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The best subset selection (or "best subsets") estimator is a classic tool for sparse regression, and developments in mathematical optimization over the past decade have made it more computationally tractable than ever. Notwithstanding its desirable statistical properties, the best subsets estimator is susceptible to outliers and can break down in the presence of a single contaminated data point. To address this issue, a robust adaption of best subsets is proposed that is highly resistant to contamination in both the response and the predictors. The adapted estimator generalizes the notion of subset selection to both predictors and observations, thereby achieving robustness in addition to sparsity. This procedure, referred to as "robust subset selection" (or "robust subsets"), is defined by a combinatorial optimization problem for which modern discrete optimization methods are applied. The robustness of the estimator in terms of the finite-sample breakdown point of its objective value is formally established. In support of this result, experiments on synthetic and real data are reported that demonstrate the superiority of robust subsets over best subsets in the presence of contamination. Importantly, robust subsets fares competitively across several metrics compared with popular robust adaptions of continuous shrinkage estimators.

Consider two or more forecasters, each making a sequence of predictions for different events over time. We ask a relatively basic question: how might we compare these forecasters, either online or post-hoc, while avoiding unverifiable assumptions on how the forecasts or outcomes were generated? This work presents a novel and rigorous answer to this question. We design a sequential inference procedure for estimating the time-varying difference in forecast quality as measured by any scoring rule. The resulting confidence intervals are nonasymptotically valid and can be continuously monitored to yield statistically valid comparisons at arbitrary data-dependent stopping times ("anytime-valid"); this is enabled by adapting variance-adaptive supermartingales, confidence sequences, and e-processes to our setting. Motivated by Shafer and Vovk's game-theoretic probability, our coverage guarantees are also distribution-free, in the sense that they make no distributional assumptions on the forecasts or outcomes. In contrast to a recent work by Henzi and Ziegel, our tools can sequentially test a weak null hypothesis about whether one forecaster outperforms another on average over time. We demonstrate their effectiveness by comparing probability forecasts on Major League Baseball (MLB) games and statistical postprocessing methods for ensemble weather forecasts.

Large-scale data analysis is growing at an exponential rate as data proliferates in our societies. This abundance of data has the advantage of allowing the decision-maker to implement complex models in scenarios that were prohibitive before. At the same time, such an amount of data requires a distributed thinking approach. In fact, Deep Learning models require plenty of resources, and distributed training is needed. This paper presents a Multicriteria approach for distributed learning. Our approach uses the Weighted Goal Programming approach in its Chebyshev formulation to build an ensemble of decision rules that optimize aprioristically defined performance metrics. Such a formulation is beneficial because it is both model and metric agnostic and provides an interpretable output for the decision-maker. We test our approach by showing a practical application in electricity demand forecasting. Our results suggest that when we allow for dataset split overlapping, the performances of our methodology are consistently above the baseline model trained on the whole dataset.

A functional dynamic factor model for time-dependent functional data is proposed. We decompose a functional time series into a predictive low-dimensional common component consisting of a finite number of factors and an infinite-dimensional idiosyncratic component that has no predictive power. The conditions under which all model parameters, including the number of factors, become identifiable are discussed. Our identification results lead to a simple-to-use two-stage estimation procedure based on functional principal components. As part of our estimation procedure, we solve the separation problem between the common and idiosyncratic functional components. In particular, we obtain a consistent information criterion that provides joint estimates of the number of factors and dynamic lags of the common component. Finally, we illustrate the applicability of our method in a simulation study and to the problem of modeling and predicting yield curves. In an out-of-sample experiment, we demonstrate that our model performs well compared to the widely used term structure Nelson-Siegel model for yield curves.

Power is an important aspect of experimental design, because it allows researchers to understand the chance of detecting causal effects if they exist. It is common to specify a desired level of power, and then compute the sample size necessary to obtain that level of power; thus, power calculations help determine how experiments are conducted in practice. Power and sample size calculations are readily available for completely randomized experiments; however, there can be many benefits to using other experimental designs. For example, in recent years it has been established that rerandomized designs, where subjects are randomized until a prespecified level of covariate balance is obtained, increase the precision of causal effect estimators. This work establishes the statistical power of rerandomized treatment-control experiments, thereby allowing for sample size calculators. Our theoretical results also clarify how power and sample size are affected by treatment effect heterogeneity, a quantity that is often ignored in power analyses. Via simulation, we confirm our theoretical results and find that rerandomization can lead to substantial sample size reductions; e.g., in many realistic scenarios, rerandomization can lead to a 25% or even 50% reduction in sample size for a fixed level of power, compared to complete randomization. Power and sample size calculators based on our results are in the R package rerandPower on CRAN.

The risk premium of a policy is the sum of the pure premium and the risk loading. In the classification ratemaking process, generalized linear models are usually used to calculate pure premiums, and various premium principles are applied to derive the risk loadings. No matter which premium principle is used, some risk loading parameters should be given in advance subjectively. To overcome this subjective problem and calculate the risk premium more reasonably and objectively, we propose a top-down method to calculate these risk loading parameters. First, we implement the bootstrap method to calculate the total risk premium of the portfolio. Then, under the constraint that the portfolio's total risk premium should equal the sum of the risk premiums of each policy, the risk loading parameters are determined. During this process, besides using generalized linear models, three kinds of quantile regression models are also applied, namely, traditional quantile regression model, fully parametric quantile regression model, and quantile regression model with coefficient functions. The empirical result shows that the risk premiums calculated by the method proposed in this study can reasonably differentiate the heterogeneity of different risk classes.

Estimating the effects of interventions on patient outcome is one of the key aspects of personalized medicine. Their inference is often challenged by the fact that the training data comprises only the outcome for the administered treatment, and not for alternative treatments (the so-called counterfactual outcomes). Several methods were suggested for this scenario based on observational data, i.e.~data where the intervention was not applied randomly, for both continuous and binary outcome variables. However, patient outcome is often recorded in terms of time-to-event data, comprising right-censored event times if an event does not occur within the observation period. Albeit their enormous importance, time-to-event data is rarely used for treatment optimization. We suggest an approach named BITES (Balanced Individual Treatment Effect for Survival data), which combines a treatment-specific semi-parametric Cox loss with a treatment-balanced deep neural network; i.e.~we regularize differences between treated and non-treated patients using Integral Probability Metrics (IPM). We show in simulation studies that this approach outperforms the state of the art. Further, we demonstrate in an application to a cohort of breast cancer patients that hormone treatment can be optimized based on six routine parameters. We successfully validated this finding in an independent cohort. BITES is provided as an easy-to-use python implementation.

In this paper we discuss a reduced basis method for linear evolution PDEs, which is based on the application of the Laplace transform. The main advantage of this approach consists in the fact that, differently from time stepping methods, like Runge-Kutta integrators, the Laplace transform allows to compute the solution directly at a given instant, which can be done by approximating the contour integral associated to the inverse Laplace transform by a suitable quadrature formula. In terms of the reduced basis methodology, this determines a significant improvement in the reduction phase - like the one based on the classical proper orthogonal decomposition (POD) - since the number of vectors to which the decomposition applies is drastically reduced as it does not contain all intermediate solutions generated along an integration grid by a time stepping method. We show the effectiveness of the method by some illustrative parabolic PDEs arising from finance and also provide some evidence that the method we propose, when applied to a simple advection equation, does not suffer the problem of slow decay of singular values which instead affects methods based on time integration of the Cauchy problem arising from space discretization.

Although Deep Neural Networks (DNNs) have shown incredible performance in perceptive and control tasks, several trustworthy issues are still open. One of the most discussed topics is the existence of adversarial perturbations, which has opened an interesting research line on provable techniques capable of quantifying the robustness of a given input. In this regard, the Euclidean distance of the input from the classification boundary denotes a well-proved robustness assessment as the minimal affordable adversarial perturbation. Unfortunately, computing such a distance is highly complex due the non-convex nature of NNs. Despite several methods have been proposed to address this issue, to the best of our knowledge, no provable results have been presented to estimate and bound the error committed. This paper addresses this issue by proposing two lightweight strategies to find the minimal adversarial perturbation. Differently from the state-of-the-art, the proposed approach allows formulating an error estimation theory of the approximate distance with respect to the theoretical one. Finally, a substantial set of experiments is reported to evaluate the performance of the algorithms and support the theoretical findings. The obtained results show that the proposed strategies approximate the theoretical distance for samples close to the classification boundary, leading to provable robustness guarantees against any adversarial attacks.

To drive purchase in online advertising, it is of the advertiser's great interest to optimize the sequential advertising strategy whose performance and interpretability are both important. The lack of interpretability in existing deep reinforcement learning methods makes it not easy to understand, diagnose and further optimize the strategy. In this paper, we propose our Deep Intents Sequential Advertising (DISA) method to address these issues. The key part of interpretability is to understand a consumer's purchase intent which is, however, unobservable (called hidden states). In this paper, we model this intention as a latent variable and formulate the problem as a Partially Observable Markov Decision Process (POMDP) where the underlying intents are inferred based on the observable behaviors. Large-scale industrial offline and online experiments demonstrate our method's superior performance over several baselines. The inferred hidden states are analyzed, and the results prove the rationality of our inference.