Bernoulli free boundary problems under uncertainty: the convex case

The present article is concerned with solving Bernoulli's exterior free boundary problem in case of an interior boundary which is random. We provide a new regularity result on the map that sends a parametrization of the inner boundary to a parametrization of the outer boundary. Moreover, by assuming that the interior boundary is convex, also the exterior boundary is convex, which enables to identify the boundaries with support functions and to determine their expectations. We in particular construct a confidence region for the outer boundary based on Aumann's expectation and provide a numerical method to compute it

Tree-based Regression for Interval-valued Data

Regression methods for interval-valued data have been increasingly studied in recent years. As most of the existing works focus on linear models, it is important to note that many problems in practice are nonlinear in nature and therefore development of nonlinear regression tools for intervalvalued data is crucial. In this project, we propose a tree-based regression method for interval-valued data, which is well applicable to both linear and nonlinear problems. Unlike linear regression models that usually require additional constraints to ensure positivity of the predicted interval length, the proposed method estimates the regression function in a nonparametric way, so the predicted length is naturally positive without any constraints. A simulation study is conducted that compares our method to popular existing regression models for interval-valued data under both linear and nonlinear settings. Furthermore, a real data example is presented where we apply our method to analyze price range data of the Dow Jones Industrial Average index and its component stocks

Empirical likelihood for random sets

We extend the method of empirical likelihood to cover hypotheses involving the Aumann expectation of random sets. By exploiting the properties of random sets, we convert the testing problem into one involving a continuum of moment restrictions for which we propose two inferential procedures. The first, which we term marked empirical likelihood, corresponds to constructing a non-parametric likelihood for each moment restriction and assessing the resulting process. The second, termed sieve empirical likelihood, corresponds to constructing a likelihood for a vector of moments with growing dimension. We derive the asymptotic distributions under the null and sequence of local alternatives for both types of tests and prove their consistency. The applicability of these inferential procedures is demonstrated in the context of two examples on the mean of interval observations and best linear predictors for interval outcomes

A linear regression model for imprecise response

A linear regression model with imprecise response and p real explanatory variables is analyzed. The imprecision of the response variable is functionally described by means of certain kinds of fuzzy sets, the LR fuzzy sets. The LR fuzzy random variables are introduced to model usual random experiments when the characteristic observed on each result can be described with fuzzy numbers of a particular class, determined by 3 random values: the center, the left spread and the right spread. In fact, these constitute a natural generalization of the interval data. To deal with the estimation problem the space of the LR fuzzy numbers is proved to be isometric to a closed and convex cone of R3 with respect to a generalization of the most used metric for LR fuzzy numbers. The expression of the estimators in terms of moments is established, their limit distribution and asymptotic properties are analyzed and applied to the determination of confidence regions and hypothesis testing procedures. The results are illustrated by means of some case-studies. © 2010 Elsevier Inc. All rights reserved

Generalized Instrumental Variable Models

This paper develops characterizations of identified sets of structures and structural features for complete and incomplete models involving continuous or discrete variables. Multiple values of unobserved variables can be associated with particular combinations of observed variables. This can arise when there are multiple sources of heterogeneity, censored or discrete endogenous variables, or inequality restrictions on functions of observed and unobserved variables. The models generalize the class of incomplete instrumental variable (IV) models in which unobserved variables are singlevalued functions of observed variables. Thus the models are referred to as generalized IV (GIV) models, but there are important cases in which instrumental variable restrictions play no significant role. Building on a definition of observational equivalence for incomplete models the development uses results from random set theory that guarantee that the characterizations deliver sharp bounds, thereby dispensing with the need for case-by-case proofs of sharpness. The use of random sets defined on the space of unobserved variables allows identification analysis under mean and quantile independence restrictions on the distributions of unobserved variables conditional on exogenous variables as well as under a full independence restriction. The results are used to develop sharp bounds on the distribution of valuations in an incomplete model of English auctions, improving on the pointwise bounds available until now. Application of many of the results of the paper requires no familiarity with random set theory

Management Information, Decision Sciences, and Financial Economics : a connection

The paper provides a brief review of the connecting literature in management information, decision sciences, and financial economics, and discusses some research that is related to the three cognate disciplines. Academics could develop theoretical models and subsequent econometric models to estimate the parameters in the associated models, and analyze some interesting issues in the three related disciplines