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Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Surprisingly, even though $B(H)$ is not a semimartingale, Cheridito proved in [Bernoulli 7 (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on a second surprise in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called mixed semimartingales, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense. The estimation of mixed semimartingales with correlation is motivated by applications to high-frequency financial data contaminated by rough noise.

Generative Adversarial Networks (GANs) have achieved a great success in unsupervised learning. Despite its remarkable empirical performance, there are limited theoretical studies on the statistical properties of GANs. This paper provides approximation and statistical guarantees of GANs for the estimation of data distributions that have densities in a H\"{o}lder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen, GANs are consistent estimators of data distributions under strong discrepancy metrics, such as the Wasserstein-1 distance. Furthermore, when the data distribution exhibits low-dimensional structures, we show that GANs are capable of capturing the unknown low-dimensional structures in data and enjoy a fast statistical convergence, which is free of curse of the ambient dimensionality. Our analysis for low-dimensional data builds upon a universal approximation theory of neural networks with Lipschitz continuity guarantees, which may be of independent interest.

We study the convergence properties, in Hellinger and related distances, of nonparametric density estimators based on measure transport. These estimators represent the measure of interest as the pushforward of a chosen reference distribution under a transport map, where the map is chosen via a maximum likelihood objective (equivalently, minimizing an empirical Kullback-Leibler loss) or a penalized version thereof. We establish concentration inequalities for a general class of penalized measure transport estimators, by combining techniques from M-estimation with analytical properties of the transport-based density representation. We then demonstrate the implications of our theory for the case of triangular Knothe-Rosenblatt (KR) transports on the $d$-dimensional unit cube, and show that both penalized and unpenalized versions of such estimators achieve minimax optimal convergence rates over H\"older classes of densities. Specifically, we establish optimal rates for unpenalized nonparametric maximum likelihood estimation over bounded H\"older-type balls, and then for certain Sobolev-penalized estimators and sieved wavelet estimators.

Many countries conduct a full census survey to report official population statistics. As no census survey ever achieves 100 per cent response rate, a post-enumeration survey (PES) is usually conducted and analysed to assess census coverage and produce official population estimates by geographic area and demographic attributes. Considering the usually small size of PES, direct estimation at the desired level of disaggregation is not feasible. Design-based estimation with sampling weight adjustment is a commonly used method but is difficult to implement when survey non-response patterns cannot be fully documented and population benchmarks are not available. We overcome these limitations with a fully model-based Bayesian approach applied to the New Zealand PES. Although theory for the Bayesian treatment of complex surveys has been described, published applications of individual level Bayesian models for complex survey data remain scarce. We provide such an application through a case study of the 2018 census and PES surveys. We implement a multilevel model that accounts for the complex design of PES. We then illustrate how mixed posterior predictive checking and cross-validation can assist with model building and model selection. Finally, we discuss potential methodological improvements to the model and potential solutions to mitigate dependence between the two surveys.

In recent years, change point detection for high dimensional data has become increasingly important in many scientific fields. Most literature develop a variety of separate methods designed for specified models (e.g. mean shift model, vector auto-regressive model, graphical model). In this paper, we provide a unified framework for structural break detection which is suitable for a large class of models. Moreover, the proposed algorithm automatically achieves consistent parameter estimates during the change point detection process, without the need for refitting the model. Specifically, we introduce a three-step procedure. The first step utilizes the block segmentation strategy combined with a fused lasso based estimation criterion, leads to significant computational gains without compromising the statistical accuracy in identifying the number and location of the structural breaks. This procedure is further coupled with hard-thresholding and exhaustive search steps to consistently estimate the number and location of the break points. The strong guarantees are proved on both the number of estimated change points and the rates of convergence of their locations. The consistent estimates of model parameters are also provided. The numerical studies provide further support of the theory and validate its competitive performance for a wide range of models. The developed algorithm is implemented in the R package LinearDetect.

We consider the problem of detecting distributional changes in a sequence of high dimensional data. Our proposed methods are nonparametric, suitable for either continuous or discrete data, and are based on weighted cumulative sums of U-statistics stemming from $L_p$ norms. We establish the asymptotic distribution of our proposed test statistics separately in cases of weakly dependent and strongly dependent coordinates as $\min\{N,d\}\to\infty$, where $N$ denotes sample size and $d$ is the dimension, and also provide sufficient conditions for consistency of the proposed test procedures under a general fixed alternative with one change point. We further assess finite sample performance of the test procedures through Monte Carlo studies, and conclude with two applications to Twitter data concerning the mentions of U.S. Governors and the frequency of tweets containing social justice keywords.

Asymmetry along with heteroscedasticity or contamination often occurs with the growth of data dimensionality. In ultra-high dimensional data analysis, such irregular settings are usually overlooked for both theoretical and computational convenience. In this paper, we establish a framework for estimation in high-dimensional regression models using Penalized Robust Approximated quadratic M-estimators (PRAM). This framework allows general settings such as random errors lack of symmetry and homogeneity, or the covariates are not sub-Gaussian. To reduce the possible bias caused by the data's irregularity in mean regression, PRAM adopts a loss function with a flexible robustness parameter growing with the sample size. Theoretically, we first show that, in the ultra-high dimension setting, PRAM estimators have local estimation consistency at the minimax rate enjoyed by the LS-Lasso. Then we show that PRAM with an appropriate non-convex penalty in fact agrees with the local oracle solution, and thus obtain its oracle property. Computationally, we demonstrate the performances of six PRAM estimators using three types of loss functions for approximation (Huber, Tukey's biweight and Cauchy loss) combined with two types of penalty functions (Lasso and MCP). Our simulation studies and real data analysis demonstrate satisfactory finite sample performances of the PRAM estimator under general irregular settings.

Since proposed in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229: 3091--3120, 2010], the Zhang--Shu framework has attracted extensive attention and motivated many bound-preserving (BP) high-order discontinuous Galerkin and finite volume schemes for various hyperbolic equations. A key ingredient in the framework is the decomposition of the cell averages of the numerical solution into a convex combination of the solution values at certain quadrature points, which helps to rewrite high-order schemes as convex combinations of formally first-order schemes. The classic convex decomposition originally proposed by Zhang and Shu has been widely used over the past decade. It was verified, only for the 1D quadratic and cubic polynomial spaces, that the classic decomposition is optimal in the sense of achieving the mildest BP CFL condition. Yet, it remained unclear whether the classic decomposition is optimal in multiple dimensions. In this paper, we find that the classic multidimensional decomposition based on the tensor product of Gauss--Lobatto and Gauss quadratures is generally not optimal, and we discover a novel alternative decomposition for the 2D and 3D polynomial spaces of total degree up to 2 and 3, respectively, on Cartesian meshes. Our new decomposition allows a larger BP time step-size than the classic one, and moreover, it is rigorously proved to be optimal to attain the mildest BP CFL condition, yet requires much fewer nodes. The discovery of such an optimal convex decomposition is highly nontrivial yet meaningful, as it may lead to an improvement of high-order BP schemes for a large class of hyperbolic or convection-dominated equations, at the cost of only a slight and local modification to the implementation code. Several numerical examples are provided to further validate the advantages of using our optimal decomposition over the classic one in terms of efficiency.

This PhD thesis contains several contributions to the field of statistical causal modeling. Statistical causal models are statistical models embedded with causal assumptions that allow for the inference and reasoning about the behavior of stochastic systems affected by external manipulation (interventions). This thesis contributes to the research areas concerning the estimation of causal effects, causal structure learning, and distributionally robust (out-of-distribution generalizing) prediction methods. We present novel and consistent linear and non-linear causal effects estimators in instrumental variable settings that employ data-dependent mean squared prediction error regularization. Our proposed estimators show, in certain settings, mean squared error improvements compared to both canonical and state-of-the-art estimators. We show that recent research on distributionally robust prediction methods has connections to well-studied estimators from econometrics. This connection leads us to prove that general K-class estimators possess distributional robustness properties. We, furthermore, propose a general framework for distributional robustness with respect to intervention-induced distributions. In this framework, we derive sufficient conditions for the identifiability of distributionally robust prediction methods and present impossibility results that show the necessity of several of these conditions. We present a new structure learning method applicable in additive noise models with directed trees as causal graphs. We prove consistency in a vanishing identifiability setup and provide a method for testing substructure hypotheses with asymptotic family-wise error control that remains valid post-selection. Finally, we present heuristic ideas for learning summary graphs of nonlinear time-series models.

Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.