Estimating spatial quantile regression with functional coefficients: A robust semiparametric framework

This paper considers an estimation of semiparametric functional (varying)-coefficient quantile regression with spatial data. A general robust framework is developed that treats quantile regression for spatial data in a natural semiparametric way. The local M-estimators of the unknown functional-coefficient functions are proposed by using local linear approximation, and their asymptotic distributions are then established under weak spatial mixing conditions allowing the data processes to be either stationary or nonstationary with spatial trends. Application to a soil data set is demonstrated with interesting findings that go beyond traditional analysis.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ480 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Blaming the exogenous environment? Conditional efficiency estimation with continuous and discrete exogenous variables

This paper proposes a fully nonparametric framework to estimate relative efficiency of entities while accounting for a mixed set of continuous and discrete (both ordered and unordered) exogenous variables. Using robust partial frontier techniques, the probabilistic and conditional characterization of the production process, as well as insights from the recent developments in nonparametric econometrics, we present a generalized approach for conditional efficiency measurement. To do so, we utilize a tailored mixed kernel function with a data-driven bandwidth selection. So far only descriptive analysis for studying the effect of heterogeneity in conditional efficiency estimation has been suggested. We show how to use and interpret nonparametric bootstrap-based significance tests in a generalized conditional efficiency framework. This allows us to study statistical significance of continuous and discrete exogenous variables on production process. The proposed approach is illustrated using simulated examples as well as a sample of British pupils from the OECD Pisa data set. The results of the empirical application show that several exogenous discrete factors have a statistically significant effect on the educational process.Nonparametric estimation, Conditional efficiency measures, Exogenous factors, Generalized kernel function, Education

Marginal integration for nonparametric causal inference

We consider the problem of inferring the total causal effect of a single variable intervention on a (response) variable of interest. We propose a certain marginal integration regression technique for a very general class of potentially nonlinear structural equation models (SEMs) with known structure, or at least known superset of adjustment variables: we call the procedure S-mint regression. We easily derive that it achieves the convergence rate as for nonparametric regression: for example, single variable intervention effects can be estimated with convergence rate $n^{-2/5}$ assuming smoothness with twice differentiable functions. Our result can also be seen as a major robustness property with respect to model misspecification which goes much beyond the notion of double robustness. Furthermore, when the structure of the SEM is not known, we can estimate (the equivalence class of) the directed acyclic graph corresponding to the SEM, and then proceed by using S-mint based on these estimates. We empirically compare the S-mint regression method with more classical approaches and argue that the former is indeed more robust, more reliable and substantially simpler.Comment: 40 pages, 14 figure

Blaming the exogenous environment? Conditional efficiency estimation with continuous and discrete environmental variables

This paper proposes a fully nonparametric framework to estimate relative efficiency of entities while accounting for a mixed set of continuous and discrete (both ordered and unordered) exogenous variables. Using robust partial frontier techniques, the probabilistic and conditional characterization of the production process, as well as insights from the recent developments in nonparametric econometrics, we present a generalized approach for conditional efficiency measurement. To do so, we utilize a tailored mixed kernel function with a data-driven bandwidth selection. So far only descriptive analysis for studying the effect of heterogeneity in conditional efficiency estimation has been suggested. We show how to use and interpret nonparametric bootstrap-based significance tests in a generalized conditional efficiency framework. This allows us to study statistical significance of continuous and discrete environmental variables. The proposed approach is illustrated by a sample of British pupils from the OECD Pisa data set. The results show that several exogenous discrete factors have a significant effect on the educational process.

Interpretable statistics for complex modelling: quantile and topological learning

As the complexity of our data increased exponentially in the last decades, so has our need for interpretable features. This thesis revolves around two paradigms to approach this quest for insights. In the first part we focus on parametric models, where the problem of interpretability can be seen as a “parametrization selection”. We introduce a quantile-centric parametrization and we show the advantages of our proposal in the context of regression, where it allows to bridge the gap between classical generalized linear (mixed) models and increasingly popular quantile methods. The second part of the thesis, concerned with topological learning, tackles the problem from a non-parametric perspective. As topology can be thought of as a way of characterizing data in terms of their connectivity structure, it allows to represent complex and possibly high dimensional through few features, such as the number of connected components, loops and voids. We illustrate how the emerging branch of statistics devoted to recovering topological structures in the data, Topological Data Analysis, can be exploited both for exploratory and inferential purposes with a special emphasis on kernels that preserve the topological information in the data. Finally, we show with an application how these two approaches can borrow strength from one another in the identification and description of brain activity through fMRI data from the ABIDE project