Papers

The Strang splitting estimator is a powerful estimator for parametric inference in multivariate stochastic differential equation models with nonlinear drift and additive noise. While the choice of splitting does not affect the asymptotic distribution of the estimator, it makes a huge impact in finite-sample settings and it has not yet been shown how the splitting can be chosen optimally. We derive error measures for the transition densities of the Strang splitting scheme, in particular calculating the bias up to the order of $h^3$, where $h$ is the length of the time step. We study the connection between these error measures and the performance of the Strang splitting estimator in the double-well potential model and the stochastic FitzHugh-Nagumo model, respectively. Our simulation studies suggest that linearization around fixed points yields accurate parameter estimates for potential models, while other splittings perform better for slow-fast excitable models.

We introduce a method for detecting multiple change-points in the mean of a high-dimensional data sequence. Unlike existing top-down (i.e. divisive) algorithms, we adopt a bottom-up (i.e. agglomerative) approach, whereby we iteratively merge neighboring segments of data starting from the finest level. This is particularly useful for signals with frequent change-points, since local evidence is assessed before segments are combined into coarser summaries. We compute $L_2$- and $L_\infty$-aggregated test statistics of neighboring segments and combine the information from their respective ranks, which makes the method adaptive in handling different degrees of change-point sparsity. We show the consistency of the estimated number and locations of change-points under both iid Gaussian and possibly dependent and/or non-Gaussian noise. The practicality of our approach is demonstrated through simulations and a real data example involving the UK House Price Index data.

Persistence intensity functions provide interpretable and informative first-order summaries of random persistence diagram distributions. We study two-sample testing for equality of persistence intensity functions, allowing the underlying diagram distributions to differ under the null. We construct a weighted-kernel statistic as an unbiased estimator of the squared reproducing-kernel Hilbert space distance between the corresponding weighted intensity embeddings, and calibrate it by studentization. For shrinking bandwidths, we establish uniform asymptotic normality under the null and thereby obtain asymptotic Type I error control. Its power is characterized in terms of the $L^2$ discrepancy between the weighted intensity functions. To accommodate persistence diagrams with possibly unbounded cardinality, we introduce regularity conditions that control the effect of cardinality variation and yield the desired moment bounds for the test statistic. We further show that every probability density on $\{(x,y)\in\mathbb{R}^2:y>x > 0\}$ can be realized as the persistence intensity function of a random diagram. Using these results, we establish that the proposed test attains minimax-optimal separation rates over anisotropic Sobolev balls. Lastly, since the optimal bandwidth is not directly accessible in practice, we adapt a bandwidth aggregation framework.

White noise testing is a fundamental problem in time series analysis. Yet it remains largely unsolved for high-dimensional functional time series, despite the growing attention this area has received in recent years, as existing tests are confined to either univariate functional time series or high-dimensional scalar time series. In this paper, we develop a general error-contamination framework for testing white noise in high-dimensional functional time series. We propose a supremum-type test statistic based on cross-autocovariance functions and develop a parametric bootstrap procedure to approximate its null distribution. By imposing a general high-level condition, we derive a new Gaussian approximation result that ensures size control, and establish an asymptotic power guarantee. We then apply our framework to two concrete applications: (i) white noise test for discretely observed functional time series, and (ii) residual-based goodness-of-fit test for functional factor model. For each problem, we verify the corresponding high-level condition to ensure the theoretical validity of our proposed method. Extensive simulations show that our proposed method achieves good finite-sample performance. The practical utility of our proposed method is further illustrated through applications to two real datasets.

We introduce a self-exciting point process with power-law intensity dynamics that admits a finite-dimensional Markovian state representation. The model is constructed to preserve the local jump and slope update structure of power-law Hawkes processes, while replacing global history dependence with a nonlinear Markov chain governing the intensity dynamics. Within a general state-space framework, we establish irreducibility, aperiodicity, and the T-chain property under mild regularity conditions on the inter-arrival time distribution. Under an explicit stability condition, we further prove that the latent state process is positive Harris recurrent, ensuring the existence of a unique invariant distribution. Simulation results based on the local Whittle estimator show that the proposed Markovian intensity model exhibits long-memory behavior near the boundary of the stability region.

This paper develops adaptive procedures for detecting and locating mean changes in high-dimensional time series. Quadratic CUSUM statistics target dense changes, whereas coordinatewise maximum statistics target sparse changes. Two weighting schemes are considered to accommodate both interior and boundary changes. Under general non-Gaussian vector dependence, we establish the limiting distributions, validate the required centering and scaling estimators, and prove asymptotic independence between matched quadratic and maximum statistics. These results justify Cauchy combination tests. We further establish single-change localization and consistent multiple-change recovery using wild binary segmentation. Numerical results illustrate the effectiveness of the proposed methods.

High-dimensional screening is commonly built on covariance, correlation, or least-squares measures. These summary measures can be unstable or even undefined, when predictors are sparse or have heavy-tailed distributions. Building on our recent work on extended codifference and the idea of Ball-covariance, we develop Ball-codifference for marginal screening in statistical modeling with heavy-tailed predictors and responses. The proposed statistic combines the rank-type geometry of random balls with the codifference as a dependency measure constructed based on the characteristic function, so it can be computed without requiring well-defined finite first or second moments. We define Ball-codifference and its normalized screening utility, and formulate a sure independence screening procedure. Large-sample normality follows from a bounded V-statistic and functional-delta-method argument under standard nondegeneracy and regularity conditions. Simulation studies under Gaussian and sub-Gaussian stable designs show that codifference-weighted Ball screening gives competitive or improved recovery of highly associated predictors, especially when tail heaviness is pronounced. Also, our data example illustrates that our variable screening method significantly improves prediction accuracy in linear regression.

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

The double-descent phenomenon observed in modern machine learning models has challenged the classical bias-variance trade-off by showing that prediction error can decrease again as model complexity exceeds the interpolation threshold. Although this phenomenon has been extensively studied for deep neural networks and other high-capacity models, its implications for classical statistical methods remain less well understood. In this paper, we investigate double descent in linear discriminant analysis (LDA), a fundamental classification method for binary classification. Using random matrix theory, we derive the asymptotic misclassification risk of LDA in the high-dimensional regime where the dimension (p) and sample size (n) satisfy (p/n \to \gamma). Specifically, we obtain an explicit asymptotic expression for the LDA risk with a general population covariance matrix when (\gamma \in (0,1)), and for the Moore-Penrose pseudo-inverse LDA classifier under an isotropic covariance model when (\gamma \in (1,\infty)). Together, these results provide a complete characterization of the double-descent risk curve of LDA across both the under- and over-parameterized regimes. Simulation studies and experiments on the ARCENE cancer classification dataset demonstrate close agreement with the theoretical predictions.

Every production model is updated, by retraining, fine-tuning, quantization, or a silent vendor swap, and each update risks being worse than what it replaced. We formalize update promotion as certified paired risk-difference auditing. Our starting point is a support identity: the risk difference between two models lives on the inputs where they disagree, observable without labels. We build DISCERN, a sequential two-tier protocol. A zero-label tier certifies benign updates whose disagreement rate is below tolerance from unlabeled traffic alone. An audited tier labels only sampled disagreements through an anytime-valid confidence sequence, valid at every stopping time and under any label-routing rule, even an adversarial judge. We prove finite-sample validity and matching label-complexity bounds of order rho^2/eps^2 at the rate level, so exploiting free disagreement provably saves a factor 1/rho over any pairing-blind auditor, and the guarantee composes across an unbounded sequence of promotions from one error budget. Across 14,000+ replayed audit streams over 785 update pairs, including LoRA fine-tunes of language models up to 1.4B parameters, miscoverage is 0.0002 (nominal 5%), power 0.986 with zero false alarms, and 56% of benign updates certify with zero labels. Each audit emits a machine-checkable evidence record for post-market monitoring.

River-flow forecasting requires predictive distributions that remain informative in both routine and extreme conditions. We develop a Bayesian quantile-based correction-and-synthesis framework built on Dynamic Quantile Linear Models (DQLMs). The framework links U.S. Geological Survey (USGS) observations, retrospective products, and ensemble forecast products through a shared latent quantile process, learns dynamic discrepancies for each external source, and combines quantile-specific posterior predictions into a single predictive distribution. We also adapt variational Bayes inference to the extended dynamic quantile linear model using Laplace–Delta approximations for non-conjugate parameters. The methodology is illustrated using daily flow for the San Lorenzo River together with products from the European Centre for Medium-Range Weather Forecasts (ECMWF) Global Flood Awareness System (GloFAS) and the National Oceanic and Atmospheric Administration (NOAA) National Weather Service (NWS), with emphasis on medium-range forecasting and uncertainty quantification across multiple quantile levels.

Scalar metrics are often used to evaluate clusterings against known classes, but they can obscure a fundamental trade-off: clusterings should be informative about class labels while avoiding unnecessary fragmentation. Here we describe normalized scores of cluster homogeneity and parsimony that quantify this trade-off. These scores build on the information bottleneck principle, modified to not reward lossy compression. We show by example and mathematical proof that our definitions of these scores have the intuitive property of varying monotonically under cluster refinement in contrast to related proposals. Extending the information-theoretic framework beyond Shannon entropies, we furthermore derive set-matching and pair-based counterparts of the homogeneity and parsimony scores. These unify commonly used evaluation criteria and show that, in the pair-based setting, the homogeneity-parsimony trade-off recovers the receiver operating characteristic of binary classifiers. We demonstrate the framework's utility for feature selection and algorithm comparison, illustrating how considering scores jointly can clarify clustering operating points and identify Pareto-optimal solutions.

Gaussian descriptions of molecular shape underpin 3D shape-based virtual screening, but existing methods evaluate Gaussian overlap analytically. The widely used first-order approximation is fast but systematically overestimates overlap, whereas the exact molecular volume requires a combinatorial inclusion-exclusion expansion. We introduce the first stochastic estimator of Gaussian shape overlap: an unbiased Monte Carlo method that importance-samples directly from a molecule's Gaussian mixture. The estimator reproduces analytic overlap without bias and extends to the exact union volume of all inclusion-exclusion orders with O(N) cost per sample. On drug-like molecules, the union estimator matches high-resolution grid quadrature with a mean relative error of 0.07 percent, while the first-order approximation overestimates the true union volume by 3.4x on average. The estimator provides analytic standard errors, enabling confidence-bounded screening that reduces sampling by 94 percent while preserving ranking. Implemented in JAX, it is fully differentiable and supports gradient-based rigid alignment on CPU, GPU, and TPU. On the DUD-E and LIT-PCBA benchmarks, the method achieves shape-only enrichment comparable to existing single-conformer approaches while additionally providing unbiased absolute volumes and uncertainty estimates.

The twoblock clustering tree (\tbtree) is introduced as a highly interpretable regression tree for multivariate responses. Twoblock trees are deterministic decision trees that have local multivariate linear models as their leaves and use dense or sparse twoblock dimension reduction as local leaf models and in the impurity. The resulting models are both computationally efficient and can be highly interpretable. The estimator's primary aim is an interpretable, regime-aligned piecewise-linear description of the data, with predictive competitiveness retained as a constraint through a split/leaf decoupling. Beyond proposing the decision tree estimator itself, this paper also introduces an estimator for the twoblock dimension reduced space based on maximizing coskewness, which facilitates identification of non-normal clusters in the data. The tree inherently produces a set of local linear models and is therefore apt to recover piecewise linear regimes, which is illustrated in a simulation. However, two real-world data examples illustrate that twoblock trees are also capable of modeling more complexly nonlinear dependencies and can perform on par with black-box modeling techniques, such as random forests. At each point, both the twoblock models that generate the splits, as well as the ones in the leaves, can be inspected and interpreted.

In this work, we study the problem of testing conditional independence between random variables $X$ and $Y$ given a confounder $Z$. The local permutation test (LPT) offers a principled approach to this problem by partitioning the $Z$-space into pre-specified bins, and permuting the $X$ and $Y$ data within each bin, to assess the significance of an observed test statistic. However, when the partitions are pre-fixed, the resulting partition can be poorly balanced, as some bins may contain most of the samples while others contain only a few. This motivates the use of data-adaptive binning strategies, such as equisized bins with a fixed (typically small) number of points. We study this natural and practically important extension of LPT, providing finite-sample bounds on the Type I error for an arbitrary test statistic, providing stronger validity results than previously known. We also show that LPT attains power comparable to the oracle likelihood ratio tests derived from the Neyman-Pearson lemma. Within a linear confounder model class, we further analyze the effect of bin size and demonstrate that constant bin sizes can match the performance of partitions with growing bin-size. These results, further supported by extensive numerical simulations, position the proposed data-adaptive strategy as both practically implementable and statistically efficient.

We study conformal prediction for regression using calibration data with outcomes that are doubly censored (clipped) at known fixed thresholds. We show that existing methods are unsatisfactory in this setting, as they yield intervals that may have higher marginal coverage than desired and yet lose conditional coverage precisely for the easier-to-predict cases whose outcomes are typically fully observed. This reveals that marginal coverage, the usual target of conformal prediction, may not be the ideal goal under clipping. We address this challenge by introducing a new nonconformity score and calibration methods at both ends of this trade-off: one for tight marginal coverage, and a two-step method that prioritizes conditional coverage. We characterize their finite-sample coverage and oracle-like asymptotic behavior under suitable consistency of the underlying model, and we compare them to more direct adaptations of existing approaches.

Likelihood-based inference in particle physics, and in the physical sciences more broadly, relies on the assumption that the model accurately describes the data. When the model is misspecified, though, standard confidence intervals shrink to zero width with increasing data, producing overconfident and potentially misleading constraints. We propose the well-tempered likelihood, which divides the likelihood-ratio test statistic by a goodness-of-fit (GOF) statistic evaluated at the best-fit point. Under correct specification, the GOF is $\mathcal{O}(1)$ and standard inference is recovered. Under misspecification, both the likelihood and the GOF scale as $\mathcal{O}(N)$ for $N$ data points, so their ratio remains $\mathcal{O}(1)$ and the resulting well-tempered confidence interval self-limits at a floor determined by the model's inadequacy, essentially reducing the effective sample size. In other words, all models are correct, as long as your dataset is small enough. We present binned and unbinned formulations of the well-tempered likelihood—the latter based on a classifier two-sample test—and demonstrate our method on a Gaussian example and on a measurement of the strong coupling constant using synthetic electron-positron collisions.

Let $X(t)$, $t\in K$, be a centred Gaussian process with continuous sample paths on a compact metric space $K$, and let $M=\min_{t\in K}X(t)$. Let $\sigma_*^2$ denote the minimum covariance energy associated with $X$, and assume that $\sigma_*^2>0$. Motivated by the results of \cite{chakrabarty2018asymptotic} for smooth Gaussian processes, we show that, conditionally on $M>u$, the scaled overshoot $u(M-u)$ converges, as $u\to\infty$, to an exponential random variable with mean $\sigma_*^2$. Moreover, every weak subsequential limit of the conditional law of a measurable minimizer of $X$ is an optimal covariance-energy measure. In particular, if this measure is unique, then the conditional law converges weakly to it. The results are illustrated by stationary Gaussian processes, fractional Brownian motion, and fractional Brownian sheet.

We propose a Bayesian emulator for extrapolating spatially gridded simulation output across resolution. The method treats each pixel as following a nonlinear resolution-response curve, while linking pixels through Gaussian process priors on the curve parameters to preserve spatial structure and quantify uncertainty. In synthetic experiments designed to mimic resolution-dependent bias, the approach recovers the high-resolution target with competitive or improved accuracy relative to several alternative emulators, particularly in lower-information settings, while maintaining near-nominal interpolative coverage. We also illustrate the method on a radiation-hydrodynamics example from Cassio, where it is used to extrapolate a derived wave-front diagnostic beyond the finest observed simulation. These results suggest that spatially regularized resolution extrapolation can provide a useful statistical tool for studying high-fidelity behavior when direct simulation is expensive.

We propose CEDAR (Causal Edge Discovery for Autoregressive Processes), a constraint-based method for lagged causal edge discovery in sparse autoregressive time series. CEDAR screens candidate cross-variable lags using AR(1)-residualized, U-centered distance correlation, then applies two targeted conditional-independence tests per significant cross-variable lag candidate and accepts at most one lag per ordered pair. A stable MCI pruning step removes indirect edges, and optional deterministic C-nodes adjust for specified trend-like nonstationarity. In sparse regimes where few lags survive screening, CEDAR requires $O(d^2)$ CI tests after screening while retaining edge-level interpretability. CEDAR is most effective when data are scarce and variables exhibit lag-1 self-dynamics; methods with richer conditioning sets become preferable as $T$ grows or when higher-order autoregressive or simultaneous multi-lag effects are common.

When does a weight-tied looped transformer – one block applied T times – implement an actual algorithm? We answer with four findings from controlled populations on group word problems. (1) The budget law: free training installs a linear computation frontier, a mechanism that solves v positions per loop, whose speed is priced by the training contract: v ~ n_train/T_train (exponent 0.98 +/- 0.04, R^2=0.99), exactly unity under T=n training. SGD selects a frontier matching the minimum the contract demands; granting more test-time loops than ever trained rescues late positions at fixed input length, yielding a principled halting rule T* = ceil(n / v-hat). (2) Architecture prior, not expressivity, picks the algorithm: standard-depth transformers learn parallel scans on this family; weight tying flips the selection to the serial frontier, even when positional addressing for a log-depth scan is supplied. At matched depth and parameters, untied models extrapolate worst and fail to learn A5 at all. (3) The walls are not where circuit complexity says: NC1-completeness costs nothing (A5 generalizes fully), while group order does (S5's 120x120 operator deadlocks joint learning) – and an operator-first curriculum dissolves the wall in every seed. (4) Mechanisms are portable, not mandatable: warm-starting across budget contracts transfers the algorithm in every seed, re-pricing its speed, while imposing seriality through the input schedule fails where free training succeeds. These results are invisible to standard instruments, which provably saturate at the fixed points trained loops converge to. We introduce a head instrument, the convergence-time scaling tau(n,i), validate it causally via damage cones whose slope reproduces v, and show in-distribution head measurements predict out-of-distribution fate where tail metrics do not. Results replicate on the public easy-to-hard benchmark.

Modern power systems increasingly require probabilistic forecasts amid interacting uncertainties from renewable intermittency, flexible demand, market volatility, and weather-dependent generation. However, existing methods often treat multi-scale decomposition, exogenous-variable alignment, and probabilistic output as separate steps, obscuring how predictable structures and uncertainty-bearing fluctuations jointly shape the forecast distribution. This paper proposes a state-space exogenous-context and temporal-frequency resolution architecture for general probabilistic energy forecasting. Its central premise is that trend-periodic components primarily determine the baseline trajectory, whereas high-frequency residuals and external perturbations govern the spread and asymmetry of forecast uncertainty. Accordingly, the architecture adaptively separates deterministic and residual streams, aligns exogenous context with both, refines the deterministic backbone through multi-resolution spectral-temporal state-space modeling, and estimates ordered quantile boundaries from their complementary representations. Experiments on load, price, solar, and wind forecasting achieve the best continuous ranked probability score in 14 of 18 settings, reducing average CRPS by 5.74% and upper-tail quantile risk by 7.27% over the strongest baselines. These results support deterministic-stochastic separation as an effective design principle for general probabilistic energy forecasting.

We study Gaussian-width complexity on statistical manifolds through a pair of functionals: the primal Fisher width $w_G(T) = w(G^{1/2}T)$, induced by the Fisher metric, and the inverse-Fisher width $w_{G^{-1}}(T) = w(G^{-1/2}T)$, induced by the inverse Fisher metric. The two widths play complementary statistical roles. On the learning side, the Fisher width measures the size of local parameter fluctuations in the geometry induced by the Fisher information. For Fisher-regular losses, we prove that the scale \(w_G(H_r)/\sqrt n\) is attained on sufficiently small Fisher balls. On the recovery side, the inverse-Fisher width captures the effect of anisotropic Gaussian measurements whose covariance is determined by the inverse Fisher information. For sparse recovery, the resulting geometry depends not only on sparsity but also on the position of the active coordinates in the Fisher spectrum. We obtain a two-sided estimate for the corresponding statistical dimension, together with support-sensitive recovery estimates and a natural ordering of supports with different curvature profiles. Finally, we establish a sharp relation between the primal and inverse-Fisher widths. On any common compact coordinate set $T$, they satisfy \[ w_G(T)w_{G^{-1}}(T)\geq w(T)^2. \] Thus, Fisher anisotropy may transfer complexity from one geometry to the other, but cannot reduce both widths relative to the Euclidean scale.

We study estimation of the p*p residual scatter (shape) matrix in a high-dimensional multivariate linear regression, where p and n grow proportionally. When the coefficient matrix obeys a known linear restriction of rank q < d, as in multivariate analysis of variance, growth-curve models, and reduced-rank regression, the restricted fit leaves additional residual degrees of freedom that sharpen estimation of the shape matrix. To accommodate heavy-tailed errors, we work with independent elliptically distributed rows under a mild scale condition, a finite second moment on the radii, which is far weaker than the usual sub-Gaussian assumptions and covers every multivariate-t law with more than two degrees of freedom. Shrinking the restricted residual sample covariance directly is unsound here, since its limiting spectrum depends on the radial distribution. We instead shrink a scale-invariant scatter of the restricted residuals, whose spectrum is distribution-free over the elliptical family and obeys the same limiting law as under Gaussian errors, at a smaller effective aspect ratio. The resulting estimator attains the rotation-equivariant oracle and is asymptotically optimal within that class, and a Stein-type combination with the unrestricted estimator dominates it while remaining safe under misspecification. We further correct for the case in which the restriction is itself selected from the data. Simulations, a growth-curve experiment, and two real-data analyses illustrate the results.

We prove strong laws of large numbers for random functions in the Lipschitz pseudometric. Our results hold under either a topological or a model-theoretic condition, with the latter encompassing functions jointly definable in o-minimal structures but extending substantially beyond this class. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. Consequently, we identify broad classes of functions for which the failure phenomena revealed by our previous negative results [Tian and Royset, arXiv:2511.16568, 2025] do not occur.

Posterior probabilities are bounded martingales, but this fact alone does not identify the experiment that generates them. We show that every time-homogeneous martingale diffusion on $[0,1]$ with strictly positive $C^1$ volatility on the interior and absorbing endpoints, which we call an autonomous win-martingale, can be realized as the Bayesian posterior of an explicit binary sequential experiment observed through a scalar diffusion. Conversely, in a general binary diffusion experiment, the current time and observation are jointly sufficient for the hidden state precisely when the difference between the signal drifts under the two hypotheses satisfies a Riccati equation. Equivalently, this holds when the likelihood ratio is generated by a Doob $h$-transform associated with a positive space-time harmonic function for the null dynamics. Within this class, when that function is spatially harmonic, the posterior is an autonomous martingale. Outside this harmonic case, autonomy forces the drift difference to be constant, recovering the classical two-state Shiryaev-Wonham filter from binary testing. These results give concrete sequential inference interpretations to a broad class of bounded martingale diffusions, including absorbed Brownian motion, the Wright-Fisher diffusion, and the Aldous martingale of the "most exciting game."

Covariate shift often occurs because, in many real applications, the source and the target observations may be generated from different distributions. In this case, the standard metric under the source distribution is not appropriate. This paper considers deep neural network estimators for nonparametric quantile and Huber regression under covariate shift and from dependent observations. We deal with a generalized Bernstein-type inequality that is satisfied by many classical models, including i.i.d. observations, $\phi$-mixing, strong mixing, and $\mathcal{C}$-mixing processes. To perform the covariate shift phenomenon, we propose a sparse-penalized deep neural network (SPDNN) estimator that takes into account the discrepancy between the source and target distributions of the data. When the density ratio (between the source and target distributions of the covariate) is unknown, a two steps pre-training procedure is carried out: the first step is devoted to the construction of a least squares SPDNN estimator of the density ratio; which is used in the second step to perform a pre-training reweighted SPDNN estimator of the regression function. For both the quantile and the Huber regression, non-asymptotic error bounds of the proposed SPDNN estimators are established in the class of Hölder smooth functions. These estimators can adaptively attain (up to a logarithmic factor) the minimax optimal convergence rate from i.i.d. data as well as from several classical time series models.

Bayesian online learning promises uncertainty-aware prediction on data streams, but its performance hinges on inferential choices, including learning rates, prior distributions and variational families, which are usually fixed before seeing the stream. We address this by treating Bayesian update rules as experts and aggregating the Bayesian experts according to sequential predictive losses. We prove that the resulting aggregate competes with the best expert in hindsight at an aggregation cost determined by how each expert's per-round performance is evaluated. We instantiate the framework in online conformal inference and Gaussian process regression. The conformal inference application yields a smoothed Bayesian counterpart of adaptive conformal inference with long-run randomized coverage, while the Gaussian process application gives an oracle inequality in cumulative predictive Kullback-Leibler risk and adaptation to unknown Hölder smoothness up to logarithmic factors. Experiments show that the aggregate tracks strong experts without oracle expert selection.

Low-rank adaptation (LoRA) has become a widely used parameter-efficient fine-tuning method for large language models. Since different modules and layers may contribute unequally to downstream adaptation, allocating rank resources under a fixed parameter budget is an important problem for balancing efficiency, expressiveness, and generalization. Existing adaptive rank methods address this problem mainly through carefully designed importance scores constructed from gradient-derived sensitivity and uncertainty measures, without an explicit statistical interpretation. In this paper, we formulate LoRA rank allocation as a statistical hypothesis testing problem and propose StatLoRA, a statistical inference-based rank allocation method. StatLoRA associates each LoRA component with a test statistic and uses estimated p-values to determine which components should be retained or pruned under a prescribed rank budget. The proposed testing procedure is supported by our central limit theory for stochastic optimizer trajectories. In particular, we establish asymptotic normality for a broad class of commonly used optimizers in deep learning, including AdamW, and derive the corresponding asymptotic distributions for the proposed component scores used in hypothesis testing. We evaluate StatLoRA on LoRA fine-tuning of DeBERTaV3-base, BART-Large, and Qwen2.5-7B across natural language understanding, natural language generation, and question answering tasks. Experiments show that StatLoRA achieves comparable or better performance than vanilla LoRA, AdaLoRA, and IGU-LoRA under matched rank budgets. Sensitivity analyses and empirical diagnostics further support the stability of the proposed hypothesis-testing-based allocation rule and provide empirical evidence for the asymptotic theory of component scores.

Do quantum kernels improve cross-sectional stock return prediction? We run a controlled horse race on the Chinese A-share market in which a quantum fidelity kernel, a projected quantum kernel, and a classical RBF control share identical training subsamples, solver, and tuning budgets, so that only the kernel is exchanged. On the main evaluation – a point-in-time universe and 170 walk-forward windows (2012-2025) – no quantum advantage exists: the fidelity kernel is indistinguishable from its RBF control ($\Delta$IC $=+0.005$, $p=0.42$), and a $2\times2$ design crossing kernel type with training budget (a Nystrom extension to the full ~38,000-observation windows) shows quantum kernels matching, but never beating, equal-budget linear models; after family-wise correction no pairwise difference among eleven models is significant, with point estimates favoring penalized linear regressions throughout. We then document how the opposite conclusion arises: a 60-window evaluation on a universe screened with full-sample information makes the same quantum kernel appear dominant on stability criteria and significantly better than neural baselines. Interaction characteristics from the anomalies literature help nothing, quantum or classical; a widened bandwidth grid reveals an interior optimum rather than the near-classical endpoint a coarse grid suggests; and the geometric difference, while large throughout ($g \gg 1$), does not predict out-of-sample gains ($\rho=-0.20$). We propose protocol standards – kernel-swap controls, budget-equalized comparisons, point-in-time universes, and multiplicity-robust inference – for empirical claims of quantum advantage in finance.