In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an $\operatorname{Exp}(1)$ asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first predictor to enter the model (i.e., testing for a single significant predictor variable against the global null) requires only weak assumptions on the predictor matrix $X$. On the other hand, our proof for a general step in the lasso path places further technical assumptions on $X$ and the generative model, but still allows for the important high-dimensional case $p>n$, and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables.

Of course, for testing the significance of an additional variable between two nested linear models, one typically uses the chi-squared test, comparing the drop in residual sum of squares (RSS) to a $\chi^{2}_{1}$ distribution. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate: adaptivity makes the drop in RSS stochastically much larger than $\chi^{2}_{1}$ under the null hypothesis. Our analysis explicitly accounts for adaptivity, as it must, since the lasso builds an adaptive sequence of linear models as the tuning parameter $\lambda$ decreases. In this analysis, shrinkage plays a key role: though additional variables are chosen adaptively, the coefficients of lasso active variables are shrunken due to the $\ell_{1}$ penalty. Therefore, the test statistic (which is based on lasso fitted values) is in a sense balanced by these two opposing properties—adaptivity and shrinkage—and its null distribution is tractable and asymptotically $\operatorname{Exp}(1)$.

Undirected graphs can be used to describe matrix variate distributions. In this paper, we develop new methods for estimating the graphical structures and underlying parameters, namely, the row and column covariance and inverse covariance matrices from the matrix variate data. Under sparsity conditions, we show that one is able to recover the graphs and covariance matrices with a single random matrix from the matrix variate normal distribution. Our method extends, with suitable adaptation, to the general setting where replicates are available. We establish consistency and obtain the rates of convergence in the operator and the Frobenius norm. We show that having replicates will allow one to estimate more complicated graphical structures and achieve faster rates of convergence. We provide simulation evidence showing that we can recover graphical structures as well as estimating the precision matrices, as predicted by theory.

Sliced inverse regression (Duan and Li [Ann. Statist. 19 (1991) 505–530], Li [J. Amer. Statist. Assoc. 86 (1991) 316–342]) is an appealing dimension reduction method for regression models with multivariate covariates. It has been extended by Ferré and Yao [Statistics 37 (2003) 475–488, Statist. Sinica 15 (2005) 665–683] and Hsing and Ren [Ann. Statist. 37 (2009) 726–755] to functional covariates where the whole trajectories of random functional covariates are completely observed. The focus of this paper is to develop sliced inverse regression for intermittently and sparsely measured longitudinal covariates. We develop asymptotic theory for the new procedure and show, under some regularity conditions, that the estimated directions attain the optimal rate of convergence. Simulation studies and data analysis are also provided to demonstrate the performance of our method.

We propose generalized additive partial linear models for complex data which allow one to capture nonlinear patterns of some covariates, in the presence of linear components. The proposed method improves estimation efficiency and increases statistical power for correlated data through incorporating the correlation information. A unique feature of the proposed method is its capability of handling model selection in cases where it is difficult to specify the likelihood function. We derive the quadratic inference function-based estimators for the linear coefficients and the nonparametric functions when the dimension of covariates diverges, and establish asymptotic normality for the linear coefficient estimators and the rates of convergence for the nonparametric functions estimators for both finite and high-dimensional cases. The proposed method and theoretical development are quite challenging since the numbers of linear covariates and nonlinear components both increase as the sample size increases. We also propose a doubly penalized procedure for variable selection which can simultaneously identify nonzero linear and nonparametric components, and which has an asymptotic oracle property. Extensive Monte Carlo studies have been conducted and show that the proposed procedure works effectively even with moderate sample sizes. A pharmacokinetics study on renal cancer data is illustrated using the proposed method.

Under the assumption that the true density is decreasing, it is well known that the Grenander estimator converges at rate $n^{1/3}$ if the true density is curved [Sankhyā Ser. A 31 (1969) 23–36] and at rate $n^{1/2}$ if the density is flat [Ann. Probab. 11 (1983) 328–345; Canad. J. Statist. 27 (1999) 557–566]. In the case that the true density is misspecified, the results of Patilea [Ann. Statist. 29 (2001) 94–123] tell us that the global convergence rate is of order $n^{1/3}$ in Hellinger distance. Here, we show that the local convergence rate is $n^{1/2}$ at a point where the density is misspecified. This is not in contradiction with the results of Patilea [Ann. Statist. 29 (2001) 94–123]: the global convergence rate simply comes from locally curved well-specified regions. Furthermore, we study global convergence under misspecification by considering linear functionals. The rate of convergence is $n^{1/2}$ and we show that the limit is made up of two independent terms: a mean-zero Gaussian term and a second term (with nonzero mean) which is present only if the density has well-specified locally flat regions.

We propose kernel estimator for the distribution function of unobserved errors in autoregressive time series, based on residuals computed by estimating the autoregressive coefficients with the Yule–Walker method. Under mild assumptions, we establish oracle efficiency of the proposed estimator, that is, it is asymptotically as efficient as the kernel estimator of the distribution function based on the unobserved error sequence itself. Applying the result of Wang, Cheng and Yang [J. Nonparametr. Stat. 25 (2013) 395–407], the proposed estimator is also asymptotically indistinguishable from the empirical distribution function based on the unobserved errors. A smooth simultaneous confidence band (SCB) is then constructed based on the proposed smooth distribution estimator and Kolmogorov distribution. Simulation examples support the asymptotic theory.

Subspace clustering refers to the task of finding a multi-subspace representation that best fits a collection of points taken from a high-dimensional space. This paper introduces an algorithm inspired by sparse subspace clustering (SSC) [In IEEE Conference on Computer Vision and Pattern Recognition, CVPR (2009) 2790–2797] to cluster noisy data, and develops some novel theory demonstrating its correctness. In particular, the theory uses ideas from geometric functional analysis to show that the algorithm can accurately recover the underlying subspaces under minimal requirements on their orientation, and on the number of samples per subspace. Synthetic as well as real data experiments complement our theoretical study, illustrating our approach and demonstrating its effectiveness.

Stimulated by the Boston house price data, in this paper, we propose a semiparametric spatial dynamic model, which extends the ordinary spatial autoregressive models to accommodate the effects of some covariates associated with the house price. A profile likelihood based estimation procedure is proposed. The asymptotic normality of the proposed estimators are derived. We also investigate how to identify the parametric/nonparametric components in the proposed semiparametric model. We show how many unknown parameters an unknown bivariate function amounts to, and propose an AIC/BIC of nonparametric version for model selection. Simulation studies are conducted to examine the performance of the proposed methods. The simulation results show our methods work very well. We finally apply the proposed methods to analyze the Boston house price data, which leads to some interesting findings.

We investigate differences between a simple Dominance Principle applied to sums of fair prices for variables and dominance applied to sums of forecasts for variables scored by proper scoring rules. In particular, we consider differences when fair prices and forecasts correspond to finitely additive expectations and dominance is applied with infinitely many prices and/or forecasts.

We propose a self-tuning $\sqrt{\mathrm {Lasso}} $ method that simultaneously resolves three important practical problems in high-dimensional regression analysis, namely it handles the unknown scale, heteroscedasticity and (drastic) non-Gaussianity of the noise. In addition, our analysis allows for badly behaved designs, for example, perfectly collinear regressors, and generates sharp bounds even in extreme cases, such as the infinite variance case and the noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds for $\sqrt{\mathrm {Lasso}} $ including prediction norm rate and sparsity. Our analysis is based on new impact factors that are tailored for bounding prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely on moderate deviation theory for self-normalized sums to achieve Gaussian-like results under weak conditions. Moreover, we derive bounds on the performance of ordinary least square (ols) applied to the model selected by $\sqrt{\mathrm {Lasso}} $ accounting for possible misspecification of the selected model. Under mild conditions, the rate of convergence of ols post $\sqrt{\mathrm {Lasso}} $ is as good as $\sqrt{\mathrm {Lasso}} $’s rate. As an application, we consider the use of $\sqrt{\mathrm {Lasso}} $ and ols post $\sqrt{\mathrm {Lasso}} $ as estimators of nuisance parameters in a generic semiparametric problem (nonlinear moment condition or $Z$-problem), resulting in a construction of $\sqrt{n}$-consistent and asymptotically normal estimators of the main parameters.

We consider a Bayesian approach to variable selection in the presence of high dimensional covariates based on a hierarchical model that places prior distributions on the regression coefficients as well as on the model space. We adopt the well-known spike and slab Gaussian priors with a distinct feature, that is, the prior variances depend on the sample size through which appropriate shrinkage can be achieved. We show the strong selection consistency of the proposed method in the sense that the posterior probability of the true model converges to one even when the number of covariates grows nearly exponentially with the sample size. This is arguably the strongest selection consistency result that has been available in the Bayesian variable selection literature; yet the proposed method can be carried out through posterior sampling with a simple Gibbs sampler. Furthermore, we argue that the proposed method is asymptotically similar to model selection with the $L_{0}$ penalty. We also demonstrate through empirical work the fine performance of the proposed approach relative to some state of the art alternatives.