Counting Process Based Dimension Reduction Methods for Censored Outcomes

We propose a class of dimension reduction methods for right censored survival data using a counting process representation of the failure process. Semiparametric estimating equations are constructed to estimate the dimension reduction subspace for the failure time model. The proposed method addresses two fundamental limitations of existing approaches. First, using the counting process formulation, it does not require any estimation of the censoring distribution to compensate the bias in estimating the dimension reduction subspace. Second, the nonparametric part in the estimating equations is adaptive to the structural dimension, hence the approach circumvents the curse of dimensionality. Asymptotic normality is established for the obtained estimators. We further propose a computationally efficient approach that simplifies the estimation equation formulations and requires only a singular value decomposition to estimate the dimension reduction subspace. Numerical studies suggest that our new approaches exhibit significantly improved performance for estimating the true dimension reduction subspace. We further conduct a real data analysis on a skin cutaneous melanoma dataset from The Cancer Genome Atlas. The proposed method is implemented in the R package "orthoDr".Comment: First versio

Estimation of semiparametric stochastic frontiers under shape constraints with application to pollution generating technologies

A number of studies have explored the semi- and nonparametric estimation of stochastic frontier models by using kernel regression or other nonparametric smoothing techniques. In contrast to popular deterministic nonparametric estimators, these approaches do not allow one to impose any shape constraints (or regularity conditions) on the frontier function. On the other hand, as many of the previous techniques are based on the nonparametric estimation of the frontier function, the convergence rate of frontier estimators can be sensitive to the number of inputs, which is generally known as “the curse of dimensionality” problem. This paper proposes a new semiparametric approach for stochastic frontier estimation that avoids the curse of dimensionality and allows one to impose shape constraints on the frontier function. Our approach is based on the singleindex model and applies both single-index estimation techniques and shape-constrained nonparametric least squares. In addition to production frontier and technical efficiency estimation, we show how the technique can be used to estimate pollution generating technologies. The new approach is illustrated by an empirical application to the environmental adjusted performance evaluation of U.S. coal-fired electric power plants.stochastic frontier analysis (SFA), nonparametric least squares, single-index model, sliced inverse regression, monotone rank correlation estimator, environmental efficiency

Semiparametric Causal Sufficient Dimension Reduction Of High Dimensional Treatments

Cause-effect relationships are typically evaluated by comparing the outcome responses to binary treatment values, representing two arms of a hypothetical randomized controlled trial. However, in certain applications, treatments of interest are continuous and high dimensional. For example, understanding the causal relationship between severity of radiation therapy, represented by a high dimensional vector of radiation exposure values and post-treatment side effects is a problem of clinical interest in radiation oncology. An appropriate strategy for making interpretable causal conclusions is to reduce the dimension of treatment. If individual elements of a high dimensional treatment vector weakly affect the outcome, but the overall relationship between the treatment variable and the outcome is strong, careless approaches to dimension reduction may not preserve this relationship. Moreover, methods developed for regression problems do not transfer in a straightforward way to causal inference due to confounding complications between the treatment and outcome. In this paper, we use semiparametric inference theory for structural models to give a general approach to causal sufficient dimension reduction of a high dimensional treatment such that the cause-effect relationship between the treatment and outcome is preserved. We illustrate the utility of our proposal through simulations and a real data application in radiation oncology