Papers

In sparse functional data analysis, where $n$ curves are each observed at $m$ random points, the covariance surface undergoes a sharp phase transition: if the covariance has smoothness $\beta$, the risk drops from the two-dimensional nonparametric rate to the parametric rate $n^{-1}$ once $m$ exceeds $m^*_n \asymp n^{1/(2\beta)}$. We determine what a symmetry of the domain does to that transition. A cyclic group of order $q$ preserving process and design displaces the threshold to $n^{1/(2\beta)}q^{-1/2}$; the exponent is a square root because symmetry acts on the number of usable pairs, which enters the variance quadratically, while the parametric floor is untouched by group averaging. The displacement saturates: once the orbit is finer than the bandwidth the reduction factor is $\min(q,c_K/h)$, by Poisson summation and positive definiteness of the kernel autocorrelation, and beyond that point the rate collapses to the one-dimensional nonparametric rate. Hence no rotation symmetry, even the full circle group, lowers the threshold below $n^{1/(4\beta)}$; this floor is attained from above by our estimator and from below, up to a polynomial factor, by our lower bounds. Symmetry thus takes one halfway on a logarithmic scale from the classical threshold to constant sampling. The uniform lower bound rests on a positivity-preserving packing of the stationary sub-class; closing the gap is left open. If the symmetry is only approximate, the risk acquires an approximation term and the design plane splits into three regimes, one unreachable by additional sampling; the symmetry spectrum governing it is explicit in Fourier coordinates, and a hold-out procedure selects the symmetry level with leading constant one below saturation and within an absolute constant beyond. All laws above are confirmed numerically. Reparametrisation of the domain, by contrast, leaves the threshold unchanged.

Kernels measure similarity or correlation in tasks such as regression and classification. The Gaussian kernel, other names of which include squared exponential and radial basis function kernel, is one of the most popular in Gaussian process regression. We argue that the Gaussian kernel is best avoided and should never be used as a default. The argument rests on two results demonstrating that the Gaussian kernel is extremely brittle. First, the Gaussian kernel gives rise to a conditional variance that is unrealistically small. If the variance is used to quantify predictive uncertainty, catastrophic overconfidence is almost inevitable. Second, a small variance goes hand in hand with numerical ill-conditioning, so that to use the Gaussian kernel in practice requires tricks such as nugget terms that effectively modify the underlying regression or classification model. These problems are caused by the unnatural smoothness of the Gaussian kernel, a fact we are far from the first to take notice of. The problem is not the Gaussian form itself but the analyticity of the kernel: Our argument is more broadly that analytic kernels are best avoided. For stationary kernels analyticity is essentially equivalent to an exponential decay of the spectral density.

Existing constant-step analysis of stochastic \Scaf{} identifies a leading $O(\gamma/N)$ stationary mean bias and shows that higher-order bias can persist as the client count increases, but does not identify the first client-independent contribution at coefficient level. For full-participation stochastic \Scaf{} with one-dimensional homogeneous clients, fixed local-step count $H$, and bounded additive gradient noise, we prove, uniformly over $N\ge2$, $$ \begin{aligned} \mathbb{E}_{\pi_{\gamma,N,H}}[x]-x^\star ={}& -\frac{f'''(x^\star)\sigma^2}{4f''(x^\star)^2}\frac{\gamma}{N}\\ &- \frac{f'''(x^\star)\sigma^2}{12f''(x^\star)} \frac{(H-1)(5H-1)}{H}\gamma^2 +O_H\!\left(\frac{\gamma^2}{N}+\gamma^3\right). \end{aligned} $$ Hence client averaging suppresses the leading $O(\gamma/N)$ bias but does not remove the client-independent $O(\gamma^2)$ component when its coefficient is nonzero. The mechanism is indirect: although the direct control contribution cancels pathwise in the linear global average, the controls still alter within-round local trajectories and their second moments. Fresh gradient noise and persistent control fluctuations therefore generate local second-moment corrections that nonquadratic curvature converts into stationary mean bias. The coefficient vanishes for quadratic objectives. Numerical experiments are consistent with the predicted coefficient, its persistence as client count increases, and the stated joint remainder. The result is restricted to the one-dimensional homogeneous fixed-$H$ setting.

The existing literature on model calibration focuses mainly on classification and probabilistic prediction. In this work, we address calibrated point predictions for the conditional mean. Although existing impossibility results preclude exact out-of-sample calibrated predictions, we develop calibrated confidence intervals that provide uncertainty quantification around such predictions. Our results come with distribution-free theoretical guarantees and are applicable in model-agnostic, finite-sample settings under exchangeability by leveraging conformal prediction. Calibrated confidence intervals rely on a general underlying binning scheme. We present two examples of such a binning scheme, one based on a data-independent partition and the other on isotonic regression. Under structural assumptions, we prove that calibrated confidence intervals based on isotonic regression come with strong asymptotic consistency properties and have an asymptotically vanishing width. We illustrate the empirical performance of our methods by applying them first in a simulated setting and then to a highly imbalanced insurance dataset.

In this paper, we study the problem of augmenting a tiny target sample with a massive auxiliary sample. Utilizing Tukey's factorization, there are two popular approaches: the inverse probability weight (IPW) and the full-likelihood (FL) methods. We show that the IPW approach suffers from the limited target sample problem while the FL method may estimate some model parameters at the rate of the massive auxiliary sample size, a phenomenon we call full efficiency gain. We study the theory behind the full efficiency gain for exponential families and mixtures of exponential families. We also study the efficiency gain for the IPW method under a nonparametric procedure and show how it can achieve a parametric rate of the target sample size. As a side note, we also discuss how one may use FL to train neural network models simultaneously for both the target distribution and the odds model.

We study the Bayesian recovery of the initial state in a semilinear time-fractional subdiffusion equation from noisy random space-time point observations. A rescaled Gaussian prior based on a Whittle–Matérn process is assigned to the unknown initial condition. We prove that the solution has \(\dot H^3\)-regularity when the nonlinearity satisfies a Lipschitz condition in the Sobolev norm. We then establish posterior contraction rates for the prediction error in the \(L^2\)-norm and for the parameter in Sobolev norms. The rates are polynomial in the sample size, with exponent depending on the prior smoothness and the spatial dimension. Moreover, we prove a minimax lower bound by constructing a wavelet-packing set and controlling the Kullback–Leibler divergences.

We develop a novel method of inference for network-dependent high-dimensional random vectors. Dependence is characterized via a functional dependence measure based on graph distance, allowing the approximation theory to capture the interaction between the decay of dependence and the growth of network neighborhoods. We establish Gaussian approximation results for the maximum norm under finite-moment and sub-Weibull conditions, providing explicit conditions under which the dimension may increase with the network size. We also propose a high-dimensional network HAC covariance estimator and establish its convergence properties, yielding a feasible procedure for simultaneous inference. Simulation studies demonstrate favorable finite-sample performance of the proposed method. We apply the procedure to study how spillover effects vary with an index of network homophily by constructing confidence bands for the conditional spillover-effect function. The application reveals heterogeneity and local significance that would be obscured by conventional low-dimensional inference.

Let $q_n(r)$ denote the tail probability at $r$ of the self-normalized sum of $n$ independent centered uniform variables. At $r=3$, the first distribution-sensitive term in the two-sided Edgeworth expansion of Student's statistic vanishes. We evaluate the expansion at the common moving boundary $r_n=3+\lambda/n$ in dimensions $n$ and $n-k$. Uniformly over deletion ranks retaining a fixed positive fraction of observations, the first nonzero difference converges to an explicit phase surface $H_\delta(\lambda)$; its zero curve unifies fixed, sublinear and fixed-fraction deletions, with tangent crossing $12/35$. Through the Khintchine scale-mixture representation, this comparison yields a single compactly supported $C^\infty$ symmetric unimodal parent, independent of $n$, whose Student tail exceeds the equal-scale uniform tail for all sufficiently large $n$ along nominal levels tending to $2\{1-\Phi(\sqrt3)\}$ from below. In contrast, a quantile-ratio order shows that the uniform parent maximizes every even moment and every convergent even power series with nonnegative coefficients. We also derive the fixed-confidence dimension expansion and an exact reversal at $n=6$.

We provide an overview of the problem of jointly testing two hypotheses and estimating a parameter of the selected model. Such problems arise in a variety of applications. First, we present a conceptual introduction to suboptimal and optimal procedures for joint detection and estimation. A numerical example illustrates the advantages of the optimal procedure over suboptimal ones. Next, we discuss how more advanced problem formulations affect the presented results. The second part covers joint detection and estimation in a sequential framework. First, we provide an introduction to sequential analysis through sequential hypothesis testing. Then, suboptimal and optimal sequential procedures for joint detection and estimation are discussed. A numerical example shows the advantages of optimal sequential procedures over suboptimal and optimal sequential procedures with a fixed number of samples. The third part discusses open problems and future research directions in joint detection and estimation.

Local differential privacy (LDP) protects individuals in a dataset by perturbing each measurement before release. For real-valued data, a widely used mechanism is additive Laplace noise. We study the problem of estimating the distribution of the latent confidential data from the privatized observations via the nonparametric maximum likelihood estimator (NPMLE) under an i.i.d. sampling model. We first show that under the Laplace convolution model, the NPMLE admits a finite-dimensional reformulation in which the support is restricted to the observation set. This reduces the original infinite-dimensional optimization over all mixing distributions to an $n$-dimensional convex optimization over mixture weights, where $n$ is the sample size. We then study the statistical convergence of the NPMLE under the 1-Wasserstein distance, and explicitly connect its convergence rate with the privacy noise scale. Allowing the privacy noise level to change with the sample size, our analysis shows that the NPMLE remains consistent when the Laplace noise grows at a rate slower than $n^{3/16}$. Conversely, when the Laplace noise is of the order $\sqrt n$ or larger, no estimator can achieve uniformly consistent recovery of the latent distribution.

Randomization testing with permutations is a very common nonparametric approach to hypothesis testing. However, randomization testing can be done with other group transformations including random rotations. In this work, we consider the problem of invariance in quadratic form statistics under a unified framework with closed form p-values. In particular, we propose a nonparametric variant of the classic Durbin-Watson test for testing for autocorrelation in time series data at arbitrary lags. Our test is performed by integrating over a group of invariances of the test statistic, and easy-to-compute analytic formulae for the p-value are derived from concentration inequalities on compact groups on a per-lag basis. Thus, the usual necessity of large-scale Monte Carlo simulations is rendered unnecessary. Our tests outperform the classic Breusch-Godfrey and Ljung-Box tests on simulated data with respect to statistical power to identify significant autocorrelation. They also can be used to identify the presence of autocorrelation at large lags such as in monthly solar intensity data, which follows an approximate 11 year (132 month) cycle. The general formulation of this approach can be easily adapted to other quadratic form statistics.

Lacey used sparse domination to study the sharp weighted norm estimate of the maximal function of predictable multipliers in discrete time filtration spaces. Domelevo, Petermichl, and Škreb developed the self similarity argument known as sparse domination in an abstract martingale setting with a continuous time parameter. In our investigation, we establish sparse domination for discrete-time martingale transforms, introducing the novel concept of conditional sparsity as a core property of our approach. The conditional sparsity framework enables derivation of sharp weighted estimates and a mixed-norm estimate \( A_p^\alpha A_r^\beta \) that improves upon known sharp \( L^p \) bounds. Moreover, we develop dedicated sparse domination specifically for Doob's maximal operator, recovering the sharp bound as a direct application. Finally, we focus on the application of sparse theory to quantitative two-weight estimates.

Monotone rearrangements provide a simple way to enforce shape constraints of an estimator, but existing distributional theory does not cover flat regions, where the target induces no local ordering. We study rearranged estimators in two canonical flat settings. First, for a histogram estimator of the uniform density, we establish functional weak convergence of its non-decreasing rearrangement at the parametric rate on compact subsets of the interval $(0,1)$ after an additional deterministic centering. This result is strikingly different from what is known for strictly monotone densities. Second, we consider two rearranged estimators of rearranged copulas under independence, based on empirical-copula increments and on a checkerboard approximation. After appropriate centering and rescaling, both estimators converge weakly on $[0,1]^2$ to an integrated Gaussian process which was not known before. We further use these results to prove asymptotic normality for a broad class of rearranged copula-based dependence measures, which were recently discussed in Strothmann et al. (2024).

Epidemic models often pool air travel, commuting, and other mobility layers into a single weighted network. Pooling keeps the total imported infections into a region but discards the transport mode and route that delivered them, the information a mode-specific intervention needs. We make this precise for directed multilayer flows with heavy-tailed variation. Pooling acts as an asymmetric filter. The magnitude of an extreme importation, its radial tail index, is an exact invariant of sampling and nonnegative aggregation, whereas its composition across layers and routes, the angular part, is reweighted by layer-specific sampling and can be irrecoverably merged. We give a necessary and sufficient condition for recovering route composition, attach a surveillance decision cost to the loss, and show that the first established route need not be the busiest. A controlled metapopulation study calibrates the cost, and two contrasting reconstructions show where it bites. In US pandemic influenza, air adds fitted information beyond commuting, and a layer-resolved ranking targets more air-import burden than a pooled one. In the early Italian COVID-19 wave, commuting improves onset reconstruction while air attribution stays unresolved. Pooling can therefore support total-importation surveillance but cannot in general identify the mode or route behind an importation.

A frequency domain bootstrap procedure for functional time series is proposed and applied to the class of spectral mean operators. The procedure works by first generating independent pseudo periodogram operators across the positive Fourier frequencies using an estimator of the spectral density operator involved. Functional replicates of the spectral mean operators of interest are then generated. Through an additive, projection-based decomposition of the bootstrapped spectral mean operator, its leading $m$-dimensional part is properly complemented to also capture the relevant fourth order characteristics of the process. The complementation is achieved by means of a resampling procedure based on convolved periodogram operators of subsamples. The resulting bootstrap spectral mean operator consistently estimates the entire second order as well as the $m$-dimensional fourth order structure of the distribution of spectral mean operators. By allowing for the decomposition parameter $m$ to increase to infinity as the sample size increases to infinity, consistency in estimating the entire fourth order structure of the process also is achieved. The asymptotic theory developed investigates properties of the procedure for fixed and for increasing $m$ and establishes validity of the frequency domain bootstrap proposal under rather weak conditions on the underlying functional process class.

A key result in statistics is the data processing inequality, originally proved by Blackwell (1951) and later refined by DeGroot (1962) in terms of statistical uncertainty. The latter statement claims that the Bayes risk achieved on raw data always undercuts the Bayes risk on a processed version of the same data. This seemingly contradicts empirical findings in machine learning: pre-training, representation learning, feature learning, data augmentation are techniques used to improve performance of machine learning models. We reconcile both worlds by simply accepting that machine learning problems are constrained learning problems: the model class used does not include all measurable functions. We present counterexamples showing that the classical data processing inequality fails to hold in such a setting. Hence, we formulate a generalized data processing inequality, requiring the constrained Bayes risk of a joint distribution (with respect to a loss function and a constrained model class) to lower bound the constrained Bayes risk on the stochastically modified data distribution, regardless of the choice of distribution. We show this inequality to be equivalent to a set containment condition on a specific function set induced by the loss and model class, called the superprediction set. Finally, we exploit our characterization, derive sufficient conditions for this containment and quantify the inequality-gap.

Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the design variable is used as the error criterion. An initially randomly pruned fully connected deep neural network with logistic squasher as activation function is fitted to the data via gradient descent, using a data-dependent choice of non-constant stepsizes during gradient descent. It is shown that this network achieves (up to a logarithmic factor) the optimal minimax rate of convergence in case that the regression function is $(p,C)$–smooth. Here the estimate is able to circumvent the curse of dimensionality provided the predictors are concentrated in the neighborhood of a low dimensional manifold. The finite sample size performance of the estimate is illustrated by applying it to the simulated data.

Estimation of a regression function from exponentially $\beta$-mixing data is considered. The $L_2$ error with integration with respect to the design is used as the error criterion. Deep neural network estimates with logistic activation function are defined, where all parameters are learned by gradient descent. The rate of convergence of the expected $L_2$ error is analyzed for $(p,C)$-smooth regression functions. In the special case that the design is concentrated on a $d^*$-dimensional manifold, it is shown that the expected $L_2$ error of the estimate achieves a rate of convergence which depends on $d^*$ and not on the dimension $d$ of the design.

Stochastic gradient descent (SGD) is typically analyzed at a deterministic horizon chosen before the algorithm is run, even though practical stopping decisions are made adaptively by inspecting the evolving trajectory. This mismatch creates a fundamental certification problem: fixed-time guarantees do not generally remain valid at data-dependent stopping times, while deterministic horizons derived from worst-case bounds can be highly conservative. We address this problem for strongly convex stochastic optimization by constructing fully observable, trajectory-adaptive upper confidence sequences for the squared distance of the last iterate to the optimizer and the suboptimality of a weighted average. These bounds hold simultaneously over time, attain the optimal $1/t$ decay rate up to iterated-logarithmic factors in the worst case, and adapt to the realized stochastic gradients, allowing SGD to stop as soon as a prescribed accuracy is certified without sacrificing statistical validity. Our approach treats the evolving SGD trajectory as a sequential experiment whose observations provide evidence about the unknown optimization error. To formalize this perspective, we develop new recursive confidence-sequence techniques and a general time-uniform empirical Bernstein inequality for adapted processes with time-varying conditional means and predictable ranges that may grow without bound. We further extend these confidence-sequence constructions to minibatch SGD, with the empirical Bernstein bounds exploiting the realized second-moment structure within each minibatch. Numerical experiments show that the resulting stopping rules can require several orders of magnitude fewer iterations than natural deterministic horizons.

Functional data analysis is an important statistical field that treats data as random functions. In practice, the random functions are often not fully observed but instead measured at discrete times. While simpler problems, such as mean and covariance estimation, have been widely studied for discretely observed data, optimal estimation of linear regression for this data type has remained unsolved for over two decades. To tackle this fundamental challenge, we propose a novel approach, referred to as pooling ridge estimation, which combines the advantages of pooling strategy and RKHS-based method by incorporating the unbiased estimation of operators based on discretely observed measurements from all subjects. This unified estimation framework enables us to achieve minimax optimality in prediction risk in arbitrary sampling schemes ranging from sparse to dense designs, for both scalar-on-function and function-on-function regression models. Such methodological and theoretical advances are obtained for the first time and accurately reveal the influence of discrete sampling. For scalar-on-function regression, the phase transition occurs once, separating the convergence behavior into two distinct regimes. Remarkably, for function-on-function regression, up to three phase transitions may occur, determined by the sampling frequencies of the predictor/response functions. Finally, simulation experiments and two real data examples provide empirical support for the proposed methods.

We establish non-asymptotic Berry–Esseen bounds and bootstrap approximations for the Hilbert-space norm of nondegenerate $U$-statistics. Our triangular-array framework allows the kernel to take values in a separable Hilbert space $H_n$ that may vary with the sample size, thereby covering both infinite-dimensional spaces and Euclidean spaces of increasing dimension. The Gaussian approximation combines a Hilbert-space version of Stein's method with refined control of the higher-order terms in the Hoeffding decomposition. Its error depends on fourth moments of the kernel and on the spectral geometry of the covariance operator of the Hájek projection, particularly the proportion of squared spectral mass lying outside the leading eigendirection. Consequently, the theory accommodates singular and approximately low-rank covariance operators, provided that sufficient spectral mass remains beyond the leading direction. For $H_n=\mathbb{R}^{d_n}$, we obtain dimension-explicit bounds under coordinatewise fourth-moment conditions. We also establish approximation bounds for the empirical, Gaussian-weighted, and jackknife multiplier bootstraps, yielding tests with asymptotically correct size and consistency against alternatives at covariance-dependent separation rates. For testing the vector of pairwise Kendall's tau coefficients, a matching minimax lower bound shows that the resulting separation rate is minimax rate-optimal over dense Gaussian correlation alternatives.

Comparing correlation matrices across time or stress scenarios is critical in quantitative finance and multivariate statistics, yet sample estimation noise often obscures whether an observed distance reflects a true structural shift. We derive the asymptotic sampling distribution of the intrinsic off-log (log-Euclidean) distance between two independently estimated full-rank correlation matrices under the null hypothesis that their population correlation matrices coincide. Under general sampling with finite fourth moments, the scaled squared distance converges to a weighted sum of independent $\chi_1^2$ variables, with weights determined by the asymptotic covariance of the Generalized Fisher Transformation (GFT) coordinates. Under Gaussian sampling at independence, this simplifies to a parameter-free $4\chi_d^2$ law. To calibrate tail probabilities, we provide closed-form cumulant generating functions, Lugannani–Rice saddlepoint quantiles, and an explicit Chernoff envelope requiring no root-finding. The first moment of the limiting law establishes a simple rule of thumb for the baseline expected distance under the null hypothesis ($\operatorname E[d_{\mathrm{LE}}] \lesssim 2\sqrt{d/n}$ near independence), quantifying the average separation induced strictly by estimation error. We establish plug-in consistency, present an explicit Gaussian covariance factorization, compare the distance statistic with coordinate Wald tests, and characterize its local power.

Exponential random graph models (ERGMs) describe dependence among network ties, but inference becomes difficult when the likelihood is intractable and candidate network statistics are strongly correlated. We introduce BERGM Elastic Net, an adaptive empirical-Bayes approach that combines lasso shrinkage with ridge stabilization in a Bayesian ERGM. A latent-variable formulation supports approximate exchange sampling, while empirical-Bayes updates adapt the amount of regularization to the observed network. We connect the proposed prior to elastic-net penalized likelihood and clarify the interpretation of thresholded reporting and coefficient grouping. The method is developed for over-specified network models containing many related structural and covariate effects.

Modern medical research, such as dose-finding studies, seamless trials, and shared control designs, often involves comparing multiple treatments simultaneously. Despite its wide applications, most research focuses on continuous endpoints, leaving the inference for general outcome types in high demand. In this article, we propose a new inference method for conducting multiple-treatment comparisons involving endpoints within the generalized linear model (GLM) framework under covariate-adaptive randomization (CAR). First, we investigate the asymptotic properties of the standard Wald z-statistics (z-scores) in multi-arm trials, highlighting issues when the working model is misspecified, particularly through omitted covariates. Our theoretical findings reveal that these \textcolor{black}{z-scores} do not consistently converge to a standard multivariate normal distribution, leading to either conservative or inflated Type I error rates, depending on the specific GLM endpoint. Second, based on these theoretical results, we develop adjusted test statistics to correct the distributional problems. To appropriately control the family-wise Type I error rate inherent in multi-arm comparisons, we incorporate our adjusted statistics with Simes-type multiple-testing procedures. This robust inference method can effectively control Type I error while potentially improving power. Extensive simulation studies and a real-world application to a metastatic breast cancer trial confirm the effectiveness and practicality of our approach.

We study estimation of the von Neumann entropy of a $d$-dimensional quantum state from independent outcomes of a fixed rank-one measurement. Let $\kappa_\nu\in[1,d+1]$ denote the normalized trace-zero frame spread; $\kappa_\nu=1$ for exact complex projective $2$-designs. For any fixed rank-one measurement and $n\le c_2d^2/\kappa_\nu$, the minimax mean-squared error is at least $c_1\log^2\{d^2/(\kappa_\nu n)\}$. For bounded frame spread and every fixed $0<a<2$, the minimax risk is of order $\log^2 d$ throughout $n\lesssim d^{2-a}$; hence fixed uniform accuracy requires $\Omega(d^2)$ observations. Existing tomography bounds give bounded risk from $O(d^3\log^3 d)$ observations for finite equal-weight exact designs and the Haar covariant measurement, while the intermediate minimax regime remains open. The results concern a fixed measurement repeated independently across copies; adaptive and collective protocols, and protocols that reuse common randomness across copies, are not covered.

We establish a lower bound for estimating the von Neumann entropy from independent outcomes of any fixed rank-one POVM. A rotation-averaged van Trees argument gives a global minimax risk of at least $(d/n)\log^2\{n/(4d)\}$ when $d\ge C$ and $n\ge Cd$, without a projective-design assumption. We also characterize risk on an operator-norm ball of radius $r$ around the maximally mixed state. We allow an approximate second moment: on the trace-zero Hermitian subspace, the measurement frame may differ by $\varepsilon<1$ from the tight projective frame. A clipped estimator based on canonical dual shadows and the complete U-statistic for purity has risk at most $d^3r^2/n+d^4/n^2+d^6r^6$. Lower bounds under the same frame control yield the local minimax rate $d^3r^2/n+d^4/n^2$ when $n\ge Cd^2$ and $r$ lies in an explicit matching range. For every fixed upper bound on $\varepsilon$, approximation changes only the constants, not the powers of $d,n,r$. At the critical radius $r=n^{-1/2}$, the local risk is asymptotically negligible relative to the global risk when $n\log^2(n/d)\gg d^3$. This separation holds for projective 2-designs, including global Clifford measurements in qubit dimensions, and for their uniformly well-conditioned frame approximations. Finite-sample experiments in dimension four illustrate the critical-radius benchmark and the effect of a nonexact frame.

We study the fundamental problem of estimating the dependence structure of a centered stationary Gaussian process under local differential privacy (LDP). In this setting, the spectral density characterizes the dependence structure of the data and is the quantity to be estimated. Our main contribution is to close the open $\alpha^2$-versus-$\alpha^4$ gap between the previously known lower and upper bounds on the minimax rate. Specifically, we establish a minimax lower bound showing that, over Sobolev-type classes of spectral densities, the effective sample size in the high-privacy regime is $N\alpha^4$, rather than the usual $N\alpha^2$ arising for independent observations. This additional privacy cost is caused by the temporal dependence between the observations rather than by their marginal distributions. The proof relies on a contraction bound for privatized dependent Gaussian observations. Our second contribution is a matching upper bound, free of the polylogarithmic losses present in previous work. Rather than applying a generic privatization scheme to classical estimators, we construct a problem-specific procedure attaining the rate identified by our lower bound. Beyond closing the gaps in spectral density estimation, we apply the tools developed for this problem to several related questions. We (i) close the logarithmic gap for fixed-lag autocovariance estimation, (ii) show that the $\alpha^4$ cost arises locally around every spectral density bounded away from zero, and (iii) establish that classical asymptotic equivalence with an independent Gaussian experiment generally fails under LDP.

Probabilistic models of Directed Acyclic Graphs (DAGs) with latent variables impose equality constraints on the observed data distribution beyond ordinary conditional independencies. These so-called Verma constraints arise in nested Markov models associated with Acyclic Directed Mixed Graphs, the latent projection of latent-variable DAGs. While nested Markov models have been extensively studied from the perspectives of graphical representation and causal identification, their implications for semiparametric efficiency theory remain less understood. We develop results toward establishing a semiparametric framework for statistical models defined by Verma constraints. Our key observation is that nested Markov constraints admit weighted conditional-moment representations under post-fixing distributions induced by graphical fixing operations. We show that fixing induces weighted orthogonality relations in L2(P), thereby converting Verma constraints into explicit tangent-space restrictions. Building on this representation, we characterize the tangent-space orthocomplement for models defined by a single nested Markov constraint through residualized weighted moment functions. This geometric formulation yields Hilbert-space characterizations of semiparametric efficient influence functions and efficiency bounds via orthogonal projection and equivalent minimum-variance formulations. We further discuss extensions to models involving multiple nested Markov constraints, for which we characterize a subspace of the orthocomplement as sums of the corresponding weighted orthogonality relations, while leaving the complete tangent-space characterization open. More broadly, our results connect nested graphical structure with semiparametric Hilbert-space geometry and provide a foundation for a general efficiency theory for nested Markov models. We illustrate the framework through several latent-variable DAGs.

We establish general upper bounds on the maximum effective dimension and information gain of kernel Gram matrices over arbitrary designs in terms of approximation properties of the kernel or, equivalently, of the corresponding reproducing kernel Hilbert space. We show that across three regularity regimes, covering as special cases the Matérn and squared exponential kernels, these bounds cannot be improved beyond constant factors.

Many time series are not stable over their full observation horizon, but may contain scientifically meaningful periods during which a signal remains stable up to a prescribed tolerance. We formulate this as an existence test for stable intervals in a non-stationary regression model with dependent, locally stationary errors. For a signal $d$ derived from the mean function, including deviations from reference levels and derivatives, stability over duration $\delta$ is encoded by $d_\infty=\inf_{t\in[0,1-\delta]}\sup_{s\in[t,t+\delta]}|d(s)|$. The hypothesis $d_\infty\ge\Delta$ states that no interval of length $\delta$ remains within the tolerance $\Delta$, while rejection provides evidence for the existence of a relevant stable period. We estimate $d$ by local linear regression and construct plug-in tests for $d_\infty$. The asymptotic distribution is determined only by near-extremal windows at which the minimax functional is attained. We formalize this localization through extremal sets and derive Gaussian and extreme value approximations for kernel estimators over possibly shrinking index sets with time-varying long-run variance. The resulting tests are consistent and have asymptotic level control. The theory also extends supremum-based relevant-change tests to time-varying long-run variance and derivative-based hypotheses. Simulations and applications to physiological and engineering time series illustrate the method.