This paper studies hypothesis testing and parameter estimation in the context of the divide-and-conquer algorithm. In a unified likelihood based framework, we propose new test statistics and point estimators obtained by aggregating various statistics from k subsamples of size n/k, where n is the sample size. In both low dimensional and sparse high dimensional settings, we address the important question of how large k can be, as n grows large, such that the loss of efficiency due to the divide-and-conquer algorithm is negligible. In other words, the resulting estimators have the same inferential efficiencies and estimation rates as an oracle with access to the full sample. Thorough numerical results are provided to back up the theory.

Frequentists' inference often delivers point estimators associated with condence intervals or sets for parameters of interest. Constructing the condence intervals or sets requires understanding the sampling distributions of the point estimators, which, in many but not all cases, are related to asymptotic Normal distributions ensured by central limit theorems. Although previous literature has established various forms of central limit theorems for statistical inference in super population models, we still need general and convenient forms of central limit theorems for some randomization-based causal analysis of experimental data, where the parameters of interests are functions of a nite population and randomness comes solely from the treatment assignment. We use central limit theorems for sample surveys and rank statistics to establish general forms of the nite population central limit theorems that are particularly useful for proving asymptotic distributions of randomization tests under the sharp null hypothesis of zero individual causal eects, and for obtaining the asymptotic repeated sampling distributions of the causal eect estimators. The new central limit theorems hold for general experimental designs with multiple treatment levels, multiple treatment factors and vector outcomes, and are immediately applicable for studying the asymptotic properties of many methods in causal inference, including instrumental variable, regression adjustment, rerandomization, clustered randomized experiments, and so on. Previously, the asymptotic properties of these problems are often based on heuristic arguments, which in fact rely on general forms of nite population central limit theorems that have not been established before. Our new theorems ll in this gap by providing more solid theoretical foundation for asymptotic randomization-based causal inference.

Large dimensional Gram type matrices are common objects in high-dimensional statistics and machine learning. In this paper, we study the limiting distribution of the edge eigenvalues for a general class of high-dimensional Gram type random matrices, including separable sample covariance matrices, sparse sample covariance matrices, bipartite stochastic block model and random Gram matrices with general variance profiles. Specifically, we prove that under (almost) sharp moment conditions and certain tractable regularity assumptions, the edge eigenvalues, i.e., the largest few eigenvalues of non-spiked Gram type random matrices or the extremal bulk eigenvalues of spiked Gram type random matrices, satisfy the Tracy-Widom distribution asymptotically. Our results can be used to construct adaptive, accurate and powerful statistics for high-dimensional statistical inference. In particular, we propose data-dependent statistics to infer the number of signals under general noise structure, test the one-sided sphericity of separable matrix, and test the structure of bipartite stochastic block model. Numerical simulations show strong support of our proposed statistics. The core of our proof is to establish the edge universality and Tracy-Widom distribution for a rectangular Dyson Brownian motion with regular initial data. This is a general strategy to study the edge statistics for high-dimensional Gram type random matrices without exploring the specific independence structure of the target matrices. It has potential to be applied to more general random matrices that are beyond the ones considered in this paper.

We provide here a framework to analyze the phase transition phenomenon of slice inverse regression (SIR), a supervised dimension reduction technique introduced by \cite{Li:1991}. Under mild conditions, the asymptotic ratio ρ=limp/n is the phase transition parameter and the SIR estimator is consistent if and only if ρ=0. When dimension p is greater than n, we propose a diagonal thresholding screening SIR (DT-SIR) algorithm. This method provides us with an estimate of the eigen-space of the covariance matrix of the conditional expectation var(E[x|y]). The desired dimension reduction space is then obtained by multiplying the inverse of the covariance matrix on the eigen-space. Under certain sparsity assumptions on both the covariance matrix of predictors and the loadings of the directions, we prove the consistency of DT-SIR in estimating the dimension reduction space in high dimensional data analysis. Extensive numerical experiments demonstrate superior performances of the proposed method in comparison to its competitors.

A decisive question in nonparametric smoothing techniques is the choice of the bandwidth or smoothing parameter. The present paper addresses this question when using local polynomial approximations for estimating the regression function and its derivatives. A fully-automatic bandwidth selection procedure has been proposed by Fan and Gijbels (1995a), and the empirical performance of it was tested in detail via a variety of examples. Those experiences supported the methodology towards a great extend. In this paper we establish asymptotic results for the proposed variable bandwidth selector. We provide the rate of convergence of the bandwidth estimate, and obtain the asymptotic distribution of its error relative to the theoretical optimal variable bandwidth. These asymptotic properties give extra support to the proposed bandwidth selection procedure. It is also demonstrated how the proposed selection