Bing Li’s research while affiliated with Pennsylvania State University and other places

The notion of relative universality with respect to a {\sigma}-field was introduced to establish the unbiasedness and Fisher consistency of an estimator in nonlinear sufficient dimension reduction. However, there is a gap in the proof of this result in the existing literature. The existing definition of relative universality seems to be too strong for the proof to be valid. In this note we modify the definition of relative universality using the concept of \k{o}-measurability, and rigorously establish the mentioned unbiasedness and Fisher consistency. The significance of this result is beyond its original context of sufficient dimension reduction, because relative universality allows us to use the regression operator to fully characterize conditional independence, a crucially important statistical relation that sits at the core of many areas and methodologies in statistics and machine learning, such as dimension reduction, graphical models, probability embedding, causal inference, and Bayesian estimation.

One of the celebrated results of Professor D. Basu is his 1955 paper on ancillary statistics, which established the well known Basu’s Theorem. A Bayesian version of this result, where the parameter $\Theta$ is treated as a random variable, is developed in this note, along with other extensions of the related classical results, such as Rao-Blackwell and Lehmann-Scheffé theorems and the relation between complete sufficiency and minimal sufficiency. These extensions shed new light on these fundamental theorems for frequentist statistical inference in the context Bayesian inference.

We introduce a sufficient graphical model by applying the recently developed nonlinear sufficient dimension reduction techniques to the evaluation of conditional independence. The graphical model is nonparametric in nature, as it does not make distributional assumptions such as the Gaussian or copula Gaussian assumptions. However, unlike a fully nonparametric graphical model, which relies on the high-dimensional kernel to characterize conditional independence, our graphical model is based on conditional independence given a set of sufficient predictors with a substantially reduced dimension. In this way we avoid the curse of dimensionality that comes with a high-dimensional kernel. We develop the population-level properties, convergence rate, and variable selection consistency of our estimate. By simulation comparisons and an analysis of the DREAM 4 Challenge data set, we demonstrate that our method outperforms the existing methods when the Gaussian or copula Gaussian assumptions are violated, and its performance remains excellent in the high-dimensional setting.

We introduce a skewed Gaussian graphical model as an extension to the Gaussian graphical model. One of the appealing properties of the Gaussian distribution is that conditional independence can be fully characterized by the sparseness in the precision matrix. The skewed Gaussian distribution adds a shape parameter to the Gaussian distribution to take into account possible skewness in the data; thus it is more flexible than the Gaussian model. Nevertheless, the appealing property of the Gaussian distribution is retained to a large degree: the conditional independence is still characterized by the sparseness in the parameters, which now include a shape parameter in addition to the precision matrix. As a result, the skewed Gaussian graphical model can be efficiently estimated through a penalized likelihood method just as the Gaussian graphical model. We develop an algorithm to maximize the penalized likelihood based on the alternating direction method of multipliers, and establish the asymptotic normality and variable selection consistency for the new estimator. Through simulations, we demonstrate that our method performs better than the Gaussian and Gaussian copula methods when these distributional assumptions are not satisfied. The method is applied to a breast cancer MicroRNA dataset to construct a gene network, which shows better interpretability than the Gaussian graphical model.

In this article, we develop a nonparametric graphical model for multivariate random functions. Most existing graphical models are restricted by the assumptions of multivariate Gaussian or copula Gaussian distributions, which also imply linear relations among the random variables or functions on different nodes. We relax those assumptions by building our graphical model based on a new statistical object – the functional additive regression operator. By carrying out regression and neighborhood selection at the operator level, our method can capture nonlinear relations without requiring any distributional assumptions. Moreover, the method is built up using only one-dimensional kernel, thus avoids the curse of dimensionality from which a fully nonparametric approach often suffers, and enables us to work with large-scale networks. We derive error bounds for the estimated regression operator and establish graph estimation consistency, while allowing the number of functions to diverge at the exponential rate of the sample size. We demonstrate the efficacy of our method by both simulations and analysis of an electroencephalography dataset.

Very often for the same scientific question, there may exist different techniques or experiments that measure the same numerical quantity. Historically, various methods have been developed to exploit the information within each type of data independently. However, statistical data fusion methods that could effectively integrate multi-source data under a unified framework are lacking. In this paper, we propose a novel data fusion method, called B-scaling, for integrating multi-source data. Consider K measurements that are generated from different sources but measure the same latent variable through some linear or nonlinear ways. We seek to find a representation of the latent variable, named B-mean, which captures the common information contained in the K measurements while takes into account the nonlinear mappings between them and the latent variable. We also establish the asymptotic property of the B-mean and apply the proposed method to integrate multiple histone modifications and DNA methylation levels for characterizing epigenomic landscape. Both numerical and empirical studies show that B-scaling is a powerful data fusion method with broad applications.

We propose a new low‐dimensional registration procedure that exploits the relationship between response and predictor in a function‐on‐function regression. In this context, Functional Covariance Components (FCC) provide a flexible and powerful tool to represent the data in a low‐dimensional space, capturing the most meaningful modes of dependency between the two set of curves. Based on this reduced representation, our procedure aligns simultaneously the two sets of curves, in a way that optimizes the subsequent regression analysis. To implement our procedure, we use both the Continuous Registration algorithm (CR) and a novel parallel algorithm coded in R. We then compare it to other common registration approaches via simulations and an application to the AneuRisk data.