InfoVis Is So Much More: A Comment on Gelman and Unwin and an Invitation to Consider the Opportunities

We propose new tools for visualizing large amounts of functional data in the form of smooth curves. The proposed tools include functional versions of the bagplot and boxplot, which make use of the first two robust principal component scores, Tukey’s data depth and highest density regions.

By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with existing methods for detecting outliers in functional data, and show that our methods are better able to identify outliers.

An R-package containing computer code and datasets is available in the online supplements.     

Robust spectral risk optimization when the subjective risk aversion is ambiguous: a moment-type approach

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker’s risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim 30(4):3198–3229, 2020), the new robust model allows one to elicit the decision maker’s risk preference through pairwise comparisons and use the elicited preference information to construct an ambiguity set of risk spectra. The robust spectral risk measure (RSRM) is based on the worst case risk spectrum from the set. To calculate RSRM and solve the associated optimal decision making problem, we use a technique from Acerbi and Simonetti (Portfolio optimization with spectral measures of risk. Working paper, 2002) to develop a new computational approach which is independent of order statistics and reformulate the robust spectral risk optimization problem as a single deterministic convex programming problem when the risk spectra in the ambiguity set are step-like. Moreover, we propose an approximation scheme when the risk spectra are not step-like and derive a bound for the model approximation error and its propagation to the optimal decision making problems. Some preliminary numerical test results are reported about the performance of the robust model and the computational scheme.

     

Influence measures and robust estimators of dependence in multivariate extremes

We develop a simple influence measure to assess whether Bayesian estimators in multivariate extreme value problems are sensitive to outliers. The proposed measure is easy to compute by importance sampling and successfully captures two effects on the functional: the “data effect” and the “parameter uncertainty effect”. We also propose a new Bayesian estimator which is easy to implement and is robust. The methods are tested and illustrated using simulated data and then applied to stock market data.     

Unsupervised Learning With Random Forest Predictors

A random forest (RF) predictor is an ensemble of individual tree predictors. As part of their construction, RF predictors naturally lead to a dissimilarity measure between the observations. One can also define an RF dissimilarity measure between unlabeled data: the idea is to construct an RF predictor that distinguishes the “observed” data from suitably generated synthetic data. The observed data are the original unlabeled data and the synthetic data are drawn from a reference distribution. Here we describe the properties of the RF dissimilarity and make recommendations on how to use it in practice.

An RF dissimilarity can be attractive because it handles mixed variable types well, is invariant to monotonic transformations of the input variables, and is robust to outlying observations. The RF dissimilarity easily deals with a large number of variables due to its intrinsic variable selection; for example, the Addcl 1 RF dissimilarity weighs the contribution of each variable according to how dependent it is on other variables.

We find that the RF dissimilarity is useful for detecting tumor sample clusters on the basis of tumor marker expressions. In this application, biologically meaningful clusters can often be described with simple thresholding rules.     

Robust nonparametric regression estimation

In this paper we define a robust conditional location functional without requiring any moment condition. We apply the nonparametric proposals considered by C. Stone (Ann. Statist. 5 (1977), 595–645) to this functional equation in order to obtain strongly consistent, robust nonparametric estimates of the regression function. We give some examples by using nearest neighbor weights or weights based on kernel methods under no assumptions whatsoever on the probability measure of the vector (X,Y). We also derive strong convergence rates and the asymptotic distribution of the proposed estimates.     

Neumann p-Laplacian problems with a reaction term on metric spaces

We use a variational approach to study existence and regularity of solutions for a Neumann p-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincaré inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.

     

Asymptotic Analysis for a Class of Infinite Systems of First-Order PDE: Nonlinear Parabolic PDE in the Singular Limit

Abstract

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.     

Robust risk management

Estimating the probabilities by which different events might occur is usually a delicate task, subject to many sources of inaccuracies. Moreover, these probabilities can change over time, leading to a very difficult evaluation of the risk induced by any particular decision. Given a set of probability measures and a set of nominal risk measures, we define in this paper the concept of robust risk measure as the worst possible of our risks when each of our probability measures is likely to occur. We study how some properties of this new object can be related with those of our nominal risk measures, such as convexity or coherence. We introduce a robust version of the Conditional Value-at-Risk (CVaR) and of entropy-based risk measures. We show how to compute and optimize the Robust CVaR using convex duality methods and illustrate its behavior using data from the New York Stock Exchange and from the NASDAQ between 2005 and 2010.     

Multivariate robust second-order stochastic dominance and resulting risk-averse optimization

ABSTRACT

By utilizing a min-biaffine scalarization function, we define the multivariate robust second-order stochastic dominance relationship to flexibly compare two random vectors. We discuss the basic properties of the multivariate robust second-order stochastic dominance and relate it to the nonpositiveness of a functional which is continuous and subdifferentiable everywhere. We study a stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and develop the necessary and sufficient conditions of optimality in the convex case. After specifying an ambiguity set based on moments information, we approximate the ambiguity set by a series of sets consisting of discrete distributions. Furthermore, we design a convex approximation to the proposed stochastic optimization problem with multivariate robust second-order stochastic dominance constraints and establish its qualitative stability under Kantorovich metric and pseudo metric, respectively. All these results lay a theoretical foundation for the modelling and solution of complex stochastic decision-making problems with multivariate robust second-order stochastic dominance constraints.     

The Dirichlet Problem for Semilinear Second-Order Degenerate Elliptic Equations and Applications to Stochastic Exit Time Control Problems

Abstract

We study nonlinear noncoercive elliptic problems with measure data, proving first that the global estimates already known when the problem is coercive are still true for noncoercive problems. We then prove new estimates, on sets far from the support of the singular part of the right-hand side, in the energy space associated to the operator, which entails additional regularity results on the solutions.     

An Algorithm for Robust Inference for the Cox Model with Frailties

Abstract

This article proposes a robust method of statistical inference for the Cox's proportional hazards model with frailties. We use the Metropolis—Hastings algorithm and the bootstrap method. We present a computationally efficient algorithm with a customized data structure to implement this method and demonstrate this technique with real data.     

A Delayed Black and Scholes Formula

Abstract

In this article we develop an explicit formula for pricing European options when the underlying stock price follows nonlinear stochastic functional differential equations with fixed and variable delays. We believe that the proposed models are sufficiently flexible to fit real market data, and yet simple enough to allow for a closed-form representation of the option price. Furthermore, the models maintain the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent local martingale measure.     

Rényi 100, Quantitative and qualitative (in)dependence

We discuss recent developments in the following important areas of Alfréd Rényi’s research interest: axiomatization of quantitative dependence measures, qualitative independence in combinatorics, conditional qualitative independence in statistics/data science and in measure theory/probability theory, and finally, prime gaps that are responsible for Rényi’s early career reputation. Most authors of this paper are main contributors to the new developments.

     

Discrete Dichotomies and Asymptotic Behavior for Abstract Retarded Functional Difference Equations in Phase Space

We study the problem of asymptotic behavior between weighted bounded solutions of a system of homogeneous linear functional difference equations and its perturbation under non-classical dichotomic properties and also we obtain some results about approximation. We apply our results to Volterra difference systems with infinite delay.     

Robust estimation in single-index models when the errors have a unimodal density with unknown nuisance parameter

This paper develops a robust profile estimation method for the parametric and nonparametric components of a single-index model when the errors have a strongly unimodal density with unknown nuisance parameter. We derive consistency results for the link function estimators as well as consistency and asymptotic distribution results for the single-index parameter estimators. Under a log-Gamma model, the sensitivity to anomalous observations is studied using the empirical influence curve. We also discuss a robust K-fold cross-validation procedure to select the smoothing parameters. A numerical study carried on with errors following a log-Gamma model and for contaminated schemes shows the good robustness properties of the proposed estimators and the advantages of considering a robust approach instead of the classical one. A real data set illustrates the use of our proposal.

     

Robust dependence measure for detecting associations in large data set

In this paper, we proposed a new statistical dependency measure for two random vectors based on copula, called copula dependency coefficient (CDC). The CDC is proved to be robust to outliers and easy to be implemented. Especially, it is powerful and applicable to high-dimensional problems. All these properties make CDC practically important in related applications. Both experimental and application results show that CDC is a good robust dependence measure for association detecting.     

Model-free conditional independence feature screening for ultrahigh dimensional data

Feature screening plays an important role in ultrahigh dimensional data analysis. This paper is concerned with conditional feature screening when one is interested in detecting the association between the response and ultrahigh dimensional predictors (e.g., genetic makers) given a low-dimensional exposure variable (such as clinical variables or environmental variables). To this end, we first propose a new index to measure conditional independence, and further develop a conditional screening procedure based on the newly proposed index. We systematically study the theoretical property of the proposed procedure and establish the sure screening and ranking consistency properties under some very mild conditions. The newly proposed screening procedure enjoys some appealing properties. (a) It is model-free in that its implementation does not require a specification on the model structure; (b) it is robust to heavy-tailed distributions or outliers in both directions of response and predictors; and (c) it can deal with both feature screening and the conditional screening in a unified way. We study the finite sample performance of the proposed procedure by Monte Carlo simulations and further illustrate the proposed method through two real data examples.     

Stability conditions to Bresse systems with indefinite memory dissipation

ABSTRACT

We study the asymptotic behavior of Bresse system with non-dissipative kernel memory acting only in the equation of longitudinal displacement. We show that the exponential stability depends on conditions regarding the decay rate of the kernel and a new relationship between the coefficients of the system. Moreover, this new condition on the constants of the system is used to prove strong stability and exponential stability for the homogeneous case (frictional dissipation in the longitudinal equation).     

Transmuted lower record type inverse rayleigh distribution: estimation,characterizations and applications

This study introduces a new lifetime distribution called the transmuted lower record type inverse Rayleigh which extends the inverse Rayleigh distribution and has the potential to model the recovery times of Covid-19 patients.The new distribution is obtained using the distributions of the first two lower record statistics of the inverse Rayleigh distribution. We discuss some statistical inferences and mathematical properties of the suggested distribution. We examine some characteristics of the proposed distribution such as density shape, hazard function,moments, moment generating function, incomplete moments,Rényi entropy, order statistics, stochastic ordering. We consider five estimation methods such as maximum likelihood, least squares, weighted least squares, Anderson-Darling, Cramér-von Mises for the point estimation of the proposed distribution. Then, a comprehensive Monte Carlo simulation study is carried out to assess the risk behavior of the examined estimators. We provide two real data applications to illustrate the fitting ability of the proposed model, and compare its fit with competitor ones. Unlike many previously proposed distributions, the introduced distribution in this paper has modeled the recovery times of Covid-19 patients.

     

Functional Cantor equation

We consider the class of entire functions of exponential type in relation to the scattering theory for the Schrödinger equation with a finite potential that is a finite Borel measure. These functions have a special self-similarity and satisfy q-difference functional equations. We study their asymptotic behavior and the distribution of zeros.