Measuring conditional dependence is an important topic in statistics with broad applications including graphical models. Under a factor model setting, a new conditional dependence measure based on projection is proposed. The corresponding conditional independence test is developed with the asymptotic null distribution unveiled where the number of factors could be high-dimensional. It is also shown that the new test has control over the asymptotic significance level and can be calculated efficiently. A generic method for building dependency graphs without Gaussian assumption using the new test is elaborated. Numerical results and real data analysis show the superiority of the new method.

In survival analysis, the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model. This model usually assumes that the covariate has a log-linear effect on the hazard function. In this paper we consider the proportional hazards regression model with a nonparametric risk effect. We discuss estimation of the risk function and its derivatives in two cases: when the baseline hazard function is parametrized and when it is not parametrized. In the case of a parametric baseline hazard function, inference is based on a local version of the likelihood function, while in the case of a nonparametric baseline hazard, we use a local version of the partial likelihood. This results in maximum local likelihood estimators and maximum local partial likelihood estimators, respectively. We establish the asymptotic normality of the estimators. It turns out that both methods have

Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries by such data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given certain number of predictors, namely, the distribution of the correlation of a response variable Y with the best s linear combinations of p covariates X, even when X and Y are independent. When the covariance matrix of X possesses the restricted eigenvalue property, we derive such distributions for both finite s and diverging s, using Gaussian approximation and empirical process

Variable selection in high-dimensional space characterizes many contemporary problems in scientific discovery and decision making. Many frequently-used techniques are based on independence screening; examples include correlation ranking (Fan and Lv, 2008) or feature selection using a two-sample t-test in high-dimensional classification (Tibshirani et al., 2003). Within the context of the linear model, Fan and Lv (2008) showed that this simple correlation ranking possesses a sure independence screening property under certain conditions and that its revision, called iteratively sure independent screening (ISIS), is needed when the features are marginally unrelated but jointly related to the response variable. In this paper, we extend ISIS, without explicit definition of residuals, to a general pseudo-likelihood framework, which includes generalized linear models as a special case. Even in the least-squares setting, the new method improves ISIS by allowing variable deletion in the iterative process. Our technique allows us to select important features in high-dimensional classification where the popularly used two-sample t-method fails. A new technique is introduced to reduce the false discovery rate in the feature screening stage. Several simulated and two real data examples are presented to illustrate the methodology.

Maximum likelihood ratio theory contributes tremendous success to parametric inferences, due to the fundamental theory of Wilks (1938). Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to nd and can not be optimal as shown in this paper. In this paper, we introduce the sieve likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks' phenomenon is unveiled. We demonstrate that the sieve likelihood statistics are asymptotically distribution free and follow 2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and generalized varying coefficient models. We further demonstrate that sieve likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster (1993). They can even be adaptively optimal in the sense of Spokoiny (1996) by using a simple choice of adaptive smoothing parameter. Our work indicates that the sieve likelihood ratio statistics are indeed general and powerful for nonparametric inferences based on function estimation.