There has been remarkable progress over the past decade in establishing finite-sample, non-asymptotic bounds on recovering unknown system parameters from observed system behavior. Surprisingly, however, we show that the current state-of-the-art bounds do not accurately capture the statistical complexity of system identification, even in the most fundamental setting of estimating a discrete-time linear dynamical system (LDS) via ordinary least-squares regression (OLS). Specifically, we utilize asymptotic normality to identify classes of problem instances for which current bounds overstate the squared parameter error, in both spectral and Frobenius norm, by a factor of the state-dimension of the system. Informed by this discrepancy, we then sharpen the OLS parameter error bounds via a novel second-order decomposition of the parameter error, where crucially the lower-order term is a matrix-valued martingale that we show correctly captures the CLT scaling. From our analysis we obtain finite-sample bounds for both (i) stable systems and (ii) the many-trajectories setting that match the instance-specific optimal rates up to constant factors in Frobenius norm, and polylogarithmic state-dimension factors in spectral norm.

High-dimensional statistical settings ($p \gg n$) pose fundamental challenges for classical inference, largely due to bias introduced by regularized estimators such as the LASSO. To address this, Javanmard and Montanari (2014) propose a debiased estimator that enables valid hypothesis testing and confidence interval construction. This report examines their debiased LASSO framework, which yields asymptotically normal estimators in high-dimensional settings. The key theoretical results underlying this approach are presented. Specifically, the construction of an optimized debiased estimator that restores asymptotic normality, which enables the computation of valid confidence intervals and $p$-values. To evaluate the claims of Javanmard and Montanari, a subset of the original simulation study and the real-data analysis is presented. The original empirical analysis is extended to the desparsified LASSO, which is referenced but not implemented in the original study. The results demonstrate that while the debiased LASSO achieves reliable coverage and controls Type I error, the LASSO projection estimator can offer improved power in idealized low-signal settings without compromising error rates. The results reveal a trade-off: the LASSO projection estimator performs well in low-signal settings, while Javanmard and Montanari's method is more robust to complex correlations, improving precision and signal detection in real data.

We study high-dimensional mediation analysis in which exposures, mediators, and outcomes are all multivariate, and both exposures and mediators may be high-dimensional. We formalize this as a many (exposures)-to-many (mediators)-to-many (outcomes) (MMM) mediation analysis problem. Methodologically, MMM mediation analysis simultaneously performs variable selection for high-dimensional exposures and mediators, estimates the indirect effect matrix (i.e., the coefficient matrices linking exposure-to-mediator and mediator-to-outcome pathways), and enables prediction of multivariate outcomes. Theoretically, we show that the estimated indirect effect matrices are consistent and element-wise asymptotically normal, and we derive error bounds for the estimators. To evaluate the efficacy of the MMM mediation framework, we first investigate its finite-sample performance, including convergence properties, the behavior of the asymptotic approximations, and robustness to noise, via simulation studies. We then apply MMM mediation analysis to data from the Alzheimer's Disease Neuroimaging Initiative to study how cortical thickness of 202 brain regions may mediate the effects of 688 genome-wide significant single nucleotide polymorphisms (SNPs) (selected from approximately 1.5 million SNPs) on eleven cognitive-behavioral and diagnostic outcomes. The MMM mediation framework identifies biologically interpretable, many-to-many-to-many genetic-neural-cognitive pathways and improves downstream out-of-sample classification and prediction performance. Taken together, our results demonstrate the potential of MMM mediation analysis and highlight the value of statistical methodology for investigating complex, high-dimensional multi-layer pathways in science. The MMM package is available at https://github.com/THELabTop/MMM-Mediation.

There remain theoretical gaps in deep neural network estimators for the nonparametric Cox proportional hazards model. In particular, it is unclear how gradient-based optimization error propagates to population risk under partial likelihood, how pointwise bias can be controlled to permit valid inference, and how ensemble-based uncertainty quantification behaves under realistic variance decay regimes. We develop an asymptotic distribution theory for deep Cox estimators that addresses these issues. First, we establish nonasymptotic oracle inequalities for general trained networks that link in-sample optimization error to population risk without requiring the exact empirical risk optimizer. We then construct a structured neural parameterization that achieves infinity-norm approximation rates compatible with the oracle bound, yielding control of the pointwise bias. Under these conditions and using the Hajek--Hoeffding projection, we prove pointwise and multivariate asymptotic normality for subsampled ensemble estimators. We derive a range of subsample sizes that balances bias correction with the requirement that the Hajek--Hoeffding projection remain dominant. This range accommodates decay conditions on the single-overlap covariance, which measures how strongly a single shared observation influences the estimator, and is weaker than those imposed in the subsampling literature. An infinitesimal jackknife representation provides analytic covariance estimation and valid Wald-type inference for relative risk contrasts such as log-hazard ratios. Finally, we illustrate the finite-sample implications of the theory through simulations and a real data application.

Stochastic network models play a central role across a wide range of scientific disciplines, and questions of statistical inference arise naturally in this context. In this paper we investigate goodness-of-fit and two-sample testing procedures for statistical networks based on the principle of maximum entropy (MaxEnt). Our approach formulates a constrained entropy-maximization problem on the space of networks, subject to prescribed structural constraints. The resulting test statistics are defined through the Lagrange multipliers associated with the constrained optimization problem, which, to our knowledge, is novel in the statistical networks literature. We establish consistency in the classical regime where the number of vertices is fixed. We then consider asymptotic regimes in which the graph size grows with the sample size, developing tests for both dense and sparse settings. In the dense case, we analyze exponential random graph models (ERGM) (including the Erdös-Rènyi models), while in the sparse regime our theory applies to Erd{ö}s-R{è}nyi graphs. Our analysis leverages recent advances in nonlinear large deviation theory for random graphs. We further show that the proposed Lagrange-multiplier framework connects naturally to classical score tests for constrained maximum likelihood estimation. The results provide a unified entropy-based framework for network model assessment across diverse growth regimes.

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.

Variational Bayes (VB) is a scalable alternative to Markov chain Monte Carlo (MCMC)forBayesian posterior inference.

We make use commute time'JWMNP' as the target The California datacenter has access to all of the features. The Texas datacenter has access to all but'AGEP', 'SCHL '. For each method that we test, we run 20 trials to form 95% confidence intervals. Optimized-Naive-Collab, described in Section 6. As the Schur complement is also p.s.d.