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Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least square estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality. 

Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least squares estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality. 

Estimation and inference in statistics pose significant challenges when data are collected adaptively. Even in linear models, the Ordinary Least Squares (OLS) estimator may fail to exhibit asymptotic normality for single coordinate estimation and have inflated error. This issue is highlighted by a recent minimax lower bound, which shows that the error of estimating a single coordinate can be enlarged by a multiple of $\sqrt{d}$ when data are allowed to be arbitrarily adaptive, compared with the case when they are i.i.d. Our work explores this striking difference in estimation performance between utilizing i.i.d. and adaptive data. We investigate how the degree of adaptivity in data collection impacts the performance of estimating a low-dimensional parameter component in high-dimensional linear models. We identify conditions on the data collection mechanism under which the estimation error for a low-dimensional parameter component matches its counterpart in the i.i.d. setting, up to a factor that depends on the degree of adaptivity. We show that OLS or OLS on centered data can achieve this matching error. In addition, we propose a novel estimator for single coordinate inference via solving a Two-stage Adaptive Linear Estimating equation (TALE). Under a weaker form of adaptivity in data collection, we establish an asymptotic normality property of the proposed estimator. 

Observations in various applications are frequently represented as a time series of multidimensional arrays, called tensor time series, preserving the inherent multidimensional structure. In this paper, we present a factor model approach, in a form similar to tensor CANDECOMP/PARAFAC (CP) decomposition, to the analysis of high-dimensional dynamic tensor time series. As the loading vectors are uniquely defined but not necessarily orthogonal, it is significantly different from the existing tensor factor models based on Tucker-type tensor decomposition. The model structure allows for a set of uncorrelated one-dimensional latent dynamic factor processes, making it much more convenient to study the underlying dynamics of the time series. A new high-order projection estimator is proposed for such a factor model, utilizing the special structure and the idea of the higher order orthogonal iteration procedures commonly used in Tucker-type tensor factor model and general tensor CP decomposition procedures. Theoretical investigation provides statistical error bounds for the proposed methods, which shows the significant advantage of utilizing the special model structure. Simulation study is conducted to further demonstrate the finite sample properties of the estimators. Real data application is used to illustrate the model and its interpretations.