Metric regularity and quantitative stability in stochastic programs with probabilistic constraints

Necessary and sufficient conditions for metric regularity of (several joint) probabilistic constraints are derived using recent results from nonsmooth analysis. The conditions apply to fairly general nonconvex, nonsmooth probabilistic constraints and extend earlier work in this direction. Further, a verifiable sufficient condition for quadratic growth of the objective function in a more specific convex stochastic program is indicated and applied in order to obtain a new result on quantitative stability of solution sets when the underlying probability distribution is subjected to perturbations. This is used to establish a large deviation estimate for solution sets when the probability measure is replaced by empirical ones

Advances in Optimization and Nonlinear Analysis

The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics

Dynamic programming with recursive preferences

There is now a considerable amount of research on the deficiencies of additively separable preferences for effective modelling of economically meaningful behaviour. Through analysis of observational data and the design of suitable experiments, economists have constructed progressively more realistic representations of agents and their choices. For intertemporal decisions, this typically involves a departure from the additively separable benchmark. A familiar example is the recursive preference framework of Epstein and Zin (1989), which has become central to the quantitative asset pricing literature, while also finding widespread use in applications range from optimal taxation to fiscal policy and business cycles. This thesis presents three essays which examine mathematical research questions within the context of recursive preferences and dynamic programming. The focus is particularly on showing existence and uniqueness of recursive utility processes under stationary and non-stationary consumption growth specifications, and on solving the closely related problem of optimality of dynamic programs with recursive preferences. On one hand, the thesis has been motivated by the availability of new and unexploited techniques for studying the aforementioned questions. The techniques in question primarily build upon an alternative version of the theory of monotone concave operators proposed by Du (1989, 1990). They are typically well suited to analysis of dynamic optimality with a variety of recursive preference specifications. On the other hand, motivation also comes from the demand side: while many useful results for dynamic programming within the context of recursive preferences have been obtained by existing literature, suitable results are still lacking for some of the most popular specifications for applied work, such as common parameterizations of the Epstein-Zin specification, or preference specifications that incorporate loss aversion and narrow framing into the Epstein-Zin framework, or the ambiguity sensitive preference specifications. In this connection, the thesis has sought to provide a new approach to dynamic optimality suitable for recursive preference specifications commonly used in modern economic analysis. The approach to examining the problems of dynamic programming exploits the theory of monotone convex operators, which, while less familiar than that of monotone concave operators, turns out to be well suited to dynamic maximization. The intuition is that convexity is preserved under maximization, while concavity is not. Meanwhile, concavity pairs well with minimization problems, since minimization preserves concavity. By applying this idea, a parallel theory for these two cases is established and it provides sufficient conditions that are easy to verify in applications

KERNEL REGRESSION SUBJECT TO INTERVAL-CENSORED RESPONSES AND QUALITATIVE CONSTRAINTS

This thesis is concerned with problems arising when one wants to apply flexible non- parametric local regression models to data when there is additional qualitative in­ formation. It is also concerned with nonparametric regression problems involving interval-censored responses. These problems are studied via asymptotic theory where possible and by simulation. Iterated conditional expectation methods and local likelihood estimation for nonparametric interval-censored regression are developed. Simulation results show that local likelihood estimation is often superior to local regression estimators when observations have been imputed using either interval midpoints or iterated conditional expectations when the censoring intervals are wide or of varying width. When the intervals are smaller and of fixed width, none of the imputation approaches dominate the others. Constrained data sharpening for nonparametric regression is applied to new situations such as where constraints are defined by convexity, concavity, and in terms of differential operators. Data sharpening is compared with competing kernel methods in terms of bias, variance and MISE. It is proved that the constrained data sharpening estimator has the same rate of convergence as the constrained weighting estimator of Hall and Huang (2001). Also, penalized data sharpening is proposed as a new form of constrained data sharpening. The sharpened responses can be computed analytically which makes the method very convenient, both for studying theoretically and for applying practicall

Examining the Functional Specification of Two-Parameter Model under Location and Scale Parameter Condition

The functional specification of mean-standard deviation approach is examined under location and scale parameter condition. Firstly, the full set of restrictions imposed on the mean-standard deviation function under the location and scale parameter condition are made clear. Secondly, the examination based on the restrictions mentioned in the previous sentence derives the new properties of the mean-standard deviation function on the applicability of additive separability and the curvature of expansion path which links the points that give the same slope of indifference curve. It reveals that attention has not been sufficiently paid to the restrictions in interpreting the linear mean-standard deviation model and the nonlinear mean-standard deviation model that have been used in previous research. Thirdly, the interpretation of the nonlinear mean-standard deviation model is reconsidered in detail and then an alternative nonlinear mean-standard deviation model is proposed. The implication of the two nonlinear mean-standard deviation models to the empirical approach called "joint analysis of risk preference structure and technology" is discussed.mean-standard deviation approach, location and scale parameter condition, functional specification, risk aversion, uncertainty, Research Methods/ Statistical Methods,

Fusion of Hard and Soft Information in Nonparametric Density Estimation

This article discusses univariate density estimation in situations when the sample (hard information) is supplemented by “soft” information about the random phenomenon. These situations arise broadly in operations research and management science where practical and computational reasons severely limit the sample size, but problem structure and past experiences could be brought in. In particular, density estimation is needed for generation of input densities to simulation and stochastic optimization models, in analysis of simulation output, and when instantiating probability models. We adopt a constrained maximum likelihood estimator that incorporates any, possibly random, soft information through an arbitrary collection of constraints. We illustrate the breadth of possibilities by discussing soft information about shape, support, continuity, smoothness, slope, location of modes, symmetry, density values, neighborhood of known density, moments, and distribution functions. The maximization takes place over spaces of extended real-valued semicontinuous functions and therefore allows us to consider essentially any conceivable density as well as convenient exponential transformations. The infinite dimensionality of the optimization problem is overcome by approximating splines tailored to these spaces. To facilitate the treatment of small samples, the construction of these splines is decoupled from the sample. We discuss existence and uniqueness of the estimator, examine consistency under increasing hard and soft information, and give rates of convergence. Numerical examples illustrate the value of soft information, the ability to generate a family of diverse densities, and the effect of misspecification of soft information.U.S. Army Research Laboratory and the U.S. Army Research Office grant 00101-80683U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-10-1-0246U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-12-1-0273U.S. Army Research Laboratory and the U.S. Army Research Office grant 00101-80683U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-10-1-0246U.S. Army Research Laboratory and the U.S. Army Research Office grant W911NF-12-1-027

Estimation of Econometric Models by Risk Minimization: A Stochastic Quasigradient Approach

The paper presents a risk minimization approach to estimate a flexible form that meets a priori restrictions on slope and curvature by means of constraints on both the estimated parameters and the function values. The resulting constrained risk minimization combines parametric and nonparametric estimation and contains integrals and implicit constraints. Within econometrics, simulation has become a common tool to solve problems of this kind. However, it appears that in our case, the simulation approach only applies when the model is linear in parameters, has simple constraints on parameters and a quadratic risk function. To deal with other cases, we use a stochastic optimization technique known as the stochastic quasi-gradient method for stationary and nonstationary problems with Cesaro averaging. This method is also applicable to an expanding series of random observations, and produces asymptotically (weakly) convergent estimates