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This paper studies the impact of bootstrap procedure on the eigenvalue distributions of the sample covariance matrix under the high-dimensional factor structure. We provide asymptotic distributions for the top eigenvalues of bootstrapped sample covariance matrix under mild conditions. After bootstrap, the spiked eigenvalues which are driven by common factors will converge weakly to Gaussian limits via proper scaling and centralization. However, the largest non-spiked eigenvalue is mainly determined by order statistics of bootstrap resampling weights, and follows extreme value distribution. Based on the disparate behavior of the spiked and non-spiked eigenvalues, we propose innovative methods to test the number of common factors. According to the simulations and a real data example, the proposed methods are the only ones performing reliably and convincingly under the existence of both weak factors and cross-sectionally correlated errors. Our technical details contribute to random matrix theory on spiked covariance model with convexly decaying density and unbounded support, or with general elliptical distributions.

Mixtures of experts (MoE) models are a popular framework for modeling heterogeneity in data, for both regression and classification problems in statistics and machine learning, due to their flexibility and the abundance of available statistical estimation and model choice tools. Such flexibility comes from allowing the mixture weights (or gating functions) in the MoE model to depend on the explanatory variables, along with the experts (or component densities). This permits the modeling of data arising from more complex data generating processes when compared to the classical finite mixtures and finite mixtures of regression models, whose mixing parameters are independent of the covariates. The use of MoE models in a high-dimensional setting, when the number of explanatory variables can be much larger than the sample size, is challenging from a computational point of view, and in particular from a theoretical point of view, where the literature is still lacking results for dealing with the curse of dimensionality, for both the statistical estimation and feature selection problems. We consider the finite MoE model with soft-max gating functions and Gaussian experts for high-dimensional regression on heterogeneous data, and its $l_1$-regularized estimation via the Lasso. We focus on the Lasso estimation properties rather than its feature selection properties. We provide a lower bound on the regularization parameter of the Lasso function that ensures an $l_1$-oracle inequality satisfied by the Lasso estimator according to the Kullback--Leibler loss.

Recently, high dimensional vector auto-regressive models (VAR), have attracted a lot of interest, due to novel applications in the health, engineering and social sciences. The presence of temporal dependence poses additional challenges to the theory of penalized estimation techniques widely used in the analysis of their iid counterparts. However, recent work (e.g., [Basu and Michailidis, 2015, Kock and Callot, 2015]) has established optimal consistency of $\ell_1$-LASSO regularized estimates applied to models involving high dimensional stable, Gaussian processes. The only price paid for temporal dependence is an extra multiplicative factor that equals 1 for independent and identically distributed (iid) data. Further, [Wong et al., 2020] extended these results to heavy tailed VARs that exhibit "$\beta$-mixing" dependence, but the rates rates are sub-optimal, while the extra factor is intractable. This paper improves these results in two important directions: (i) We establish optimal consistency rates and corresponding finite sample bounds for the underlying model parameters that match those for iid data, modulo a price for temporal dependence, that is easy to interpret and equals 1 for iid data. (ii) We incorporate more general penalties in estimation (which are not decomposable unlike the $\ell_1$ norm) to induce general sparsity patterns. The key technical tool employed is a novel, easy-to-use concentration bound for heavy tailed linear processes, that do not rely on "mixing" notions and give tighter bounds.

For discretely observed functional data, estimating eigenfunctions with diverging index is essential in nearly all methods based on functional principal components analysis. In this paper, we propose a new approach to handle each term appeared in the perturbation series and overcome the summability issue caused by the estimation bias. We obtain the moment bounds for eigenfunctions and eigenvalues for a wide range of the sampling rate. We show that under some mild assumptions, the moment bound for the eigenfunctions with diverging indices is optimal in the minimax sense. This is the first attempt at obtaining an optimal rate for eigenfunctions with diverging index for discretely observed functional data. Our results fill the gap in theory between the ideal estimation from fully observed functional data and the reality that observations are taken at discrete time points with noise, which has its own merits in models involving inverse problem and deserves further investigation.

The paper studies non-stationary high-dimensional vector autoregressions of order $k$, VAR($k$). Additional deterministic terms such as trend or seasonality are allowed. The number of time periods, $T$, and number of coordinates, $N$, are assumed to be large and of the same order. Under such regime the first-order asymptotics of the Johansen likelihood ratio (LR), Pillai-Barlett, and Hotelling-Lawley tests for cointegration is derived: Test statistics converge to non-random integrals. For more refined analysis, the paper proposes and analyzes a modification of the Johansen test. The new test for the absence of cointegration converges to the partial sum of the Airy$_1$ point process. Supporting Monte Carlo simulations indicate that the same behavior persists universally in many situations beyond our theorems. The paper presents an empirical implementation of the approach to the analysis of stocks in S$\&$P$100$ and of cryptocurrencies. The latter example has strong presence of multiple cointegrating relationships, while the former is consistent with the null of no cointegration.

The CP decomposition for high dimensional non-orthogonal spiked tensors is an important problem with broad applications across many disciplines. However, previous works with theoretical guarantee typically assume restrictive incoherence conditions on the basis vectors for the CP components. In this paper, we propose new computationally efficient composite PCA and concurrent orthogonalization algorithms for tensor CP decomposition with theoretical guarantees under mild incoherence conditions. The composite PCA applies the principal component or singular value decompositions twice, first to a matrix unfolding of the tensor data to obtain singular vectors and then to the matrix folding of the singular vectors obtained in the first step. It can be used as an initialization for any iterative optimization schemes for the tensor CP decomposition. The concurrent orthogonalization algorithm iteratively estimates the basis vector in each mode of the tensor by simultaneously applying projections to the orthogonal complements of the spaces generated by other CP components in other modes. It is designed to improve the alternating least squares estimator and other forms of the high order orthogonal iteration for tensors with low or moderately high CP ranks, and it is guaranteed to converge rapidly when the error of any given initial estimator is bounded by a small constant. Our theoretical investigation provides estimation accuracy and convergence rates for the two proposed algorithms. Both proposed algorithms are applicable to deterministic tensor, its noisy version, and the order-$2K$ covariance tensor of order-$K$ tensor data in a factor model with uncorrelated factors. Our implementations on synthetic data demonstrate significant practical superiority of our approach over existing methods.

In data science, vector autoregression (VAR) models are popular in modeling multivariate time series in the environmental sciences and other applications. However, these models are computationally complex with the number of parameters scaling quadratically with the number of time series. In this work, we propose a so-called neighborhood vector autoregression (NVAR) model to efficiently analyze large-dimensional multivariate time series. We assume that the time series have underlying neighborhood relationships, e.g., spatial or network, among them based on the inherent setting of the problem. When this neighborhood information is available or can be summarized using a distance matrix, we demonstrate that our proposed NVAR method provides a computationally efficient and theoretically sound estimation of model parameters. The performance of the proposed method is compared with other existing approaches in both simulation studies and a real application of stream nitrogen study.

As a special infinite-order vector autoregressive (VAR) model, the vector autoregressive moving average (VARMA) model can capture much richer temporal patterns than the widely used finite-order VAR model. However, its practicality has long been hindered by its non-identifiability, computational intractability, and relative difficulty of interpretation. This paper introduces a novel infinite-order VAR model which, with only a little sacrifice of generality, inherits the essential temporal patterns of the VARMA model but avoids all of the above drawbacks. As another attractive feature, the temporal and cross-sectional dependence structures of this model can be interpreted separately, since they are characterized by different sets of parameters. For high-dimensional time series, this separation motivates us to impose sparsity on the parameters determining the cross-sectional dependence. As a result, greater statistical efficiency and interpretability can be achieved, while no loss of temporal information is incurred by the imposed sparsity. We introduce an $\ell_1$-regularized estimator for the proposed model and derive the corresponding nonasymptotic error bounds. An efficient block coordinate descent algorithm and a consistent model order selection method are developed. The merit of the proposed approach is supported by simulation studies and a real-world macroeconomic data analysis.

We consider observations $(X,y)$ from single index models with unknown link function, Gaussian covariates and a regularized M-estimator $\hat\beta$ constructed from convex loss function and regularizer. In the regime where sample size $n$ and dimension $p$ are both increasing such that $p/n$ has a finite limit, the behavior of the empirical distribution of $\hat\beta$ and the predicted values $X\hat\beta$ has been previously characterized in a number of models: The empirical distributions are known to converge to proximal operators of the loss and penalty in a related Gaussian sequence model, which captures the interplay between ratio $p/n$, loss, regularization and the data generating process. This connection between$(\hat\beta,X\hat\beta)$ and the corresponding proximal operators require solving fixed-point equations that typically involve unobservable quantities such as the prior distribution on the index or the link function. This paper develops a different theory to describe the empirical distribution of $\hat\beta$ and $X\hat\beta$: Approximations of $(\hat\beta,X\hat\beta)$ in terms of proximal operators are provided that only involve observable adjustments. These proposed observable adjustments are data-driven, e.g., do not require prior knowledge of the index or the link function. These new adjustments yield confidence intervals for individual components of the index, as well as estimators of the correlation of $\hat\beta$ with the index. The interplay between loss, regularization and the model is thus captured in a data-driven manner, without solving the fixed-point equations studied in previous works. The results apply to both strongly convex regularizers and unregularized M-estimation. Simulations are provided for the square and logistic loss in single index models including logistic regression and 1-bit compressed sensing with 20\% corrupted bits.

We consider a potential outcomes model in which interference may be present between any two units but the extent of interference diminishes with spatial distance. The causal estimand is the global average treatment effect, which compares outcomes under the counterfactuals that all or no units are treated. We study a class of designs in which space is partitioned into clusters that are randomized into treatment and control. For each design, we estimate the treatment effect using a Horvitz-Thompson estimator that compares the average outcomes of units with all or no neighbors treated, where the neighborhood radius is of the same order as the cluster size dictated by the design. We derive the estimator's rate of convergence as a function of the design and degree of interference and use this to obtain estimator-design pairs that achieve near-optimal rates of convergence under relatively minimal assumptions on interference. We prove that the estimators are asymptotically normal and provide a variance estimator. For practical implementation of the designs, we suggest partitioning space using clustering algorithms.