge on the distribution of the sensor trajectory. Probabilistic bounds on the convergence rates of the estimator are established. These rates are better than the traditional "root n"-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, the good behaviour of the procedure is demonstrated through simulations and an application to the evaluation of the average temperature of oceans is considered.

math.ST↗

arXiv2017

On the weak convergence of the empirical conditional copula under a simplifying assumption

When the copula of the conditional distribution of two random variables given a covariate does not depend on the value of the covariate, two conflicting intuitions arise about the best possible rate of convergence attainable by nonparametric estimators of that copula. In the end, any such estimator must be based on the marginal conditional distribution functions of the two dependent variables given the covariate, and the best possible rates for estimating such localized objects is slower than the parametric one. However, the invariance of the conditional copula given the value of the covariate suggests the possibility of parametric convergence rates. The more optimistic intuition is shown to be correct, confirming a conjecture supported by extensive Monte Carlo simulations by I. Hobaek Haff and J. Segers [Computational Statistics and Data Analysis 84:1--13, 2015] and improving upon the nonparametric rate obtained theoretically by I. Gijbels, M. Omelka and N. Veraverbeke [Scandinavian Journal of Statistics 2015, to appear]. The novelty of the proposed approach lies in the double smoothing procedure employed for the estimator of the marginal cumulative distribution functions. Under mild conditions on the bandwidth sequence, the estimator is shown to take values in a certain class of smooth functions, the class having sufficiently small entropy for empirical process arguments to work. The copula estimator itself is asymptotically undistinguishable from a kind of oracle empirical copula, making it appear as if the marginal conditional distribution functions were known.

math.ST↗

arXiv2016

Integral approximation by kernel smoothing

Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in $\mathbb{R}^d$, $d\geq 1$. We show that, for any function $φ:\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, \[n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{φ(X_i)}{\widehat{f}^(X_i)}- \int φ(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,\] where $\widehat{f}$ is the classical kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$. Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions $φ$ and $f$, the dimension $d$ and the bandwidth of the kernel estimator $\widehat{f}$. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of $φ$. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.

math.ST↗

arXiv2015

Efficiency of Z-estimators indexed by the objective functions

We study the convergence of $Z$-estimators $\widehat θ(η)\in \mathbb R^p$ for which the objective function depends on a parameter $η$ that belongs to a Banach space $\mathcal H$. Our results include the uniform consistency over $\mathcal H$ and the weak convergence in the space of bounded $\mathbb R^p$-valued functions defined on $\mathcal H$. Furthermore when $η$ is a tuning parameter optimally selected at $η_0$, we provide conditions under which an estimated $\widehat η$ can be replaced by $η_0$ without affecting the asymptotic variance. Interestingly, these conditions are free from any rate of convergence of $\widehat η$ to $η_0$ but they require the space described by $\widehat η$ to be not too large. We highlight several applications of our results and we study in detail the case where $η$ is the weight function in weighted regression.

math.ST↗

arXiv2015

Continuous inverse regression

We provide new theoretical results in the field of inverse regression methods for dimension reduction. Our approach is based on the study of some empirical processes that lie close to a certain dimension reduction subspace, called the central subspace. The study of these processes essentially includes weak convergence results and the consistency of some general bootstrap procedures. While such properties are used to obtain new results about sliced inverse regression, they mainly allow to define a natural family of methods for dimension reduction. First the estimation methods are shown to have root $n$ rates and the bootstrap is proved to be valid. Second, we describe a family of Cramér-von Mises test statistics that can be used in testing structural properties of the central subspace or the significancy of some sets of predictors. We show that the quantiles of those tests could be computed by bootstrap. Most of the existing methods related to inverse regression involve a slicing of the response that is difficult to select in practice. While our approach guarantee a comprehensive estimation, the slicing is no longer needed.

math.ST↗

arXiv2013

On the acceleration of some empirical means with application to nonparametric regression

Let $(X_1,\ldots ,X_n)$ be an i.i.d. sequence of random variables in $\R^d$, $d\geq 1$, for some function $φ:\R^d\r \R$, under regularity conditions, we show that \begin{align*} n^{1/2} \left(n^{-1} \sum_{i=1}^n \frac{φ(X_i)}{\w f^{(i)}(X_i)}-\int_{} φ(x)dx \right) \overset¶{\lr} 0, \end{align*} where $\w f^{(i)}$ is the classical leave-one-out kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\w f^{(i)}=f$. As a consequence, it improves the classical Monte Carlo procedure for integral approximation. The paper mainly addressed with theoretical issues related to the later result (rates of convergence, bandwidth choice, regularity of $φ$) but also interests some statistical applications dealing with random design regression. In particular, we provide the asymptotic normality of the estimation of the linear functionals of a regression function on which the only requirement is the Hölder regularity. This leads us to a new version of the \textit{average derivative estimator} introduced by Härdle and Stoker in \cite{hardle1989} which allows for \textit{dimension reduction} by estimating the \textit{index space} of a regression.

math.ST↗

arXiv2013

Bootstrap Testing of the Rank of a Matrix via Least Squared Constrained Estimation

In order to test if an unknown matrix has a given rank (null hypothesis), we consider the family of statistics that are minimum squared distances between an estimator and the manifold of fixed-rank matrix. Under the null hypothesis, every statistic of this family converges to a weighted chi-squared distribution. In this paper, we introduce the constrained bootstrap to build bootstrap estimate of the law under the null hypothesis of such statistics. As a result, the constrained bootstrap is employed to estimate the quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light one the accuracy of the constrained bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test if an unknown parameter belongs to a sub-manifold locally smooth. Finally, the constrained bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.

math.ST↗

arXiv2011

Test function: A new approach for covering the central subspace

In this paper we offer a complete methodology for sufficient dimension reduction called the test function (TF). TF provides a new family of methods for the estimation of the central subspace (CS) based on the introduction of a nonlinear transformation of the response. Theoretical background of TF is developed under weaker conditions than the existing methods. By considering order 1 and 2 conditional moments of the predictor given the response, we divide TF in two classes. In each class we provide conditions that guarantee an exhaustive estimation of the CS. Besides, the optimal members are calculated via the minimization of the asymptotic mean squared error deriving from the distance between the CS and its estimate. This leads us to two plug-in methods which are evaluated with several simulations.

math.ST↗