Bootstrap inference in functional linear regression models with scalar response under heteroscedasticity

Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean (projection of the slope function) estimates have complicated sampling distributions due to the infinite dimensional regressors, which create truncation bias and scaling problems that are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated projection under general dependent errors, and subsequently we develop a paired bootstrap method to approximate sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for projections and for conducting hypothesis tests for the slope function. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with real data examples

Mono3D++: Monocular 3D Vehicle Detection with Two-Scale 3D Hypotheses and Task Priors

We present a method to infer 3D pose and shape of vehicles from a single image. To tackle this ill-posed problem, we optimize two-scale projection consistency between the generated 3D hypotheses and their 2D pseudo-measurements. Specifically, we use a morphable wireframe model to generate a fine-scaled representation of vehicle shape and pose. To reduce its sensitivity to 2D landmarks, we jointly model the 3D bounding box as a coarse representation which improves robustness. We also integrate three task priors, including unsupervised monocular depth, a ground plane constraint as well as vehicle shape priors, with forward projection errors into an overall energy function.Comment: Proc. of the AAAI, September 201