Autocorrelations in MCMC chains increase the variance of the estimators they produce. We propose the occlusion process to mitigate this problem. It is a process that sits upon an existing MCMC sampler, and occasionally replaces its samples with ones that are decorrelated from the chain. We show that this process inherits many desirable properties from the underlying MCMC sampler, such as a Law of Large Numbers, convergence in a normed function space, and geometric ergodicity, to name a few. We show how to simulate the occlusion process at no additional time-complexity to the underlying MCMC chain. This requires a threaded computer, and a variational approximation to the target distribution. We demonstrate empirically the occlusion process' decorrelation and variance reduction capabilities on two target distributions. The first is a bimodal Gaussian mixture model in 1d and 100d. The second is the Ising model on an arbitrary graph, for which we propose a novel variational distribution.

Non-pharmaceutical interventions (NPIs) in response to the COVID-19 pandemic necessitated a trade-off between the health impacts of viral spread and the social and economic costs of restrictions. We conduct a cost-effectiveness analysis of NPI policies enacted at the state level in the United States in 2020. Although school closures reduced viral transmission, their social impact in terms of student learning loss was too costly, depriving the nation of \$2 trillion (USD2020), conservatively, in future GDP. Moreover, this marginal trade-off between school closure and COVID deaths was not inescapable: a combination of other measures would have been enough to maintain similar or lower mortality rates without incurring such profound learning loss. Optimal policies involve consistent implementation of mask mandates, public test availability, contact tracing, social distancing orders, and reactive workplace closures, with no closure of schools beyond the usual 16 weeks of break per year. Their use would have reduced the gross impact of the pandemic in the US in 2020 from \$4.6 trillion to \$1.9 trillion and, with high probability, saved over 100,000 lives. Our results also highlight the need to address the substantial global learning deficit incurred during the pandemic.

Given a collection of feature maps indexed by a set $\mathcal{T}$, we study the performance of empirical risk minimization (ERM) on regression problems with square loss over the union of the linear classes induced by these feature maps. This setup aims at capturing the simplest instance of feature learning, where the model is expected to jointly learn from the data an appropriate feature map and a linear predictor. We start by studying the asymptotic quantiles of the excess risk of sequences of empirical risk minimizers. Remarkably, we show that when the set $\mathcal{T}$ is not too large and when there is a unique optimal feature map, these quantiles coincide, up to a factor of two, with those of the excess risk of the oracle procedure, which knows a priori this optimal feature map and deterministically outputs an empirical risk minimizer from the associated optimal linear class. We complement this asymptotic result with a non-asymptotic analysis that quantifies the decaying effect of the global complexity of the set $\mathcal{T}$ on the excess risk of ERM, and relates it to the size of the sublevel sets of the suboptimality of the feature maps. As an application of our results, we obtain new guarantees on the performance of the best subset selection procedure in sparse linear regression under general assumptions.

Here is the revised abstract, ensuring all characters are ASCII-compatible: In this work, we introduce a new framework for active experimentation, the Prediction-Guided Active Experiment (PGAE), which leverages predictions from an existing machine learning model to guide sampling and experimentation. Specifically, at each time step, an experimental unit is sampled according to a designated sampling distribution, and the actual outcome is observed based on an experimental probability. Otherwise, only a prediction for the outcome is available. We begin by analyzing the non-adaptive case, where full information on the joint distribution of the predictor and the actual outcome is assumed. For this scenario, we derive an optimal experimentation strategy by minimizing the semi-parametric efficiency bound for the class of regular estimators. We then introduce an estimator that meets this efficiency bound, achieving asymptotic optimality. Next, we move to the adaptive case, where the predictor is continuously updated with newly sampled data. We show that the adaptive version of the estimator remains efficient and attains the same semi-parametric bound under certain regularity assumptions. Finally, we validate PGAE's performance through simulations and a semi-synthetic experiment using data from the US Census Bureau. The results underscore the PGAE framework's effectiveness and superiority compared to other existing methods.

Neural posterior estimation (NPE) and neural likelihood estimation (NLE) are machine learning approaches that provide accurate posterior, and likelihood, approximations in complex modeling scenarios, and in situations where conducting amortized inference is a necessity. While such methods have shown significant promise across a range of diverse scientific applications, the statistical accuracy of these methods is so far unexplored. In this manuscript, we give, for the first time, an in-depth exploration on the statistical behavior of NPE and NLE. We prove that these methods have similar theoretical guarantees to common statistical methods like approximate Bayesian computation (ABC) and Bayesian synthetic likelihood (BSL). While NPE and NLE methods are just as accurate as ABC and BSL, we prove that this accuracy can often be achieved at a vastly reduced computational cost, and will therefore deliver more attractive approximations than ABC and BSL in certain problems. We verify our results theoretically and in several examples from the literature.

Many data sets cannot be accurately described by standard probability distributions due to the excess number of zero values present. For example, zero-inflation is prevalent in microbiome data and single-cell RNA sequencing data, which serve as our real data examples. Several models have been proposed to address zero-inflated datasets including the zero-inflated negative binomial, hurdle negative binomial model, and the truncated latent Gaussian copula model. This study aims to compare various models and determine which one performs optimally under different conditions using both simulation studies and real data analyses. We are particularly interested in investigating how dependence among the variables, level of zero-inflation or deflation, and variance of the data affects model selection.

Correlated observations are ubiquitous phenomena in a plethora of scientific avenues. Tackling this dependence among test statistics has been one of the pertinent problems in simultaneous inference. However, very little literature exists that elucidates the effect of correlation on different testing procedures under general distributional assumptions. In this work, we address this gap in a unified way by considering the multiple testing problem under a general correlated framework. We establish an upper bound on the family-wise error rate(FWER) of Bonferroni's procedure for equicorrelated test statistics. Consequently, we find that for a quite general class of distributions, Bonferroni FWER asymptotically tends to zero when the number of hypotheses approaches infinity. We extend this result to general positively correlated elliptically contoured setups. We also present examples of distributions for which Bonferroni FWER has a strictly positive limit under equicorrelation. We extend the limiting zero results to the class of step-down procedures under quite general correlated setups. Specifically, the probability of rejecting at least one hypothesis approaches zero asymptotically for any step-down procedure. The results obtained in this work generalize existing results for correlated Normal test statistics and facilitate new insights into the performances of multiple testing procedures under dependence.

In recent years, signSGD has garnered interest as both a practical optimizer as well as a simple model to understand adaptive optimizers like Adam. Though there is a general consensus that signSGD acts to precondition optimization and reshapes noise, quantitatively understanding these effects in theoretically solvable settings remains difficult. We present an analysis of signSGD in a high dimensional limit, and derive a limiting SDE and ODE to describe the risk. Using this framework we quantify four effects of signSGD: effective learning rate, noise compression, diagonal preconditioning, and gradient noise reshaping. Our analysis is consistent with experimental observations but moves beyond that by quantifying the dependence of these effects on the data and noise distributions. We conclude with a conjecture on how these results might be extended to Adam.

We propose and analyze TRAiL (Tangential Randomization in Linear Bandits), a computationally efficient regret-optimal forced exploration algorithm for linear bandits on action sets that are sublevel sets of strongly convex functions. TRAiL estimates the governing parameter of the linear bandit problem through a standard regularized least squares and perturbs the reward-maximizing action corresponding to said point estimate along the tangent plane of the convex compact action set before projecting back to it. Exploiting concentration results for matrix martingales, we prove that TRAiL ensures a $\Omega(\sqrt{T})$ growth in the inference quality, measured via the minimum eigenvalue of the design (regressor) matrix with high-probability over a $T$-length period. We build on this result to obtain an $\mathcal{O}(\sqrt{T} \log(T))$ upper bound on cumulative regret with probability at least $ 1 - 1/T$ over $T$ periods, and compare TRAiL to other popular algorithms for linear bandits. Then, we characterize an $\Omega(\sqrt{T})$ minimax lower bound for any algorithm on the expected regret that covers a wide variety of action/parameter sets and noise processes. Our analysis not only expands the realm of lower-bounds in linear bandits significantly, but as a byproduct, yields a trade-off between regret and inference quality. Specifically, we prove that any algorithm with an $\mathcal{O}(T^\alpha)$ expected regret growth must have an $\Omega(T^{1-\alpha})$ asymptotic growth in expected inference quality. Our experiments on the $L^p$ unit ball as action sets reveal how this relation can be violated, but only in the short-run, before returning to respect the bound asymptotically. In effect, regret-minimizing algorithms must have just the right rate of inference -- too fast or too slow inference will incur sub-optimal regret growth.

Complex engineering systems are often subject to multiple failure modes. Developing a remaining useful life (RUL) prediction model that does not consider the failure mode causing degradation is likely to result in inaccurate predictions. However, distinguishing between causes of failure without manually inspecting the system is nontrivial. This challenge is increased when the causes of historically observed failures are unknown. Sensors, which are useful for monitoring the state-of-health of systems, can also be used for distinguishing between multiple failure modes as the presence of multiple failure modes results in discriminatory behavior of the sensor signals. When systems are equipped with multiple sensors, some sensors may exhibit behavior correlated with degradation, while other sensors do not. Furthermore, which sensors exhibit this behavior may differ for each failure mode. In this paper, we present a simultaneous clustering and sensor selection approach for unlabeled training datasets of systems exhibiting multiple failure modes. The cluster assignments and the selected sensors are then utilized in real-time to first diagnose the active failure mode and then to predict the system RUL. We validate the complete pipeline of the methodology using a simulated dataset of systems exhibiting two failure modes and on a turbofan degradation dataset from NASA.

We address the issue of the testability of instrumental variables derived from observational data. Most existing testable implications are centered on scenarios where the treatment is a discrete variable, e.g., instrumental inequality (Pearl, 1995), or where the effect is assumed to be constant, e.g., instrumental variables condition based on the principle of independent mechanisms (Burauel, 2023). However, treatments can often be continuous variables, such as drug dosages or nutritional content levels, and non-constant effects may occur in many real-world scenarios. In this paper, we consider an additive nonlinear, non-constant effects model with unmeasured confounders, in which treatments can be either discrete or continuous, and propose an Auxiliary-based Independence Test (AIT) condition to test whether a variable is a valid instrument. We first show that if the candidate instrument is valid, then the AIT condition holds. Moreover, we illustrate the implications of the AIT condition and demonstrate that, in certain conditions, AIT conditions are necessary and sufficient to detect all invalid IVs. We also extend the AIT condition to include covariates and introduce a practical testing algorithm. Experimental results on both synthetic and three different real-world datasets show the effectiveness of our proposed condition.

The rapid deployment of distributed energy resources (DER) has introduced significant spatio-temporal uncertainties in power grid management, necessitating accurate multilevel forecasting methods. However, existing approaches often produce overly conservative uncertainty intervals at individual spatial units and fail to properly capture uncertainties when aggregating predictions across different spatial scales. This paper presents a novel hierarchical spatio-temporal model based on the conformal prediction framework to address these challenges. Our approach generates circuit-level DER growth predictions and efficiently aggregates them to the substation level while maintaining statistical validity through a tailored non-conformity score. Applied to a decade of DER installation data from a local utility network, our method demonstrates superior performance over existing approaches, particularly in reducing prediction interval widths while maintaining coverage.

Modeling and forecasting air quality plays a crucial role in informed air pollution management and protecting public health. The air quality data of a region, collected through various pollution monitoring stations, display nonlinearity, nonstationarity, and highly dynamic nature and detain intense stochastic spatiotemporal correlation. Geometric deep learning models such as Spatiotemporal Graph Convolutional Networks (STGCN) can capture spatial dependence while forecasting temporal time series data for different sensor locations. Another key characteristic often ignored by these models is the presence of extreme observations in the air pollutant levels for severely polluted cities worldwide. Extreme value theory is a commonly used statistical method to predict the expected number of violations of the National Ambient Air Quality Standards for air pollutant concentration levels. This study develops an extreme value theory-based STGCN model (E-STGCN) for air pollution data to incorporate extreme behavior across pollutant concentrations. Along with spatial and temporal components, E-STGCN uses generalized Pareto distribution to investigate the extreme behavior of different air pollutants and incorporate it inside graph convolutional networks. The proposal is then applied to analyze air pollution data (PM2.5, PM10, and NO2) of 37 monitoring stations across Delhi, India. The forecasting performance for different test horizons is evaluated compared to benchmark forecasters (both temporal and spatiotemporal). It was found that E-STGCN has consistent performance across all the seasons in Delhi, India, and the robustness of our results has also been evaluated empirically. Moreover, combined with conformal prediction, E-STGCN can also produce probabilistic prediction intervals.

The capture of changes in dynamic systems, especially ordinary differential equations (ODEs), is an important and challenging task, with multiple applications in biomedical research and other scientific areas. This article proposes a fast and mathematically rigorous online method, called ODE-informed MAnifold-constrained Gaussian process Inference for Change point detection(O-MAGIC), to detect changes of parameters in the ODE system using noisy and sparse observation data. O-MAGIC imposes a Gaussian process prior to the time series of system components with a latent manifold constraint, induced by restricting the derivative process to satisfy ODE conditions. To detect the parameter changes from the observation, we propose a procedure based on a two-sample generalized likelihood ratio (GLR) test that can detect multiple change points in the dynamic system automatically. O-MAGIC bypasses conventional numerical integration and achieves substantial savings in computation time. By incorporating the ODE structures through manifold constraints, O-MAGIC enjoys a significant advantage in detection delay, while following principled statistical construction under the Bayesian paradigm, which further enables it to handle systems with missing data or unobserved components. O-MAGIC can also be applied to general nonlinear systems. Simulation studies on three challenging examples: SEIRD model, Lotka-Volterra model and Lorenz model are provided to illustrate the robustness and efficiency of O-MAGIC, compared with numerical integration and other popular time-series-based change point detection benchmark methods.

This paper proposes a sparse regression method that continuously interpolates between Forward Stepwise selection (FS) and the LASSO. When tuned appropriately, our solutions are much sparser than typical LASSO fits but, unlike FS fits, benefit from the stabilizing effect of shrinkage. Our method, Adaptive Forward Stepwise Regression (AFS) addresses this need for sparser models with shrinkage. We show its connection with boosting via a soft-thresholding viewpoint and demonstrate the ease of adapting the method to classification tasks. In both simulations and real data, our method has lower mean squared error and fewer selected features across multiple settings compared to popular sparse modeling procedures.

This paper is motivated by modeling the cycle-to-cycle variability associated with the resistive switching operation behind memristors. As the data are by nature curves, functional principal component analysis is a suitable candidate to explain the main modes of variability. Taking into account this data-driven motivation, in this paper we propose two new forecasting approaches based on studying the sequential cross-dependence between and within a multivariate functional time series in terms of vector autoregressive modeling of the most explicative functional principal component scores. The main difference between the two methods lies in whether a univariate or multivariate PCA is performed so that we have a different set of principal component scores for each functional time series or the same one for all of them. Finally, the sample performance of the proposed methodologies is illustrated by an application on a bivariate functional time series of reset-set curves.

Our model for the lifespan of an enterprise is the geometric distribution. We do not formulate a model for enterprise foundation, but assume that foundations and lifespans are independent. We aim to fit the model to information about foundation and closure of German enterprises in the AFiD panel. The lifespan for an enterprise that has been founded before the first wave of the panel is either left truncated, when the enterprise is contained in the panel, or missing, when it already closed down before the first wave. Marginalizing the likelihood to that part of the enterprise history after the first wave contributes to the aim of a closed-form estimate and standard error. Invariance under the foundation distribution is achived by conditioning on observability of the enterprises. The conditional marginal likelihood can be written as a function of a martingale. The later arises when calculating the compensator, with respect some filtration, of a process that counts the closures. The estimator itself can then also be written as a martingale transform and consistency as well as asymptotic normality are easily proven. The life expectancy of German enterprises, estimated from the demographic information about 1.4 million enterprises for the years 2018 and 2019, are ten years. The width of the confidence interval are two months. Closure after the last wave is taken into account as right censored.

Multilevel compositional data, such as data sampled over time that are non-negative and sum to a constant value, are common in various fields. However, there is currently no software specifically built to model compositional data in a multilevel framework. The R package multilevelcoda implements a collection of tools for modelling compositional data in a Bayesian multivariate, multilevel pipeline. The user-friendly setup only requires the data, model formula, and minimal specification of the analysis. This paper outlines the statistical theory underlying the Bayesian compositional multilevel modelling approach and details the implementation of the functions available in multilevelcoda, using an example dataset of compositional daily sleep-wake behaviours. This innovative method can be used to gain robust answers to scientific questions using the increasingly available multilevel compositional data from intensive, longitudinal studies.

We propose a nonstationary functional time series forecasting method with an application to age-specific mortality rates observed over the years. The method begins by taking the first-order differencing and estimates its long-run covariance function. Through eigen-decomposition, we obtain a set of estimated functional principal components and their associated scores for the differenced series. These components allow us to reconstruct the original functional data and compute the residuals. To model the temporal patterns in the residuals, we again perform dynamic functional principal component analysis and extract its estimated principal components and the associated scores for the residuals. As a byproduct, we introduce a geometrically decaying weighted approach to assign higher weights to the most recent data than those from the distant past. Using the Swedish age-specific mortality rates from 1751 to 2022, we demonstrate that the weighted dynamic functional factor model can produce more accurate point and interval forecasts, particularly for male series exhibiting higher volatility.

Causal effect estimation is a critical task in statistical learning that aims to find the causal effect on subjects by identifying causal links between a number of predictor (or, explanatory) variables and the outcome of a treatment. In a regressional framework, we assign a treatment and outcome model to estimate the average causal effect. Additionally, for high dimensional regression problems, variable selection methods are also used to find a subset of predictor variables that maximises the predictive performance of the underlying model for better estimation of the causal effect. In this paper, we propose a different approach. We focus on the variable selection aspects of high dimensional causal estimation problem. We suggest a cautious Bayesian group LASSO (least absolute shrinkage and selection operator) framework for variable selection using prior sensitivity analysis. We argue that in some cases, abstaining from selecting (or, rejecting) a predictor is beneficial and we should gather more information to obtain a more decisive result. We also show that for problems with very limited information, expert elicited variable selection can give us a more stable causal effect estimation as it avoids overfitting. Lastly, we carry a comparative study with synthetic dataset and show the applicability of our method in real-life situations.

Sparse linear regression is one of the classic problems in the field of statistics, which has deep connections and high intersections with optimization, computation, and machine learning. To address the effective handling of high-dimensional data, the diversity of real noise, and the challenges in estimating standard deviation of the noise, we propose a novel and general graph-based square-root estimation (GSRE) model for sparse linear regression. Specifically, we use square-root-loss function to encourage the estimators to be independent of the unknown standard deviation of the error terms and design a sparse regularization term by using the graphical structure among predictors in a node-by-node form. Based on the predictor graphs with special structure, we highlight the generality by analyzing that the model in this paper is equivalent to several classic regression models. Theoretically, we also analyze the finite sample bounds, asymptotic normality and model selection consistency of GSRE method without relying on the standard deviation of error terms. In terms of computation, we employ the fast and efficient alternating direction method of multipliers. Finally, based on a large number of simulated and real data with various types of noise, we demonstrate the performance advantages of the proposed method in estimation, prediction and model selection.

Under a multinormal distribution with an arbitrary unknown covariance matrix, the main purpose of this paper is to propose a framework to achieve the goal of reconciliation of Bayesian, frequentist, and Fisher's reporting $p$-values, Neyman-Pearson's optimal theory and Wald's decision theory for the problems of testing mean against restricted alternatives (closed convex cones). To proceed, the tests constructed via the likelihood ratio (LR) and the union-intersection (UI) principles are studied. For the problems of testing against restricted alternatives, first, we show that the LRT and the UIT are not the proper Bayes tests, however, they are shown to be the integrated LRT and the integrated UIT, respectively. For the problem of testing against the positive orthant space alternative, both the null distributions of the LRT and the UIT depend on the unknown nuisance covariance matrix. Hence we have difficulty adopting Fisher's approach to reporting $p$-values. On the other hand, according to the definition of the level of significance, both the LRT and the UIT are shown to be power-dominated by the corresponding LRT and UIT for testing against the half-space alternative, respectively. Hence, both the LRT and the UIT are $\alpha$-inadmissible, these results are against the common statistical sense. Neither Fisher's approach of reporting $p$-values alone nor Neyman-Pearson's optimal theory for power function alone is a satisfactory criterion for evaluating the performance of tests. Wald's decision theory via $d$-admissibility may shed light on resolving these challenging issues of imposing the balance between type 1 error and power.

Untreated periodontitis causes inflammation within the supporting tissue of the teeth and can ultimately lead to tooth loss. Modeling periodontal outcomes is beneficial as they are difficult and time consuming to measure, but disparities in representation between demographic groups must be considered. There may not be enough participants to build group specific models and it can be ineffective, and even dangerous, to apply a model to participants in an underrepresented group if demographic differences were not considered during training. We propose an extension to RECaST Bayesian transfer learning framework. Our method jointly models multivariate outcomes, exhibiting significant improvement over the previous univariate RECaST method. Further, we introduce an online approach to model sequential data sets. Negative transfer is mitigated to ensure that the information shared from the other demographic groups does not negatively impact the modeling of the underrepresented participants. The Bayesian framework naturally provides uncertainty quantification on predictions. Especially important in medical applications, our method does not share data between domains. We demonstrate the effectiveness of our method in both predictive performance and uncertainty quantification on simulated data and on a database of dental records from the HealthPartners Institute.

Statistical process monitoring (SPM) methods are essential tools in quality management to check the stability of industrial processes, i.e., to dynamically classify the process state as in control (IC), under normal operating conditions, or out of control (OC), otherwise. Traditional SPM methods are based on unsupervised approaches, which are popular because in most industrial applications the true OC states of the process are not explicitly known. This hampered the development of supervised methods that could instead take advantage of process data containing labels on the true process state, although they still need improvement in dealing with class imbalance, as OC states are rare in high-quality processes, and the dynamic recognition of unseen classes, e.g., the number of possible OC states. This article presents a novel stream-based active learning strategy for SPM that enhances partially hidden Markov models to deal with data streams. The ultimate goal is to optimize labeling resources constrained by a limited budget and dynamically update the possible OC states. The proposed method performance in classifying the true state of the process is assessed through a simulation and a case study on the SPM of a resistance spot welding process in the automotive industry, which motivated this research.

This paper introduces the partial Gini covariance, a novel dependence measure that addresses the challenges of high-dimensional inference with heavy-tailed errors, often encountered in fields like finance, insurance, climate, and biology. Conventional high-dimensional regression inference methods suffer from inaccurate type I errors and reduced power in heavy-tailed contexts, limiting their effectiveness. Our proposed approach leverages the partial Gini covariance to construct a robust statistical inference framework that requires minimal tuning and does not impose restrictive moment conditions on error distributions. Unlike traditional methods, it circumvents the need for estimating the density of random errors and enhances the computational feasibility and robustness. Extensive simulations demonstrate the proposed method's superior power and robustness over standard high-dimensional inference approaches, such as those based on the debiased Lasso. The asymptotic relative efficiency analysis provides additional theoretical insight on the improved efficiency of the new approach in the heavy-tailed setting. Additionally, the partial Gini covariance extends to the multivariate setting, enabling chi-square testing for a group of coefficients. We illustrate the method's practical application with a real-world data example.

In the age of digital healthcare, passively collected physical activity profiles from wearable sensors are a preeminent tool for evaluating health outcomes. In order to fully leverage the vast amounts of data collected through wearable accelerometers, we propose to use quantile functional regression to model activity profiles as distributional outcomes through quantile responses, which can be used to evaluate activity level differences across covariates based on any desired distributional summary. Our proposed framework addresses two key problems not handled in existing distributional regression literature. First, we use spline mixed model formulations in the basis space to model nonparametric effects of continuous predictors on the distributional response. Second, we address the underlying missingness problem that is common in these types of wearable data but typically not addressed. We show that the missingness can induce bias in the subject-specific distributional summaries that leads to biased distributional regression estimates and even bias the frequently used scalar summary measures, and introduce a nonparametric function-on-function modeling approach that adjusts for each subject's missingness profile to address this problem. We evaluate our nonparametric modeling and missing data adjustment using simulation studies based on realistically simulated activity profiles and use it to gain insights into adolescent activity profiles from the Teen Environment and Neighborhood study.

The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step in learning problems involving observations far away from the center. In this paper, we test the goodness-of-fit of a given parametric model to the extremal dependence structure of a bivariate random sample. The proposed test statistic consists of a weighted $L_1$-Wasserstein distance between a nonparametric, rank-based estimator of the true angular measure obtained by maximizing a Euclidean likelihood on the one hand, and a parametric estimator of the angular measure on the other hand. The asymptotic distribution of the test statistic under the null hypothesis is derived and is used to obtain critical values for the proposed testing procedure via a parametric bootstrap. Consistency of the bootstrap algorithm is proved. A simulation study illustrates the finite-sample performance of the test for the logistic and H\"usler--Reiss models. We apply the method to test for the H\"usler--Reiss model in the context of river discharge data.

This PhD Thesis presents an investigation into the analysis of financial returns using mixture models, focusing on mixtures of generalized normal distributions (MGND) and their extensions. The study addresses several critical issues encountered in the estimation process and proposes innovative solutions to enhance accuracy and efficiency. In Chapter 2, the focus lies on the MGND model and its estimation via expectation conditional maximization (ECM) and generalized expectation maximization (GEM) algorithms. A thorough exploration reveals a degeneracy issue when estimating the shape parameter. Several algorithms are proposed to overcome this critical issue. Chapter 3 extends the theoretical perspective by applying the MGND model on several stock market indices. A two-step approach is proposed for identifying turmoil days and estimating returns and volatility. Chapter 4 introduces constrained mixture of generalized normal distributions (CMGND), enhancing interpretability and efficiency by imposing constraints on parameters. Simulation results highlight the benefits of constrained parameter estimation. Finally, Chapter 5 introduces generalized normal distribution-hidden Markov models (GND-HMMs) able to capture the dynamic nature of financial returns. This manuscript contributes to the statistical modelling of financial returns by offering flexible, parsimonious, and interpretable frameworks. The proposed mixture models capture complex patterns in financial data, thereby facilitating more informed decision-making in financial analysis and risk management.

Mangroves play a crucial role in maintaining coastal ecosystem health and protecting biodiversity. Therefore, continuous mapping of mangroves is essential for understanding their dynamics. Earth observation imagery typically provides a cost-effective way to monitor mangrove dynamics. However, there is a lack of regional studies on mangrove areas in the UAE. This study utilizes the UNet++ deep learning model combined with Sentinel-2 multispectral data and manually annotated labels to monitor the spatiotemporal dynamics of densely distributed mangroves (coverage greater than 70%) in the UAE from 2017 to 2024, achieving an mIoU of 87.8% on the validation set. Results show that the total mangrove area in the UAE in 2024 was approximately 9,142.21 hectares, an increase of 2,061.33 hectares compared to 2017, with carbon sequestration increasing by approximately 194,383.42 tons. Abu Dhabi has the largest mangrove area and plays a dominant role in the UAE's mangrove growth, increasing by 1,855.6 hectares between 2017-2024, while other emirates have also contributed to mangrove expansion through stable and sustainable growth in mangrove areas. This comprehensive growth pattern reflects the collective efforts of all emirates in mangrove restoration.

The objective of the paper is to price weather derivative contracts based on temperature and precipitation as underlying climate variables. We use a neural network approach combined with time series forecast to value Pacific Rim index in Toronto and Chicago

Linear regression is often deemed inherently interpretable; however, challenges arise for high-dimensional data. We focus on further understanding how linear regression approximates nonlinear responses from high-dimensional functional data, motivated by predicting cycle life for lithium-ion batteries. We develop a linearization method to derive feature coefficients, which we compare with the closest regression coefficients of the path of regression solutions. We showcase the methods on battery data case studies where a single nonlinear compressing feature, $g\colon \mathbb{R}^p \to \mathbb{R}$, is used to construct a synthetic response, $\mathbf{y} \in \mathbb{R}$. This unifying view of linear regression and compressing features for high-dimensional functional data helps to understand (1) how regression coefficients are shaped in the highly regularized domain and how they relate to linearized feature coefficients and (2) how the shape of regression coefficients changes as a function of regularization to approximate nonlinear responses by exploiting local structures.

We introduce a new method for learning Bayesian neural networks, treating them as a stack of multivariate Bayesian linear regression models. The main idea is to infer the layerwise posterior exactly if we know the target outputs of each layer. We define these pseudo-targets as the layer outputs from the forward pass, updated by the backpropagated gradients of the objective function. The resulting layerwise posterior is a matrix-normal distribution with a Kronecker-factorized covariance matrix, which can be efficiently inverted. Our method extends to the stochastic mini-batch setting using an exponential moving average over natural-parameter terms, thus gradually forgetting older data. The method converges in few iterations and performs as well as or better than leading Bayesian neural network methods on various regression, classification, and out-of-distribution detection benchmarks.

We propose a new approach for fine-grained uncertainty quantification (UQ) using a collision matrix. For a classification problem involving $K$ classes, the $K\times K$ collision matrix $S$ measures the inherent (aleatoric) difficulty in distinguishing between each pair of classes. In contrast to existing UQ methods, the collision matrix gives a much more detailed picture of the difficulty of classification. We discuss several possible downstream applications of the collision matrix, establish its fundamental mathematical properties, as well as show its relationship with existing UQ methods, including the Bayes error rate. We also address the new problem of estimating the collision matrix using one-hot labeled data. We propose a series of innovative techniques to estimate $S$. First, we learn a contrastive binary classifier which takes two inputs and determines if they belong to the same class. We then show that this contrastive classifier (which is PAC learnable) can be used to reliably estimate the Gramian matrix of $S$, defined as $G=S^TS$. Finally, we show that under very mild assumptions, $G$ can be used to uniquely recover $S$, a new result on stochastic matrices which could be of independent interest. Experimental results are also presented to validate our methods on several datasets.

We develop a new approach for clustering non-spherical (i.e., arbitrary component covariances) Gaussian mixture models via a subroutine, based on the sum-of-squares method, that finds a low-dimensional separation-preserving projection of the input data. Our method gives a non-spherical analog of the classical dimension reduction, based on singular value decomposition, that forms a key component of the celebrated spherical clustering algorithm of Vempala and Wang [VW04] (in addition to several other applications). As applications, we obtain an algorithm to (1) cluster an arbitrary total-variation separated mixture of $k$ centered (i.e., zero-mean) Gaussians with $n\geq \operatorname{poly}(d) f(w_{\min}^{-1})$ samples and $\operatorname{poly}(n)$ time, and (2) cluster an arbitrary total-variation separated mixture of $k$ Gaussians with identical but arbitrary unknown covariance with $n \geq d^{O(\log w_{\min}^{-1})} f(w_{\min}^{-1})$ samples and $n^{O(\log w_{\min}^{-1})}$ time. Here, $w_{\min}$ is the minimum mixing weight of the input mixture, and $f$ does not depend on the dimension $d$. Our algorithms naturally extend to tolerating a dimension-independent fraction of arbitrary outliers. Before this work, the techniques in the state-of-the-art non-spherical clustering algorithms needed $d^{O(k)} f(w_{\min}^{-1})$ time and samples for clustering such mixtures. Our results may come as a surprise in the context of the $d^{\Omega(k)}$ statistical query lower bound [DKS17] for clustering non-spherical Gaussian mixtures. While this result is usually thought to rule out $d^{o(k)}$ cost algorithms for the problem, our results show that the lower bounds can in fact be circumvented for a remarkably general class of Gaussian mixtures.

Stochastic processes model various natural phenomena from disease transmission to stock prices, but simulating and quantifying their uncertainty can be computationally challenging. For example, modeling a Gaussian Process with standard statistical methods incurs an $\mathcal{O}(n^3)$ penalty, and even using state-of-the-art Neural Processes (NPs) incurs an $\mathcal{O}(n^2)$ penalty due to the attention mechanism. We introduce the Transformer Neural Process - Kernel Regression (TNP-KR), a new architecture that incorporates a novel transformer block we call a Kernel Regression Block (KRBlock), which reduces the computational complexity of attention in transformer-based Neural Processes (TNPs) from $\mathcal{O}((n_C+n_T)^2)$ to $O(n_C^2+n_Cn_T)$ by eliminating masked computations, where $n_C$ is the number of context, and $n_T$ is the number of test points, respectively, and a fast attention variant that further reduces all attention calculations to $\mathcal{O}(n_C)$ in space and time complexity. In benchmarks spanning such tasks as meta-regression, Bayesian optimization, and image completion, we demonstrate that the full variant matches the performance of state-of-the-art methods while training faster and scaling two orders of magnitude higher in number of test points, and the fast variant nearly matches that performance while scaling to millions of both test and context points on consumer hardware.

We present a polynomial-time reduction from solving noisy linear equations over $\mathbb{Z}/q\mathbb{Z}$ in dimension $\Theta(k\log n/\mathsf{poly}(\log k,\log q,\log\log n))$ with a uniformly random coefficient matrix to noisy linear equations over $\mathbb{Z}/q\mathbb{Z}$ in dimension $n$ where each row of the coefficient matrix has uniformly random support of size $k$. This allows us to deduce the hardness of sparse problems from their dense counterparts. In particular, we derive hardness results in the following canonical settings. 1) Assuming the $\ell$-dimensional (dense) LWE over a polynomial-size field takes time $2^{\Omega(\ell)}$, $k$-sparse LWE in dimension $n$ takes time $n^{\Omega({k}/{(\log k \cdot (\log k + \log \log n))})}.$ 2) Assuming the $\ell$-dimensional (dense) LPN over $\mathbb{F}_2$ takes time $2^{\Omega(\ell/\log \ell)}$, $k$-sparse LPN in dimension $n$ takes time $n^{\Omega(k/(\log k \cdot (\log k + \log \log n)^2))}~.$ These running time lower bounds are nearly tight as both sparse problems can be solved in time $n^{O(k)},$ given sufficiently many samples. We further give a reduction from $k$-sparse LWE to noisy tensor completion. Concretely, composing the two reductions implies that order-$k$ rank-$2^{k-1}$ noisy tensor completion in $\mathbb{R}^{n^{\otimes k}}$ takes time $n^{\Omega(k/ \log k \cdot (\log k + \log \log n))}$, assuming the exponential hardness of standard worst-case lattice problems.

Tax administrative cost reduction is an economically and socially desirable goal for public policy. This article proposes total administrative cost as percentage of total tax revenue as a vivid measurand, also useful for cross-jurisdiction comparisons. Statistical data, surveys and a novel approach demonstrate: Germany's 2021 tax administrative costs likely exceeded 20% of total tax revenue, indicating need for improvement of Germany's taxation system - and for the many jurisdictions with similar tax regimes. In addition, this article outlines possible reasons for and implications of the seemingly high tax administrative burden as well as solutions.

Point processes and, more generally, random measures are ubiquitous in modern statistics. However, they can only take positive values, which is a severe limitation in many situations. In this work, we introduce and study random signed measures, also known as real-valued random measures, and apply them to constrcut various Bayesian non-parametric models. In particular, we provide an existence result for random signed measures, allowing us to obtain a canonical definition for them and solve a 70-year-old open problem. Further, we provide a representation of completely random signed measures (CRSMs), which extends the celebrated Kingman's representation for completely random measures (CRMs) to the real-valued case. We then introduce specific classes of random signed measures, including the Skellam point process, which plays the role of the Poisson point process in the real-valued case, and the Gaussian random measure. We use the theoretical results to develop two Bayesian nonparametric models -- one for topic modeling and the other for random graphs -- and to investigate mean function estimation in Bayesian nonparametric regression.

The predict-then-optimize (PTO) framework is indispensable for addressing practical stochastic decision-making tasks. It consists of two crucial steps: initially predicting unknown parameters of an optimization model and subsequently solving the problem based on these predictions. Elmachtoub and Grigas [1] introduced the Smart Predict-then-Optimize (SPO) loss for the framework, which gauges the decision error arising from predicted parameters, and a convex surrogate, the SPO+ loss, which incorporates the underlying structure of the optimization model. The consistency of these different loss functions is guaranteed under the assumption of i.i.d. training data. Nevertheless, various types of data are often dependent, such as power load fluctuations over time. This dependent nature can lead to diminished model performance in testing or real-world applications. Motivated to make intelligent predictions for time series data, we present an autoregressive SPO method directly targeting the optimization problem at the decision stage in this paper, where the conditions of consistency are no longer met. Therefore, we first analyze the generalization bounds of the SPO loss within our autoregressive model. Subsequently, the uniform calibration results in Liu and Grigas [2] are extended in the proposed model. Finally, we conduct experiments to empirically demonstrate the effectiveness of the SPO+ surrogate compared to the absolute loss and the least squares loss, especially when the cost vectors are determined by stationary dynamical systems and demonstrate the relationship between normalized regret and mixing coefficients.

Continually evaluating large generative models provides a unique challenge. Often, human annotations are necessary to evaluate high-level properties of these models (e.g. in text or images). However, collecting human annotations of samples can be resource intensive, and using other machine learning systems to provide the annotations, or automatic evaluation, can introduce systematic errors into the evaluation. The Prediction Powered Inference (PPI) framework provides a way of leveraging both the statistical power of automatic evaluation and a small pool of labelled data to produce a low-variance, unbiased estimate of the quantity being evaluated for. However, most work on PPI considers a relatively sizable set of labelled samples, which is not always practical to obtain. To this end, we present two new PPI-based techniques that leverage robust regressors to produce even lower variance estimators in the few-label regime.

We introduce OrigamiPlot, an open-source R package and Shiny web application designed to enhance the visualization of multivariate data. This package implements the origami plot, a novel visualization technique proposed by Duan et al. in 2023, which improves upon traditional radar charts by ensuring that the area of the connected region is invariant to the ordering of attributes, addressing a key limitation of radar charts. The software facilitates multivariate decision-making by supporting comparisons across multiple objects and attributes, offering customizable features such as auxiliary axes and weighted attributes for enhanced clarity. Through the R package and user-friendly Shiny interface, researchers can efficiently create and customize plots without requiring extensive programming knowledge. Demonstrated using network meta-analysis as a real-world example, OrigamiPlot proves to be a versatile tool for visualizing multivariate data across various fields. This package opens new opportunities for simplifying decision-making processes with complex data.

We revisit the problem of distribution learning within the framework of learning-augmented algorithms. In this setting, we explore the scenario where a probability distribution is provided as potentially inaccurate advice on the true, unknown distribution. Our objective is to develop learning algorithms whose sample complexity decreases as the quality of the advice improves, thereby surpassing standard learning lower bounds when the advice is sufficiently accurate. Specifically, we demonstrate that this outcome is achievable for the problem of learning a multivariate Gaussian distribution $N(\boldsymbol{\mu}, \boldsymbol{\Sigma})$ in the PAC learning setting. Classically, in the advice-free setting, $\tilde{\Theta}(d^2/\varepsilon^2)$ samples are sufficient and worst case necessary to learn $d$-dimensional Gaussians up to TV distance $\varepsilon$ with constant probability. When we are additionally given a parameter $\tilde{\boldsymbol{\Sigma}}$ as advice, we show that $\tilde{O}(d^{2-\beta}/\varepsilon^2)$ samples suffices whenever $\| \tilde{\boldsymbol{\Sigma}}^{-1/2} \boldsymbol{\Sigma} \tilde{\boldsymbol{\Sigma}}^{-1/2} - \boldsymbol{I_d} \|_1 \leq \varepsilon d^{1-\beta}$ (where $\|\cdot\|_1$ denotes the entrywise $\ell_1$ norm) for any $\beta > 0$, yielding a polynomial improvement over the advice-free setting.

We explore the behaviour emerging from learning agents repeatedly interacting strategically for a wide range of learning dynamics that includes projected gradient, replicator and log-barrier dynamics. Going beyond the better-understood classes of potential games and zero-sum games, we consider the setting of a general repeated game with finite recall, for different forms of monitoring. We obtain a Folk Theorem-like result and characterise the set of payoff vectors that can be obtained by these dynamics, discovering a wide range of possibilities for the emergence of algorithmic collusion.

We present LazyDINO, a transport map variational inference method for fast, scalable, and efficiently amortized solutions of high-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable (PtO) maps. Our method consists of an offline phase in which we construct a derivative-informed neural surrogate of the PtO map using joint samples of the PtO map and its Jacobian. During the online phase, when given observational data, we seek rapid posterior approximation using surrogate-driven training of a lazy map [Brennan et al., NeurIPS, (2020)], i.e., a structure-exploiting transport map with low-dimensional nonlinearity. The trained lazy map then produces approximate posterior samples or density evaluations. Our surrogate construction is optimized for amortized Bayesian inversion using lazy map variational inference. We show that (i) the derivative-based reduced basis architecture [O'Leary-Roseberry et al., Comput. Methods Appl. Mech. Eng., 388 (2022)] minimizes the upper bound on the expected error in surrogate posterior approximation, and (ii) the derivative-informed training formulation [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] minimizes the expected error due to surrogate-driven transport map optimization. Our numerical results demonstrate that LazyDINO is highly efficient in cost amortization for Bayesian inversion. We observe one to two orders of magnitude reduction of offline cost for accurate posterior approximation, compared to simulation-based amortized inference via conditional transport and conventional surrogate-driven transport. In particular, LazyDINO outperforms Laplace approximation consistently using fewer than 1000 offline samples, while other amortized inference methods struggle and sometimes fail at 16,000 offline samples.