Summary The biased coin has long been a central device in covariate-adaptive randomization: an imbalance measure is defined, and the next assignment is biased toward the treatment that improves balance. Although this biased coin minimization paradigm is highly effective under equal allocation, its natural extensions to unequal allocation can cause a shift problem, where additional covariate imbalances are not centered at zero. We propose a new covariate-adaptive randomization procedure that moves beyond this paradigm by designing the allocation function directly. The resulting procedure achieves strong marginal balance for covariates used in allocation and eliminates the shift asymptotically for additional covariates. The allocation form is derived from a mechanism analysis of the shift problem and its elimination. An explicit variance formula further reveals favorable variability in additional covariate imbalance.
Recent advances in deep learning have highlighted the phenomenon of benign overfitting in overparameterized statistical models, sparking significant interest in understanding its foundations. Owing to its simplicity and practical relevance, the ordinary least-squares interpolator has become a key object of study for gaining theoretical insight into this phenomenon. While the properties of ordinary least squares are well understood in classical underparameterized settings, its behaviour in the overparameterized regime, unlike that of ridge regression or the lasso, remains comparatively less explored. We contribute to this growing literature by deriving new algebraic and statistical results for the minimum $ \ell_{2} $-norm ordinary least-squares interpolator. In contrast to much of the existing work, which focuses on prediction risk, we centre our analysis on parameter estimation and inference, which are fundamental to many statistics and causal inference applications. Specifically, we establish overparameterized analogues of (i) the leave-$ k $-out formulas, (ii) the omitted-variable bias formula and (iii) the Frisch-Waugh-Lovell theorem. Under the Gauss-Markov model, we further extend the Gauss-Markov theorem and analyse variance estimation under homoscedasticity in the overparameterized setting. Collectively,these results provide a systematic framework for studying parameter estimation and inference in overparameterized linear models, offering a novel perspective on benign overfitting beyond its implications for prediction.
Summary We consider statistical inference under a semi-supervised setting where we have access to both labelled and unlabelled observations. We ask the question: under what circumstances, and by how much, can incorporating the unlabelled data improve upon inference using the labelled data? To answer this question, we investigate semi-supervised inference through the lens of semiparametric efficiency theory. For general pathwise differentiable targets under semiparametric models, we derive the efficiency lower bound for asymptotically linear semi-supervised estimators. Perhaps not surprisingly, this bound coincides with the results in classical missing data theory when ‘positivity’ holds, i.e., when the limiting proportion of labelled observations is strictly positive. However, unlike missing data theory, our theory can handle the boundary regime in which the proportion of labelled observations converges to zero, which is in fact the setting of primary interest in the context of semi-supervised learning. We further characterize a broad class of well-specified targets for which unlabelled data cannot yield efficiency gains. These results shed insight into the promise and limitations of the very active field of prediction-powered inference. We propose an efficient semi-supervised estimator that attains the efficiency bound, and a prediction-powered estimator that incorporates black-box predictions and is guaranteed to be no worse than the supervised baseline in asymptotic variance. We illustrate our theoretical results via simulation studies and a real-data application.
Abstract We consider nonparametric estimation of the nonstationary density of streaming data collected continuously over time. Those types of data are often available only for a limited time after being observed, for example due to storage or data privacy constraints. Therefore, analyzing them requires dynamic approaches that do not require repeated access to past data. A suitable iterative kernel estimator of nonstationary densities has been developed by Rutkowski (1989); it requires choosing important tuning parameters each time a new observation arises, for which no adequate practical selection criterion has been developed yet.We propose a data-driven dynamic selection of those parameters, which can be implemented iteratively and requires only sequential access to consecutive blocks of the most recent data. To choose the sizes of the blocks, we propose a theoretically founded approach based on properties of cross-validation criteria. We illustrate the broader potential of our ideas by suggesting a dynamic blockwise practical implementation of another kernel estimator of nonstationary densities developed by Hall et al. (2006).We apply our technique to simulated data and to light curve data from the NASA Kepler mission.
Abstract In this paper, we introduce a generalized likelihood ratio test for assessing whether the factor loading matrix in high-dimensional data exhibits a Kronecker structure, thereby offering a rigorous tool for validating two-way matrix factor models. A central challenge in likelihood-based inference for factor models arises from inherent rotational non-identifiability, which prevents the likelihood function from admitting a standard second-order Taylor expansion. To address this issue, we augment the quasi-likelihood with a quadratic identification penalty that removes rotational degeneracy locally. Building on this construction, we derive the asymptotic distribution of the proposed statistic under the null hypothesis and unveil a new Wilks-type phenomenon in a high-dimensional setting. Under the alternative hypothesis, we further establish the consistency of the test, demonstrating its ability to detect deviations from the Kronecker structure. Simulation studies demonstrate the finite-sample effectiveness of the proposed method. Furthermore, applications to real datasets illustrate the practical utility of the proposed procedure by assessing the presence of Kronecker structure in factor loading matrices.
