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In this manuscript, we consider a finite multivariate nonparametric mixture model where the dependence between the marginal densities is modeled using the copula device. Pseudo EM stochastic algorithms were recently proposed to estimate all of the components of this model under a location-scale constraint on the marginals. Here, we introduce a deterministic algorithm that seeks to maximize a smoothed semiparametric likelihood. No location-scale assumption is made about the marginals. The algorithm is monotonic in one special case, and, in another, leads to ``approximate monotonicity'' -- whereby the difference between successive values of the objective function becomes non-negative up to an additive term that becomes negligible after a sufficiently large number of iterations. The behavior of this algorithm is illustrated on several simulated datasets. The results suggest that, under suitable conditions, the proposed algorithm may indeed be monotonic in general. A discussion of the results and some possible future research directions round out our presentation.

Quantile regression is increasingly encountered in modern big data applications due to its robustness and flexibility. We consider the scenario of learning the conditional quantiles of a specific target population when the available data may go beyond the target and be supplemented from other sources that possibly share similarities with the target. A crucial question is how to properly distinguish and utilize useful information from other sources to improve the quantile estimation and inference at the target. We develop transfer learning methods for high-dimensional quantile regression by detecting informative sources whose models are similar to the target and utilizing them to improve the target model. We show that under reasonable conditions, the detection of the informative sources based on sample splitting is consistent. Compared to the naive estimator with only the target data, the transfer learning estimator achieves a much lower error rate as a function of the sample sizes, the signal-to-noise ratios, and the similarity measures among the target and the source models. Extensive simulation studies demonstrate the superiority of our proposed approach. We apply our methods to tackle the problem of detecting hard-landing risk for flight safety and show the benefits and insights gained from transfer learning of three different types of airplanes: Boeing 737, Airbus A320, and Airbus A380.

Risk management in many environmental settings requires an understanding of the mechanisms that drive extreme events. Useful metrics for quantifying such risk are extreme quantiles of response variables conditioned on predictor variables that describe, e.g., climate, biosphere and environmental states. Typically these quantiles lie outside the range of observable data and so, for estimation, require specification of parametric extreme value models within a regression framework. Classical approaches in this context utilise linear or additive relationships between predictor and response variables and suffer in either their predictive capabilities or computational efficiency; moreover, their simplicity is unlikely to capture the truly complex structures that lead to the creation of extreme wildfires. In this paper, we propose a new methodological framework for performing extreme quantile regression using artificial neutral networks, which are able to capture complex non-linear relationships and scale well to high-dimensional data. The ``black box" nature of neural networks means that they lack the desirable trait of interpretability often favoured by practitioners; thus, we unify linear, and additive, regression methodology with deep learning to create partially-interpretable neural networks that can be used for statistical inference but retain high prediction accuracy. To complement this methodology, we further propose a novel point process model for extreme values which overcomes the finite lower-endpoint problem associated with the generalised extreme value class of distributions. Efficacy of our unified framework is illustrated on U.S. wildfire data with a high-dimensional predictor set and we illustrate vast improvements in predictive performance over linear and spline-based regression techniques.

An inference procedure is proposed to provide consistent estimators of parameters in a modal regression model with a covariate prone to measurement error. A score-based diagnostic tool exploiting parametric bootstrap is developed to assess adequacy of parametric assumptions imposed on the regression model. The proposed estimation method and diagnostic tool are applied to synthetic data generated from simulation experiments and data from real-world applications to demonstrate their implementation and performance. These empirical examples illustrate the importance of adequately accounting for measurement error in the error-prone covariate when inferring the association between a response and covariates based on a modal regression model that is especially suitable for skewed and heavy-tailed response data.

Expected Shortfall (ES), also known as superquantile or Conditional Value-at-Risk, has been recognized as an important measure in risk analysis and stochastic optimization, and is also finding applications beyond these areas. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a recently proposed joint regression framework that simultaneously models the quantile and the ES of a response variable given a set of covariates, for which the state-of-the-art approach is based on minimizing a joint loss function that is non-differentiable and non-convex. This inevitably raises numerical challenges and limits its applicability for analyzing large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity with respect to nuisance parameters, we propose a statistically robust (to highly skewed and heavy-tailed data) and computationally efficient two-step procedure for fitting joint quantile and ES regression models. With increasing covariate dimensions, we establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors, which lay the foundation for statistical inference. Finally, we demonstrate through numerical experiments and two data applications that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression.

Imputing missing potential outcomes using an estimated regression function is a natural idea for estimating causal effects. In the literature, estimators that combine imputation and regression adjustments are believed to be comparable to augmented inverse probability weighting. Accordingly, people for a long time conjectured that such estimators, while avoiding directly constructing the weights, are also doubly robust (Imbens, 2004; Stuart, 2010). Generalizing an earlier result of the authors (Lin et al., 2021), this paper formalizes this conjecture, showing that a large class of regression-adjusted imputation methods are indeed doubly robust for estimating the average treatment effect. In addition, they are provably semiparametrically efficient as long as both the density and regression models are correctly specified. Notable examples of imputation methods covered by our theory include kernel matching, (weighted) nearest neighbor matching, local linear matching, and (honest) random forests.

We propose a monitoring strategy for efficient and robust estimation of disease prevalence and case numbers within closed and enumerated populations such as schools, workplaces, or retirement communities. The proposed design relies largely on voluntary testing, notoriously biased (e.g., in the case of COVID-19) due to non-representative sampling. The approach yields unbiased and comparatively precise estimates with no assumptions about factors underlying selection of individuals for voluntary testing, building on the strength of what can be a small random sampling component. This component unlocks a previously proposed "anchor stream" estimator, a well-calibrated alternative to classical capture-recapture (CRC) estimators based on two data streams. We show here that this estimator is equivalent to a direct standardization based on "capture", i.e., selection (or not) by the voluntary testing program, made possible by means of a key parameter identified by design. This equivalency simultaneously allows for novel two-stream CRC-like estimation of general means (e.g., of continuous variables such as antibody levels or biomarkers). For inference, we propose adaptations of a Bayesian credible interval when estimating case counts and bootstrapping when estimating means of continuous variables. We use simulations to demonstrate significant precision benefits relative to random sampling alone.

Denoising diffusion models have recently marked a milestone in high-quality image generation. One may thus wonder if they are suitable for neural image compression. This paper outlines an end-to-end optimized image compression framework based on a conditional diffusion model, drawing on the transform-coding paradigm. Besides the latent variables inherent to the diffusion process, this paper introduces an additional discrete ``content'' latent variable to condition the denoising process. This variable is equipped with a hierarchical prior for entropy coding. The remaining ``texture'' latent variables characterizing the diffusion process are synthesized (either stochastically or deterministically) at decoding time. We furthermore show that the performance can be tuned toward perceptual metrics of interest. Our extensive experiments involving five datasets and sixteen image quality assessment metrics show that our approach not only compares favorably in rate-perceptual quality but also shows close distortion performance with state-of-the-art models.

This paper proposes a novel multivariate definition of statistical dependence using a functional methodology inspired by Alfred R\'enyi. We define a new symmetric and self-adjoint cross density kernel through a recursive bidirectional statistical mapping between conditional densities of continuous random processes, which estimates their statistical dependence. Therefore, the kernel eigenspectrum is proposed as a new multivariate statistical dependence measure, and the formulation requires fewer assumptions about the data generation model than current methods. The measure can also be estimated from realizations. The proposed functional maximum correlation algorithm (FMCA) is applied to a learning architecture with two multivariate neural networks. The FMCA optimal solution is an equilibrium point that estimates the eigenspectrum of the cross density kernel. Preliminary results with synthetic data and medium size image datasets corroborate the theory. Four different strategies of applying the cross density kernel are thoroughly discussed and implemented to show the versatility and stability of the methodology, and it transcends supervised learning. When two random processes are high-dimensional real-world images and white uniform noise, respectively, the algorithm learns a factorial code i.e., the occurrence of a code guarantees that a certain input in the training set was present, which is quite important for feature learning.

Graphical models have long been studied in statistics as a tool for inferring conditional independence relationships among a large set of random variables. The most existing works in graphical modeling focus on the cases that the data are Gaussian or mixed and the variables are linearly dependent. In this paper, we propose a double regression method for learning graphical models under the high-dimensional nonlinear and non-Gaussian setting, and prove that the proposed method is consistent under mild conditions. The proposed method works by performing a series of nonparametric conditional independence tests. The conditioning set of each test is reduced via a double regression procedure where a model-free sure independence screening procedure or a sparse deep neural network can be employed. The numerical results indicate that the proposed method works well for high-dimensional nonlinear and non-Gaussian data.