Additive models for quantile regression: model selection and confidence bandaids

Additive models for conditional quantile functions provide an attractive framework for nonparametric regression applications focused on features of the response beyond its central tendency. Total variation roughness penalities can be used to control the smoothness of the additive components much as squared Sobelev penalties are used for classical L 2 smoothing splines. We describe a general approach to estimation and inference for additive models of this type. We focus attention primarily on selection of smoothing parameters and on the construction of confidence bands for the nonparametric components. Both pointwise and uniform confidence bands are introduced; the uniform bands are based on the Hotelling (1939) tube approach. Some simulation evidence is presented to evaluate finite sample performance and the methods are also illustrated with an application to modeling childhood malnutrition in India.

Selective machine learning of doubly robust functionals

While model selection is a well-studied topic in parametric and nonparametric regression or density estimation, selection of possibly high-dimensional nuisance parameters in semiparametric problems is far less developed. In this paper, we propose a selective machine learning framework for making inferences about a finite-dimensional functional defined on a semiparametric model, when the latter admits a doubly robust estimating function and several candidate machine learning algorithms are available for estimating the nuisance parameters. We introduce two new selection criteria for bias reduction in estimating the functional of interest, each based on a novel definition of pseudo-risk for the functional that embodies the double robustness property and thus is used to select the pair of learners that is nearest to fulfilling this property. We establish an oracle property for a multi-fold cross-validation version of the new selection criteria which states that our empirical criteria perform nearly as well as an oracle with a priori knowledge of the pseudo-risk for each pair of candidate learners. We also describe a smooth approximation to the selection criteria which allows for valid post-selection inference. Finally, we apply the approach to model selection of a semiparametric estimator of average treatment effect given an ensemble of candidate machine learners to account for confounding in an observational study

Choosing the Right Spatial Weighting Matrix in a Quantile Regression Model

This paper proposes computationally tractable methods for selecting the appropriate spatial weighting matrix in the context of a spatial quantile regression model. This selection is a notoriously difficult problem even in linear spatial models and is even more difficult in a quantile regression setup. The proposal is illustrated by an empirical example and manages to produce tractable models. One important feature of the proposed methodology is that by allowing different degrees and forms of spatial dependence across quantiles it further relaxes the usual quantile restriction attributable to the linear quantile regression. In this way we can obtain a more robust, with regard to potential functional misspecification, model, but nevertheless preserve the parametric rate of convergence and the established inferential apparatus associated with the linear quantile regression approach

Most Likely Transformations

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set-up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterisation thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models allowing the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.Comment: Accepted for publication by the Scandinavian Journal of Statistics 2017-06-1