Expectiles were introduced to statistics around 40 years ago, but have recently gained renewed interest due to their relevance in financial risk management. In particular, the 2007–2009 global financial crisis highlighted the need for more robust risk evaluation tools, leading to the adoption of inference methods for expectiles. While first-order asymptotic inference results for expectiles are well established, higher-order asymptotic results remain underdeveloped. This study aims to fill that gap by deriving higher-order asymptotic results for expectiles, ultimately improving the accuracy of confidence intervals. The paper outlines the derivation of the Edgeworth expansion for both the standardized and studentized versions of the kernel-based estimator of the expectile, using large deviation results on U-statistics. The expansion is then inverted to construct more precise confidence intervals for the expectile. These theoretical results were applied to moderate sample sizes ranging from 20 to 200. To demonstrate the advantages of this methodology, an example from risk management is presented. The enhanced confidence intervals consistently outperformed those based on the first-order normal approximation. The methodology introduced in this paper can also be extended to other contexts.

This paper focuses on semiparametric regression based on minimizing the localized Bregman divergence. A local parametric model derived from the framework of the generalized linear model with multiple covariates and a linear predictor is utilized. The parameter vector included in the model is estimated under localization. The asymptotic behavior of both the locally estimated parameter vector and the induced regression estimator is investigated. Theoretical comparisons of estimators by using the divergence risk measure are also addressed. Further generalization, including a multivariate polynomial predictor, is explored, where Faa di Bruno's theorem concerning the derivative of a composition of multivariate functions is efficiently utilized. Simulations and application to a real dataset demonstrate that the proposed regression estimator works efficiently.

The benefits of the wavelet approach for density estimation are well established in the literature, especially when the density to estimate is irregular or heterogeneous in smoothness. However, wavelet density estimates are typically not bona fide densities. In Aya-Moreno et al (2018), a `shape-preserving' wavelet density estimator was introduced, including as main step the estimation of the square-root of the density. A natural concept involving square-root of densities is the Hellinger distance - or equivalently, the Bhattacharyya affinity coefficient. In this paper, we deliver a fully data-driven version of the above 'shape-preserving' wavelet density estimator, where all user-defined parameters, such as resolution level or thresholding specifications, are selected by optimising an original leave-one-out version of the Hellinger-Bhattacharyya criterion. The theoretical optimality of the proposed procedure is established, while simulations show the strong practical performance of the estimator. Within that framework, we also propose a novel but natural 'jackknife thresholding' scheme, which proves superior to other, more classical thresholding options.

In recent years, distributionally robust optimization (DRO) has received a lot of interest due to its ability to reduce the worst-case risk when there is a perturbation to the data-generating distribution. A good statistical model is expected to perform well on both the normal and the perturbed data. On the other hand, variable selection and regularization is a research area that aims to identify the important features and remove the redundant ones. It helps to improve the prediction accuracy as well as the interpretability of the model. In this paper, we propose an optimization model that is a regularized version of the canonical distributionally robust optimization (DRO) problem where the ambiguity set is described by a general class of divergence measures that admit a suitable conic structure. The divergence measures we examined include several popular divergence measures used in the literature such as the Kullback–Leibler divergence, total variation, and the Chi-divergence. By exploiting the conic representability of the divergence measure, we show that the regularized DRO problem can be equivalently reformulated as a nonlinear conic programming problem. In the case where the regularization is convex and semi-definite programming representable, the reformulation can be further simplified as a tractable linear conic program and hence can be efficiently solved via existing software. More generally, if the regularization can be written as a difference of convex functions, we demonstrate that a solution for the regularized DRO problem can be found by solving a sequence of conic linear programming problems. Finally, we apply the proposed regularized DRO model to both simulated and real financial data and demonstrate its superior performance in comparison with some non-robust models.

This paper is concerned with the derivation of the Edgeworth expansion for the standardized and the studentized version of the kernel-based estimator of the expectile. Inverting the expansion allows us to construct accurate confidence intervals for the expectile. The theoretical asymptotic results are applied for moderate sample sizes. The methodology is illustrated with an application in risk management in finance for the estimation and accurate confidence interval construction for the coherent risk measure expectile-VaR.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as \emph{properness}, which asserts that Bayes' rule is optimal. Recent works have sought to \emph{learn losses} and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps $\mathbb{R}$ to $[0,1]$ to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between $\mathbb{R}^{C-1}$ and the projected probability simplex $\tilde{\Delta}^{C-1}$ by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns \emph{proper canonical losses} and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a $t$-test at 99% significance on all datasets with greater than 10 classes.

Wave load cycles, wet-deck slamming events, accelerations and motion comfort are important considerations for high-speed catamarans operating in moderate to large waves. Although developing a hull monitoring system according to classification guidelines for such vessels is broadly acceptable, the data processing requirements for outputs such as rainflow counting, filtering, probability distribution, fatigue damage estimation and warning due to slamming can be as sophisticated to implement as the system components themselves. Advanced analytics such as machine learning and deep learning data pipelines will also create more complexities for such systems, if included. This paper provides an overview of data analytics methods and cloud computing resources for remotely monitoring motions and structural responses of a 111 m high-speed catamaran. To satisfy the data processing requirements, MATLAB Reference Architectures on Amazon Web Services (AWS) were used. Such combination enabled fast parallel computing and advanced feature engineering in a time-efficient manner. A MATLAB Production Server on AWS has been set up for near real-time analytics and execution of functions developed according to the class guidelines. A case study using Long Short‑Term Memory (LSTM) networks for ship speed and Motion Sickness Incidence (MSI) is provided and discussed. Such data architecture provides a flexible and scalable solution, leading to deeper insights through big data processing and machine learning, which supports hull monitoring functions as a service.

Optimization under uncertainty and risk is ubiquitous in business, engineering, and finance. Typically, we use observed or simulated data in our decision models, which aim to control risk, and result in composite risk functionals. The paper addresses the stability of the decision problems when the composite risk functionals are subjected to measure perturbations at multiple levels of potentially different nature. We analyze data-driven formulations with empirical or smoothing estimators such as kernels or wavelets applied to some or to all functions of the compositions and establish laws of large numbers and consistency of the optimal values and solutions. This is the first study to propose and analyze smoothing in data-driven composite optimization problems. It is shown that kernel-based and wavelet estimation provide less biased estimation of the risk compared with the empirical plug-in estimators under some assumptions.

Index tracking is a popular form of asset management. Typically, a quadratic function is used to define the tracking error of a portfolio and the look back approach is applied to solve the index tracking problem. We argue that a forward looking approach is more suitable, whereby the tracking error is expressed as an expectation of a function of the difference between the returns of the index and of the portfolio. We also assume that there is model uncertainty in the distribution of the assets, hence a robust version of the optimization problem needs to be adopted. We use Bregman divergence in describing the deviation between the nominal and actual (true) distribution of the components of the index. In this scenario, we derive the optimal robust index tracking portfolio in a semi-analytical form as a solution of a system of nonlinear equations. Several numerical results are presented that allow us to compare the performance of this robust portfolio with the optimal non-robust portfolio. We show that, especially during market downturns, the robust portfolio can be very advantageous.

Climate change is typically modeled using sophisticated mathematical models (climate models) of physical processes that range in temporal and spatial scales. Multi-model ensemble means of climate models show better correlation with the observations than any of the models separately. Currently, an open research question is how climate models can be combined to create an ensemble mean in an optimal way. We present a novel stochastic approach based on Markov chains to estimate model weights in order to obtain ensemble means. The method was compared to existing alternatives by measuring its performance on training and validation data, as well as model-as-truth experiments. The Markov chain method showed improved performance over those methods when measured by the root mean squared error in validation and comparable performance in model-as-truth experiments. The results of this comparative analysis should serve to motivate further studies in applications of Markov chain and other nonlinear methods that address the issues of finding optimal model weight for constructing ensemble means.

An onboard monitoring system can measure features such as stress cycles counts and provide warnings due to slamming. Considering current technology trends there is the opportunity of incorporating machine learning methods into monitoring systems. A hull monitoring system has been developed and installed on a 111 m wave piercing catamaran (Hull 091) to remotely monitor the ship kinematics and hull structural responses. Parallel to that, an existing dataset of a similar vessel (Hull 061) was analysed using unsupervised and supervised learning models; these were found to be beneficial for the classification of bow entry events according to key kinematic parameters. A comparison of different algorithms including linear support vector machines, naïve Bayes and decision tree for the bow entry classification were conducted. In addition, using empirical probability distributions, the likelihood of wet-deck slamming was estimated given a vertical bow acceleration threshold of 1 in head seas, clustering the feature space with the approximate probabilities of 0.001, 0.030 and 0.25.

Multiple imputation and maximum likelihood estimation (via the expectation‐maximization algorithm) are two well‐known methods readily used for analyzing data with missing values. While these two methods are often considered as being distinct from one another, multiple imputation (when using improper imputation) is actually equivalent to a stochastic expectation‐maximization approximation to the likelihood. In this article, we exploit this key result to show that familiar likelihood‐based approaches to model selection, such as Akaike's information criterion (AIC) and the Bayesian information criterion (BIC), can be used to choose the imputation model that best fits the observed data. Poor choice of imputation model is known to bias inference, and while sensitivity analysis has often been used to explore the implications of different imputation models, we show that the data can be used to choose an appropriate imputation model via conventional model selection tools. We show that BIC can be consistent for selecting the correct imputation model in the presence of missing data. We verify these results empirically through simulation studies, and demonstrate their practicality on two classical missing data examples. An interesting result we saw in simulations was that not only can parameter estimates be biased by misspecifying the imputation model, but also by overfitting the imputation model. This emphasizes the importance of using model selection not just to choose the appropriate type of imputation model, but also to decide on the appropriate level of imputation model complexity.

This paper is concerned with a unified approach to estimating regression methods based on a certain divergence and its localisation. Some past papers have demonstrated theoretically and numerically that infusing a little localisation in the likelihood-based methods for regression and for density estimation can actually improve the resulting estimators with respect to suitably defined global risk measures. Thus a variety of local likelihood methods have been suggested. We demonstrate that similar effect can also be observed in the general framework discussed in this paper and with respect to robust estimation procedures. Localised versions of robust regression estimation procedures perform better with respect to global risk measures based on minimisation of Bregman divergence measures. An intricate relationship between regression model’s inadequacy and its robustness can be better analysed by using the local approach developed in this paper. We support our claims with a short simulation study.

Wave load cycles, wet-deck slamming events, accelerations and motion comfort are important considerations for high- speed catamarans operating in moderate to large waves. This paper provides an overview of data analytics methods and cloud computing resources for remotely monitoring motions and structural responses of a 111 m high-speed catamaran. To satisfy the data processing requirements, MATLAB Reference Architectures on Amazon Web Services (AWS) were used. Such combination enabled fast parallel computing and advanced feature engineering in a time-efficient manner. A MATLAB Production Server on AWS has been set up for near real-time analytics and execution of functions developed according to the class guidelines. A case study using Long Short-Term Memory (LSTM) networks for ship speed and Motion Sickness Incidence (MSI) is provided and discussed. Such data architecture provides a flexible and scalable solution, leading to deeper insights through big data processing and machine learning, which supports hull monitoring functions as a service.

In many applications involving regression analysis, explanatory variables (or covariates) may be imprecisely measured or may contain missing values. Although there exists a vast literature on measurement error modeling to account for errors-in-variables, and on missing data methodology to handle missingness, very few methods have been developed to simultaneously address both. In this paper, we consider likelihood-based multiple imputation to handle missing data, and combine this with two well-known functional measurement error methods: simulation-extrapolation and corrected score. This unified approach has several appealing characteristics: the model fitting procedure is easy to understand and off-the-shelf software can be incorporated into the modeling framework; no calibration data or a validation subset is required in the model fitting procedure; and the missing data component of the proposed approach is likelihood-based which allows standard likelihood machinery. We demonstrate our methods on simulated datasets and apply them to daily ozone pollution measurements in Los Angeles where observed covariates consist of missing data and imprecise measurements. We conclude that the proposed methods substantially reduce bias and mean squared errors in regression coefficients, in comparison to methods that ignore either measurement error or missingness in covariates.

An onboard monitoring system can measure features such as stress cycles counts and provide warnings due to slamming. Considering current technology trends there is the opportunity of incorporating machine learning methods into monitoring systems. A hull monitoring system has been developed and installed on a 111 m wave piercing catamaran (Hull 091) to remotely monitor the ship kinematics and hull structural responses. Parallel to that, an existing dataset of a geometrically similar vessel (Hull 061) was analysed using unsupervised and supervised learning models; these were found to be beneficial for the classification of bow entry events according to the kinematic parameters. A comparison of different algorithms including linear support vector machines, naïve Bayes and decision tree for the bow entry classification were conducted. In addition, using empirical probability distributions, the likelihood of wet-deck slamming was estimated given vertical bow acceleration thresholds.

We quantify model risk of a financial portfolio whereby a multi-period mean-standard-deviation criterion is used as a selection criterion. In this work, model risk is defined as the loss due to uncertainty of the underlying distribution of the returns of the assets in the portfolio. The uncertainty is measured by the Kullback-Leibler divergence, i.e., the relative entropy. In the worst case scenario, the optimal robust strategy can be obtained in a semi-analytical form as a solution of a system of nonlinear equations. Several numerical results are presented which allow us to compare the performance of this robust strategy with the optimal non-robust strategy. For illustration, we also quantify the model risk associated with an empirical dataset.

It has already been shown theoretically and numerically that infusing a little localization in the likelihood-based methods for regression and for density estimation can actually improve the resulting estimators with respect to suitably defined global risk measures. Thus various local likelihood methods have been suggested. In this paper, we demonstrate that a similar effect can also be observed with respect to robust estimation procedures. Localized versions of robust density estimation procedures perform better with respect to global risk measures based on minimization of Bregman divergence measures.

Wavelet estimators for a probability density f enjoy many good properties, however they are not "shape-preserving" in the sense that the final estimate may not be non-negative or integrate to unity. A solution to negativity issues may be to estimate first the square-root of f and then square this estimate up. This paper proposes and investigates such an estimation scheme, generalising to higher dimensions some previous constructions which are valid only in one dimension. The estimation is mainly based on nearest-neighbour-balls. The theoretical properties of the proposed estimator are obtained, and it is shown to reach the optimal rate of convergence uniformly over large classes of densities under mild conditions. Simulations show that the new estimator performs as well in general as the classical wavelet estimator, while automatically producing estimates which are bona fide densities.