High-dimensional semi-supervised learning: in search of optimal inference of the mean

A fundamental challenge in semi-supervised learning lies in the observed data’s disproportional size when compared with the size of the data collected with missing outcomes. An implicit understanding is that the dataset with missing outcomes, being significantly larger, ought to improve estimation and inference. However, it is unclear to what extent this is correct. We illustrate one clear benefit: root-n inference of the outcome’s mean is possible while only requiring a consistent estimation of the outcome, possibly at a rate slower than root-n. This is achieved by a novel k-fold cross-fitted, double robust estimator. We discuss both linear and nonlinear outcomes. Such an estimator is particularly suited for models that naturally do not admit root-n consistency, such as high-dimensional, nonparametric, or semiparametric models. We apply our methods to the heterogeneous treatment effects.

Regional Modeling of Forest Fuels and Structural Attributes Using Airborne Laser Scanning Data in Oregon

Airborne laser scanning (ALS) acquisitions provide piecemeal coverage across the western US, as collections are organized by local managers of individual project areas. In this study, we analyze different factors that can contribute to developing a regional strategy to use information from completed ALS data acquisitions and develop maps of multiple forest attributes in new ALS project areas in a rapid manner. This study is located in Oregon, USA, and analyzes six forest structural attributes for differences between: (1) synthetic (i.e., not-calibrated), and calibrated predictions, (2) parametric linear and semiparametric models, and (3) models developed with predictors computed for point clouds enclosed in the areas where field measurements were taken, i.e., “point-cloud predictors”, and models developed using predictors extracted from pre-rasterized layers, i.e., “rasterized predictors”. Forest structural attributes under consideration are aboveground biomass, downed woody biomass, canopy bulk density, canopy height, canopy base height, and canopy fuel load. Results from our study indicate that semiparametric models perform better than parametric models if no calibration is performed. However, the effect of the calibration is substantial in reducing the bias of parametric models but minimal for the semiparametric models and, once calibrations are performed, differences between parametric and semiparametric models become negligible for all responses. In addition, minimal differences between models using point-cloud predictors and models using rasterized predictors were found. We conclude that the approach that applies semiparametric models and rasterized predictors, which represents the easiest workflow and leads to the most rapid results, is justified with little loss in accuracy or precision even if no calibration is performed.

Theory of Weak Identification in Semiparametric Models

We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, that is, when it is locally homogeneous of degree zero. When this happens, consistent or equivariant estimation is shown to be impossible. We then show that there exists an underlying regular parameter that fully characterizes the weakly regular parameter. While this parameter is not unique, concepts of sufficiency and minimality help pin down a desirable one. If estimation of minimal sufficient underlying parameters is inefficient, it introduces noise in the corresponding estimation of weakly regular parameters, whence we can improve the estimators by local asymptotic Rao–Blackwellization. We call an estimator weakly efficient if it does not admit such improvement. New weakly efficient estimators are presented in linear IV and nonlinear regression models. Simulation of a linear IV model demonstrates how 2SLS and optimal IV estimators are improved.