On kernel smoothing for extremal quantile regression

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far enough into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Specifically, we elucidate their limit distributions when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their (intermediate or extreme) order with the oscillations of the quantile function and a von-Mises property of the conditional distribution. A simulation experiment and an illustration on real data were presented. The real data are the American electric data where the estimation of conditional extremes is found to be of genuine interest.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ466 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Frontier estimation and extreme value theory

In this paper, we investigate the problem of nonparametric monotone frontier estimation from the perspective of extreme value theory. This enables us to revisit the asymptotic theory of the popular free disposal hull estimator in a more general setting, to derive new and asymptotically Gaussian estimators and to provide useful asymptotic confidence bands for the monotone boundary function. The finite-sample behavior of the suggested estimators is explored via Monte Carlo experiments. We also apply our approach to a real data set based on the production activity of the French postal services.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ256 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Insight in the brain:a multimodal approach investigating insight in individuals with a psychotic disorder and healthy individuals

Insight is impaired in the majority of patients with schizophrenia. This group of patients – more or less – does not consider themselves to be ill and does not recognize the need for treatment. The negative outcomes are immense: they have more problems sticking to medication, a lower quality of life and poorer outcome in general, compared to patients with good insight. Studies in this dissertation showed that abnormalities of numerous brain regions across the brain are associated with impaired insight. This suggests that to have good insight, several complex brain functions have to work unimpededly. That requires orchestrated communication of numerous brain regions across the brain. We also investigated an ability that might be important for having insight, namely self-reflectiveness. Our findings show that brain networks of individuals with lower self-reflectiveness abilities are less stable with regard to brain function and structure. On the other hand, one network was overly present in individuals with lower self-reflectiveness. This is a network that is activated when an individual is not engaged in a task, and that has been shown to be involved in mind wandering. The results of this thesis are important for guiding future treatments of impaired insight such as: 1) strengthening networks that are important for insight, or 2) diminishing function of mind wandering networks. The first could be achieved with the incorporation of several aspects of therapies into treatment that aim to improve neurocognitive, social cognitive and metacognitive functions, while the latter could be achieved with mindfulness meditation training

Frontier Estimation and Extreme Values Theory

In this paper we investigate the problem of nonparametric monotone frontier estimation from an extreme-values theory perspective. This allows to revisit the asymptotic theory of the popular Free Disposal Hull estimator in a general setup, to derive new and asymptotically Gaussian estimators and to provide useful asymptotic confidence bands for the monotone boundary function. The finite sample behavior of the suggested estimators is explored through Monte-Carlo experiments. We also apply our approach to a real data set on the production activity of the French postal services.

Regularization of Nonparametric Frontier Estimators

In production theory and efficiency analysis, we are interested in estimating the production frontier which is the locus of the maximal attainable level of an output (the production), given a set of inputs (the production factors). In other setups, we are rather willing to estimate an input (or cost) frontier that is defined as the minimal level of the input (cost) attainable for a given set of outputs (goods or services produced). In both cases the problem can be viewed as estimating a surface under shape constraints (monotonicity, . . . ). In this paper we derive the theory of an estimator of the frontier having an asymptotic normal distribution. The basic tool is the order-m partial frontier where we let the order m to converge to infinity when n ! 1 but at a slow rate. The final estimator is then corrected for its inherent bias. We thus can view our estimator as a regularized frontier estimator which, in addition, is more robust to extreme values and outliers than the usual nonparametric frontier estimators, like FDH. The performances of our estimators are evaluated in finite samples through some Monte-Carlo experiments. We illustrate also how to provide, in an easy way, confidence intervals for the frontier function both with a simulated data set and a real data set.