No penalty no tears: Least squares in high-dimensional linear models

Ordinary least squares (OLS) is the default method for fitting linear models, but is not applicable for problems with dimensionality larger than the sample size. For these problems, we advocate the use of a generalized version of OLS motivated by ridge regression, and propose two novel three-step algorithms involving least squares fitting and hard thresholding. The algorithms are methodologically simple to understand intuitively, computationally easy to implement efficiently, and theoretically appealing for choosing models consistently. Numerical exercises comparing our methods with penalization-based approaches in simulations and data analyses illustrate the great potential of the proposed algorithms.Comment: Added results for non-sparse models; Added results for elliptical distribution; Added simulations for adaptive lass

Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Numerical studies confirm our theoretical findings.Comment: Published at http://dx.doi.org/10.1214/15-AOS1343 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dynamic Linear Discriminant Analysis in High Dimensional Space

High-dimensional data that evolve dynamically feature predominantly in the modern data era. As a partial response to this, recent years have seen increasing emphasis to address the dimensionality challenge. However, the non-static nature of these datasets is largely ignored. This paper addresses both challenges by proposing a novel yet simple dynamic linear programming discriminant (DLPD) rule for binary classification. Different from the usual static linear discriminant analysis, the new method is able to capture the changing distributions of the underlying populations by modeling their means and covariances as smooth functions of covariates of interest. Under an approximate sparse condition, we show that the conditional misclassification rate of the DLPD rule converges to the Bayes risk in probability uniformly over the range of the variables used for modeling the dynamics, when the dimensionality is allowed to grow exponentially with the sample size. The minimax lower bound of the estimation of the Bayes risk is also established, implying that the misclassification rate of our proposed rule is minimax-rate optimal. The promising performance of the DLPD rule is illustrated via extensive simulation studies and the analysis of a breast cancer dataset.Comment: 34 pages; 3 figure

High-dimensional ordinary least-squares projection for screening variables

Variable selection is a challenging issue in statistical applications when the number of predictors p far exceeds the number of observations n. In this ultra-high dimensional setting, the sure independence screening (SIS) procedure was introduced to significantly reduce the dimensionality by preserving the true model with overwhelming probability, before a refined second stage analysis. However, the aforementioned sure screening property strongly relies on the assumption that the important variables in the model have large marginal correlations with the response, which rarely holds in reality. To overcome this, we propose a novel and simple screening technique called the high-dimensional ordinary least-squares projection (HOLP). We show that HOLP possesses the sure screening property and gives consistent variable selection without the strong correlation assumption, and has a low computational complexity. A ridge type HOLP procedure is also discussed. Simulation study shows that HOLP performs competitively compared to many other marginal correlation based methods. An application to a mammalian eye disease data illustrates the attractiveness of HOLP

Optimal Subsampling Bootstrap for Massive Data

The bootstrap is a widely used procedure for statistical inference because of its simplicity and attractive statistical properties. However, the vanilla version of bootstrap is no longer feasible computationally for many modern massive datasets due to the need to repeatedly resample the entire data. Therefore, several improvements to the bootstrap method have been made in recent years, which assess the quality of estimators by subsampling the full dataset before resampling the subsamples. Naturally, the performance of these modern subsampling methods is influenced by tuning parameters such as the size of subsamples, the number of subsamples, and the number of resamples per subsample. In this paper, we develop a novel hyperparameter selection methodology for selecting these tuning parameters. Formulated as an optimization problem to find the optimal value of some measure of accuracy of an estimator subject to computational cost, our framework provides closed-form solutions for the optimal hyperparameter values for subsampled bootstrap, subsampled double bootstrap and bag of little bootstraps, at no or little extra time cost. Using the mean square errors as a proxy of the accuracy measure, we apply our methodology to study, compare and improve the performance of these modern versions of bootstrap developed for massive data through simulation study. The results are promising