Expectiles provide a class of asymmetric location functionals that incorporate the magnitude of deviations and admit a natural geometric interpretation. Building on their structural consistency with the convex stochastic order, this paper introduces a family of integrated expectile functionals for measuring risk, dispersion, and inequality. The proposed functionals admit analytical representations as integrals of expectiles across asymmetry levels. For a distinguished subclass of these constructions, a geometric representation is available: the resulting quantities can be expressed as weighted areas of star-shaped sets encoding the distributional asymmetry of a random variable. This approach yields a new class of expectile-based inequality indices, constituting a natural counterpart to classical Gini-type measures while preserving desirable monotonicity and consistency properties. Empirical counterparts are derived in closed form and admit explicit decompositions over finite samples. The framework extends naturally to multivariate settings through directional expectile constructions, leading to measures capable of capturing genuinely joint forms of multivariate dispersion and inequality.

We develop new inference procedures for a linear IV model that are robust to identification strength and heteroskedasticity of unknown form, and nonparametric with respect to the first-stage equation. Our first test is tailored for inference on parameters of endogenous explanatory variables. Our new statistic modifies that of Antoine and Lavergne (2003) to directly account for heteroskedasticity of unknown form. As a result, it is asymptotically pivotal, so that inference is greatly facilitated in practice. We also develop (i) an identification-robust subvector inference procedure that does not rely on the knowledge of identification strength for the remaining parameters, and (ii) a pure specification test. In both cases, the tests are conservative but powerful. We show that our procedures are computationally friendly and competitive with existing ones in simulations and an application.

This paper develops a fully discrete cumulative distribution transform (CDT) for atomic probability measures on the real line. The transform is defined through monotone quantile maps and admits explicit linear-time algorithms for both forward transformation and inverse reconstruction based solely on cumulative mass matching. Unlike the classical continuous setting, deterministic transport between atomic measures cannot generally split masses, so exact reconstruction may fail at finite resolution. We establish a precise cumulative-mass compatibility criterion for exact finite-resolution recovery and prove weak convergence of reconstructed measures under reference refinement. Several structural properties of the discrete CDT are derived, including translation, composition, and scaling laws, and the framework is extended to a discrete signed cumulative distribution transform with thresholded stabilization near zero crossings. By avoiding continuous interpolation, the proposed framework provides a simple fixed-reference transport representation for discrete data. Numerical examples illustrate translation linearization, compatibility-controlled reconstruction, refinement consistency, and stabilization of the signed transform.

The article deals with the problem of testing conditional axial symmetry within a~multivariate linear heteroscedastic regression framework. A new test based on integrated rank scores is introduced and its asymptotic distribution is derived. The proposed method extends a similar procedure developed for multivariate data to the regression setting. The test may also be employed to assess specific hypotheses concerning distributional properties of the error term. Its performance and application is illustrated in a small simulation study and with real economic data. The article also contains a few theoretical results regarding axial symmetry that may be of independent interest.

This document is a pdf generated from old plain-TeX files of 1986, of Nils Lid Hjort's `Notes on the Theory of Statistical Symbol Recogntion', a limited circulation 207-pages monograph published at the Norwegian Computing Centre, as Report no. 778/1986. It gives the basics of the statistical pattern recognition theory developed to suit the needs of several industrial projects, related to contracts with the Norwegian-German firm SysScan, the Royal Norwegian Research Council, and yet others, involving symbol recognition and classification analysis from noisy images, related to maps, documents, satellite imaging, etc. The methods and algorithms developed also needed to fit the technology of that time, anno c. 1986, with machines scanning documents, converting these to vector representation, within computational and machine system boundaries. There is an accompanying and also limited circulation booklet, `Statistical Symbol Recognition: Development of a System', by Knut Bråten, Erik Holbæk-Hanssen, and Torfinn Taxt (Report No.777/1986, Norwegian Computing Centre, Oslo), detailing the system developments. Thus developments took form and shape on two frontiers, in close collaboration, Hjort's statistical methods and getting the technology to work, with its multiple components, hardware and software.

We propose a new approach to unbiased estimation of the gradients of the stationary means associated with parametrized families of Markov chains. Our estimators are particularly efficient when the Markov chains have slow mixing rate. Our approach does not require a specific parametrization except for an oracle to evaluate the transition density and its gradient at a given data point without any additional knowledge about the density function itself. It makes our estimator suitable for parametrizations associated with neural networks. The estimator can potentially achieve large improvement in terms of efficiency. Numerical experiments confirm the good performance predicted by the theory.

We study the task of density estimation, where we hope to accurately estimate a probability density from $n$ samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.

We prove that optimally weighted second-order least squares (SLS) and the degree-two generalized polynomial maximization method (PMM) are the same population estimating equation for linear regression with conditionally homoskedastic non-Gaussian errors: they choose the same optimal linear combination of the first two centered residual moments, solve one population normal system, share one influence function, and attain the common asymptotic variance $c_2g_2/N$ -- the ordinary-least-squares slope-variance factor $c_2$ scaled by the PMM variance-reduction coefficient $g_2=1-\gamma_3^2/(2+\gamma_4)$ (with $\gamma_3,\gamma_4$ the error skewness and excess kurtosis). Feasible plug-in implementations are therefore first-order equivalent, with only higher-order finite-sample differences. The identity is sharp: under heteroskedasticity the unconditional PMM body and the conditional SLS weighting separate, costing efficiency for symmetric errors and consistency for asymmetric errors. Beyond degree two, PMM holds an efficiency reserve that SLS cannot reach within its second-moment span. For symmetric platykurtic errors SLS collapses to ordinary least squares for the slope, while degree-three PMM exploits kurtosis information outside the SLS moment span through a closed-form coefficient $g_3$; for canonical asymmetric laws this reserve is $30$--$50\%$ within the degree-three polynomial moment class. The Lean 4 development machine-checks the degree-specific algebraic core -- the closed forms for $g_2$ and $g_3$, the $g_2\le1$ result, the design cancellations, and the symmetric collapse -- while the general monotonicity $g_{S+1}\le g_S\le1$ is proved analytically by nesting. A Monte Carlo study illustrates the equivalence, the reserve, and the heteroskedastic boundary at finite samples.

Several recent works have addressed the problem of deconvolution under a linear model, where the goal is to estimate a completely unknown $G_0$ from a vector of noisy observations $\boldsymbol{Y} = X\boldsymbol{\beta} + \boldsymbol{\epsilon}$, assuming the coefficients $\beta_j$ are i.i.d. unobserved realizations from $G_0$. Assuming $G_0$ has a density $g_0$, we study theoretically a Bayesian nonparametric method proposed in Weinstein et al. (2025) that postulates a Pólya tree prior $\Pi$ on $g_0$ and bases a deconvolution estimate on the posterior distribution $\Pi(\cdot|\boldsymbol{Y})$. Our main result asserts that under the true model (fixed and unknown $g_0$), and under a suitable condition on the minimum eigenvalue of $X^\top X$, the posterior $\Pi(\cdot|\boldsymbol{Y})$ concentrates around $g_0$ in sup-norm. The analysis presented builds on and extends results from Castillo (2017), where posterior consistency of Pólya trees was proved for density estimation, the simpler problem of estimating $g_0$ when observing the coefficients $\beta_j$ directly.

This monograph develops probability and stochastic-process ideas as a translation language for statistics: from designed observations and data objects to targets, stability statements, inference, and use. The chapters move from motivating examples and randomization through probability measures, kernels, likelihoods, data objects, weak convergence, empirical fields, functional data, M- and Z-estimation, testing, local approximations, event-time processes, and prediction. Historical and biomedical examples are used to keep abstract objects tied to records, mechanisms, and decisions. The aim is to give readers a common grammar for classical probability, modern data structures, and statistical practice.

Spectral methods rely fundamentally on the stability of principal eigenspaces under random perturbations. Classically, this stability is quantified by the Davis-Kahan and Wedin theorems, which bound the eigenspace error using the operator norm of the noise and the relevant spectral gaps. While these worst-case bounds are sharp for arbitrary deterministic perturbations, they can be wasteful in the low-rank signal-plus-random-noise setting, as they fail to capture the fine-grained interaction between the signal geometry and the noise distribution. In this paper, we study the spectral perturbation of signal-plus-noise matrices corrupted by sparse, random noise with an arbitrary, inhomogeneous variance profile. We demonstrate that under heterogeneous noise variances, the empirical eigenvectors suffer a systematic, deterministic geometric bias that is entirely invisible to classical perturbation bounds. By leveraging the Quadratic Vector Equation (QVE) and establishing fine-grained isotropic local laws, we derive near-optimal, non-asymptotic perturbation bounds for the leading eigenspaces in the operator and $2\to\infty$ norms. The bounds separate the usual signal-to-noise contribution, stochastic fluctuations, and structured geometric bias terms determined by the alignment between the signal eigenspaces and the row-wise variance profile.

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

Fractional Gaussian noise and $\FARIMA(0,d,0)$ have the same long-memory pole $|\theta|^{-2d}$ and hence the same leading PACF law $\alpha(n)\sim d/n$. We show that this agreement breaks at the first non-universal order. For $0<d<1/2$, the pure fGn PACF satisfies $$ \alpha_{\fGn}(n)=\frac d n+\frac{C_{\fGn}(d)}{n^2}+o(n^{-2}), \qquad C_{\fGn}(d)<d^2, $$ The proof uses the Bingham--Inoue--Kasahara representation, a phase-coefficient expansion for fGn, and a Hankel-operator perturbation argument. Thus the fGn spectral envelope is invisible at first order but visible in second-order finite prediction, explaining why short-memory order selection can differ when fGn data are fitted by FARIMA-type models.

This paper proposes estimation and inference procedures for quantiles of the heterogeneous individual-specific coefficients in panel data. Unlike conventional panel quantile regression, which focuses on outcome heterogeneity, our approach targets the $\tau$-quantile of the cross-sectional distribution of individual-specific slopes. We establish the asymptotic theory under both stochastic and deterministic designs, with convergence rates $\sqrt{N}$ and $\sqrt{N\sqrt{T}}$, respectively. We also develop two corresponding bootstrap procedures for practical inference, and formally establish their validity. The suggested methods are of practical interest since they require weaker sample size growth conditions than standard fixed-effect quantile regression, and accommodate large $N$ settings. Numerical simulations and an empirical application illustrate the empirical effectiveness of the methods under both designs.

We study causal inference in experiments and quasi-experiments, where the economic outcome is imperfectly measured by a remotely sensed variable. The remotely sensed variable is low-cost, scalable, and predictive of the economic outcome in observational data; examples include satellite imagery and mobile phone activity. We model the remotely sensed variable as post-outcome: variation in the economic outcome causes variation in the remotely sensed variable. For example, changes in environmental quality cause changes in satellite imagery, not vice versa. Under this assumption, we propose a formula to nonparametrically identify the causal parameter by combining experimental and observational data. We develop a method for n^{-1/2} inference that is robust to misspecification and that does not restrict the algorithms used to process remotely sensed variables.

It is well-known that maximum likelihood estimates converge faster than the classic square root rate if the Fisher information is infinite. This is often the case when the effective region depends on the estimated parameters, or when density has a singularity inside the effective region at a point dependent on the estimated parameters. We present a one-parameter family of bivariate absolutely continuous distributions on the half-space with smooth densities. The effective domain is always the same half-space and does not depend on this parameter. The order of magnitude for the weighted least squares estimate is asymptotically smaller than the classic square root rate. For the Gaussian variance mixture case, the maximum likelihood estimate coincides with this weighted least squares estimate.

Bayesian online learning provides a coherent framework for sequential inference. However, its theoretical understanding remains limited, particularly in the one-pass setting. Existing theoretical guarantees typically require the mini-batch sample size to diverge, a condition that fails in the one-pass regime. In this paper, we propose a new Bayesian online learning algorithm tailored to the one-pass setting, which incorporates a warm-start phase to ensure stable sequential updates. For this algorithm, we show that the sequentially updated posterior attains the optimal convergence rate. Building on this, we establish an online analogue of the Bernstein-von Mises theorem, which guarantees valid uncertainty quantification without diverging mini-batch sample sizes. Our analysis is based on a novel theoretical framework that differs fundamentally from existing approaches in the online learning literature. Numerical experiments on generalized linear models show that the proposed method matches the performance of the batch estimator while outperforming existing online procedures.

The analysis of extremal dependence in high dimensions is a key challenge in modern extreme-value statistics. Existing methodology primarily focuses on modeling and estimation of extremal dependence structures, often supported by concentration bounds for empirical tail quantities. However, comparatively little is known about general inferential procedures in high-dimensional extremes. In this paper, we develop foundational results that enable inference for rank-based empirical tail dependence coefficients, stable tail dependence functions, and functionals derived from them. We start by establishing finite-sample probability bounds that quantify the linearization error for such estimators uniformly over collections of coordinates. Moreover, we derive high-dimensional central limit theorems and establish the validity of multiplier bootstrap procedures for collections of empirical tail dependence statistics. Within an asymptotic framework, our results allow the dimension to grow exponentially with the effective sample size. We illustrate the usefulness of the results through two applications: uniform expansions and normal approximations for M-estimators of tail dependence parameters and inference for spatial isotropy based on collections of tail dependence functions.

The empirical Bayes $g$-modeling approach based on the nonparametric maximum likelihood estimator (NPMLE) has been central to large-scale estimation and inference in the normal means problem. However, theoretical guarantees for uncertainty quantification remain scarce. A key obstacle is that the NPMLE is necessarily discrete, which yields discrete posterior credible sets and a slow logarithmic deconvolution rate. We address both limitations by introducing a hierarchical Gaussian smoothing layer that restricts the mixing distribution to a Gaussian location mixture. Our smooth NPMLE inherits the favorable properties of the classical NPMLE: it is computable via convex optimization and achieves nearly parametric denoising performance. Moreover, it achieves a polynomial deconvolution rate that is asymptotically minimax over the corresponding class. Our procedure also leads to estimated smooth posteriors that converge to the true posteriors at a polynomial rate. Further, we characterize marginal coverage sets that are optimal in expected length, construct plug-in estimators of these sets, and establish theoretical guarantees for the estimated sets in terms of both coverage probability and expected length. We also extend the theory to settings with model misspecification and heteroscedastic Gaussian observations, and study identifiability of the proposed hierarchical model.

The Cauchy combination test (CCT) is widely used because it yields a closed-form combined $p$-value and is known to be asymptotically valid as the nominal level $\alpha\downarrow0$ under broad dependence structures. We study a different asymptotic question: whether the usual Cauchy cutoff remains accurate at an ordinary fixed level when the number $K$ of combined $p$-values grows under dependence. Under a canonical one-factor equicorrelated Gaussian copula model, we show that the raw CCT is generally not asymptotically exact at fixed $\alpha$. With fixed positive correlation, the statistic converges to a random latent-factor limit, so there is no universal fixed-level reference law. When the common correlation $\rho_K$ weakens with $K$, fixed-level behaviour is governed by the boundary-layer scale $s_K=\sqrt{\rho_K}(\log K)^{3/2}$, and the raw CCT is asymptotically exact if and only if $\rho_K(\log K)^3\to0$. Because the size distortion arises entirely from the reference law and not from the statistic, it can be corrected without modifying the test statistic itself. We propose the boundary-layer calibrated CCT (BL-CCT), which replaces the standard Cauchy reference by a one-parameter Gaussian-smoothed Cauchy family. Unlike recent variants that modify the test statistic, BL-CCT leaves the statistic unchanged and corrects only the reference law. BL-CCT is asymptotically exact under the weaker condition $\rho_K\log K\to0$ and provides a useful finite-$K$ approximation on bounded boundary layers. We also conduct several power analyses: although BL-CCT only raises the cutoff, it incurs no first-order power loss relative to the raw CCT on the exactness scale, under local dense, sparse, and dense Gaussian alternatives. Numerical experiments support the calibration theory.

We construct a new class of infinite-dimensional diffusions with values in a generalized Kingman simplex with finitely many marks. The model describes the temporal evolution of the relative frequencies of infinitely many types that are labeled by a finite number $H$ of marks, but unlabeled within each mark. We first establish a blockwise skew-product representation for a finite-type Wright-Fisher diffusion, extending the aggregation-renormalization self-similarity property of Dirichlet laws. The decomposition separates an $H$-dimensional Wright-Fisher diffusion governing the evolving random mark masses, from $H$ Wright-Fisher diffusions, each run on its own random clock, which describe the evolution of the relative frequencies within each mark. After ranking the within-mark frequencies in decreasing order, we identify the distributional limit as the number of types per mark tends to infinity and we derive an explicit form of its infinitesimal generator on a suitable domain. The limiting diffusion admits the multiple Poisson-Dirichlet distribution as a stationary distribution; it recovers the infinitely-many-neutral-alleles diffusion when all types share the same mark and yields a diffusion on the Thoma simplex when there are two marks.

We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our en-to-end formal infrastructure implement the missing contents in latest Lean library, including a complete development of Gaussian Lipschitz concentration, Dudley's entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is provided in this https URL.

This paper proposes methods for producing compound selection decisions in a Gaussian sequence model. Given unknown, fixed parameters $\mu_ {1:n}$ and known $\sigma_{1:n}$ with observations $Y_i \sim \textsf{N}(\mu_i, \sigma_i^2)$, the decision maker would like to select a subset of indices $S$ so as to maximize utility $\frac{1}{n}\sum_{i\in S} (\mu_i - K_i)$, for known costs $K_i$. Inspired by Stein's unbiased risk estimate (SURE), we introduce an almost unbiased estimator, called ASSURE, for the expected utility of a proposed decision rule. ASSURE allows a user to choose a welfare-maximizing rule from a pre-specified class by optimizing the estimated welfare, thereby producing selection decisions that borrow strength across noisy estimates. We show that ASSURE produces decision rules that are asymptotically no worse than the optimal but infeasible decision rule in the pre-specified class. We apply ASSURE to the selection of Census tracts for economic opportunity, the identification of discriminating firms, and the analysis of $p$-value decision procedures in A/B testing.

This work presents sparse invariant coordinate selection, SICS, a new method for sparse and robust independent component analysis. SICS is based on classical invariant coordinate selection, which is presented in such a form that a LASSO-type penalty can be applied to promote sparsity. Robustness is achieved by using robust scatter matrices. In the first part of the paper, the background and building blocks: scatter matrices, measures of robustness, ICS and independent component analysis, are carefully introduced. Then the proposed new method and its algorithm are derived and presented. This part also includes consistency and breakdown point results for a general case of sparse ICS-like methods. The performance of SICS in identifying sparse independent component loadings is investigated with multiple simulations. The method is illustrated with an example in constructing sparse causal graphs and we also propose a graphical tool for selecting the appropriate sparsity level in SICS.

This article considers the problem of testing conditional independence between two random vectors \(bm X\) and \(\bm Y\) given a confounding random vector \(\bm Z\). An exchangeable-pairs framework is introduced through which the conditional independence testing problem is reformulated as a two-sample testing problem. The framework is motivated by ideas from the model-X literature and is based on a fundamental exchangeability property that holds under the null hypothesis of conditional independence. An energy-distance/maximum mean discrepancy type measure is employed on the resulting exchangeable pairs to quantify departures from conditional independence. A consistent estimator of the proposed discrepancy measure is constructed and its theoretical properties are established under general assumptions. A conditional independence test is then developed using this estimator as a test statistic and is calibrated through a suitable resampling procedure. It is shown that the proposed test is consistent against fixed alternatives, possesses nontrivial asymptotic power against local contiguous alternatives, attains the minimax separation rate for detecting alternatives characterized by the proposed discrepancy measure, and remains consistent when the data dimension diverges with the sample size. The effect of estimating the conditional distribution used to generate the exchangeable pairs is also investigated, and condition under which validity and power properties are preserved is established. Extensive simulation studies demonstrate that the proposed procedure performs competitively with some state-of-the-art methods.

Characteristic function-based goodness-of-fit tests are suggested for multivariate distributions. The test statistics, which are straightforward to compute, are defined as two-sample criteria measuring discrepancy of empirical characteristic functions between multivariate ranks of the original observations and the ranks obtained from an artificial sample generated from the reference distribution under test. Multivariate ranks are constructed using the theory of the optimal measure transport, thus rendering the tests of a simple null hypothesis distribution-free, while bootstrap approximations are still necessary for testing composite null hypotheses. Asymptotic theory is developed and a simulation study, concentrating on comparisons with previously proposed tests of multivariate normality, demonstrates that the method performs well in finite samples. The broad applicability of the proposed methods is illustrated through an application to a real dataset.

As machine learning (ML) enters high-stakes domains, trustworthy uncertainty quantification (UQ) is essential for safety. In this paper we introduce PCS-UQ, a framework based on the Predictability, Computability, and Stability (PCS) principles for veridical data science. Starting with a candidate set of models or algorithms, PCS-UQ integrates a rigorous prediction-check to screen out unsuitable models in the set and utilizes bootstrap samples, in order to capture both inter-sample variability and algorithmic instability for the prediction-checked algorithms. We then introduce a novel multiplicative calibration scheme to enhance local adaptivity, which basically corresponds to a new score in conformal prediction. Moreover, we produce a compilation of 17 real-world regression datasets with manually-constructed subgroups. On this benchmark, PCS-UQ maintains the target coverage while outperforming or matching conformal methods equipped with oracle-selected algorithms in interval width. PCS-UQ achieves consistent subgroup coverage, outperforming these oracle-selected conformal methods. Notably, PCS-UQ stands out in achieving both competitive interval widths and consistent subgroup this http URL 6 classification datasets, PCS-UQ reduces prediction set sizes by 20\%. To scale the framework for deep learning, we propose computationally efficient variants that bypass expensive retraining. On three computer vision benchmarks, these variants reduce prediction set sizes by 20\% over conformal baselines. Finally, we provide theoretical proof that a modified PCS-UQ algorithm preserves valid coverage under exchangeability as a form of split conformal inference.

The hierarchical Dirichlet process is the cornerstone of Bayesian nonparametric multilevel models. Its generative model can be described through a set of latent variables, commonly referred to as tables within the popular restaurant franchise metaphor. The latent tables simplify the expression of the posterior and allow for the implementation of Gibbs sampling algorithms to approximately draw posterior samples. However, managing their assignments can become computationally expensive, especially as the size of the dataset and the number of levels increase. In this work, we identify a prior for the concentration parameter of the hierarchical Dirichlet process that (i) induces a quasi-conjugate posterior distribution, and (ii) removes the need for tables, leading to more interpretable expressions for the posterior, with both a scalable and an exact algorithm to sample from it. Remarkably, this construction extends beyond the Dirichlet process, leading to a new framework for defining normalized hierarchical random measures and a new class of algorithms to sample from their posteriors. The key analytical tool is the independence of multivariate increments, that is, their representation as completely random vectors.

This paper proposes a new approach to the asymptotic analysis of the finite predictor for stationary sequences. Our method yields the exact asymptotics of both the relative prediction error and the partial correlation coefficients. The underlying assumptions are analytic in nature, making the approach applicable to processes with long-range dependence. The ARMA-type process driven by fractional Gaussian noise (fGn), which had previously remained elusive, is used as a case study.

This paper studies the design of cluster experiments to estimate the global treatment effect in the presence of network spillovers. We provide a framework to choose the clustering that minimizes the worst-case mean-squared error of the estimated global effect. We show that optimal clustering solves a novel penalized min-cut optimization problem computed via off-the-shelf semi-definite programming algorithms. Our analysis also characterizes simple conditions to choose between any two cluster designs, including choosing between a cluster or individual-level randomization. We illustrate the method's properties using unique network data from the universe of Facebook's users and existing data from a field experiment.