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Author: Ilya Molchanov Publisher: Cambridge University Press ISBN: 1108651356 Category : Business & Economics Languages : en Pages : 199
Book Description
Random set theory is a fascinating branch of mathematics that amalgamates techniques from topology, convex geometry, and probability theory. Social scientists routinely conduct empirical work with data and modelling assumptions that reveal a set to which the parameter of interest belongs, but not its exact value. Random set theory provides a coherent mathematical framework to conduct identification analysis and statistical inference in this setting and has become a fundamental tool in econometrics and finance. This is the first book dedicated to the use of the theory in econometrics, written to be accessible for readers without a background in pure mathematics. Molchanov and Molinari define the basics of the theory and illustrate the mathematical concepts by their application in the analysis of econometric models. The book includes sets of exercises to accompany each chapter as well as examples to help readers apply the theory effectively.
Author: Ilya Molchanov Publisher: Cambridge University Press ISBN: 1108651356 Category : Business & Economics Languages : en Pages : 199
Book Description
Random set theory is a fascinating branch of mathematics that amalgamates techniques from topology, convex geometry, and probability theory. Social scientists routinely conduct empirical work with data and modelling assumptions that reveal a set to which the parameter of interest belongs, but not its exact value. Random set theory provides a coherent mathematical framework to conduct identification analysis and statistical inference in this setting and has become a fundamental tool in econometrics and finance. This is the first book dedicated to the use of the theory in econometrics, written to be accessible for readers without a background in pure mathematics. Molchanov and Molinari define the basics of the theory and illustrate the mathematical concepts by their application in the analysis of econometric models. The book includes sets of exercises to accompany each chapter as well as examples to help readers apply the theory effectively.
Author: Ilya Molchanov Publisher: Springer Science & Business Media ISBN: 9781852338923 Category : Mathematics Languages : en Pages : 508
Book Description
This is the first systematic exposition of random sets theory since Matheron (1975), with full proofs, exhaustive bibliographies and literature notes Interdisciplinary connections and applications of random sets are emphasized throughout the book An extensive bibliography in the book is available on the Web at http://liinwww.ira.uka.de/bibliography/math/random.closed.sets.html, and is accompanied by a search engine
Author: Hung T. Nguyen Publisher: CRC Press ISBN: 1420010611 Category : Mathematics Languages : en Pages : 268
Book Description
The study of random sets is a large and rapidly growing area with connections to many areas of mathematics and applications in widely varying disciplines, from economics and decision theory to biostatistics and image analysis. The drawback to such diversity is that the research reports are scattered throughout the literature, with the result that i
Author: Ilya Molchanov Publisher: Springer ISBN: 144717349X Category : Mathematics Languages : en Pages : 688
Book Description
This monograph, now in a thoroughly revised second edition, offers the latest research on random sets. It has been extended to include substantial developments achieved since 2005, some of them motivated by applications of random sets to econometrics and finance. The present volume builds on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time fixes terminology and notation that often vary in the literature, establishing it as a natural part of modern probability theory and providing a platform for future development. It is completely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight. Aimed at research level, Theory of Random Sets will be an invaluable reference for probabilists; mathematicians working in convex and integral geometry, set-valued analysis, capacity and potential theory; mathematical statisticians in spatial statistics and uncertainty quantification; specialists in mathematical economics, econometrics, decision theory, and mathematical finance; and electronic and electrical engineers interested in image analysis.
Author: Publisher: ISBN: Category : Languages : en Pages :
Book Description
Abstract: In many statistical applications, the observed data take the form of sets rather than points. Examples include bracket data in survey analysis, tumor growth and rock grain images in morphology analysis, and noisy measurements on the support function of a convex set in medical imaging and robotic vision. Additionally, in studies of treatment effects, researchers often wish to conduct inference on nonparametric bounds for the effects which can be expressed by means of random sets. This article develops the concept of nonparametric likelihood for random sets and its mean, known as the Aumann expectation, and proposes general inference methods by adapting the theory of empirical likelihood. Several examples, such as regression with bracket income data, Boolean models for tumor growth, bound analysis on treatment effects, and image analysis via support functions, illustrate the usefulness of the proposed methods. Supplementary materials for this article are available online.
Book Description
This short book provides a unified view of the history and theory of random sets and fuzzy random variables, with special emphasis on its use for representing higher-order non-statistical uncertainty about statistical experiments. The authors lay bare the existence of two streams of works using the same mathematical ground, but differing form their use of sets, according to whether they represent objects of interest naturally taking the form of sets, or imprecise knowledge about such objects. Random (fuzzy) sets can be used in many fields ranging from mathematical morphology, economics, artificial intelligence, information processing and statistics per se, especially in areas where the outcomes of random experiments cannot be observed with full precision. This book also emphasizes the link between random sets and fuzzy sets with some techniques related to the theory of imprecise probabilities. This small book is intended for graduate and doctoral students in mathematics or engineering, but also provides an introduction for other researchers interested in this area. It is written from a theoretical perspective. However, rather than offering a comprehensive formal view of random (fuzzy) sets in this context, it aims to provide a discussion of the meaning of the proposed formal constructions based on many concrete examples and exercises. This book should enable the reader to understand the usefulness of representing and reasoning with incomplete information in statistical tasks. Each chapter ends with a list of exercises.
Author: Ilya Molchanov Publisher: ISBN: Category : Languages : en Pages :
Book Description
In recent years, the econometrics literature has shown a growing interest in the study of partially identified models, in which the object of economic and statistical interest is a set rather than a point. The characterization of this set and the development of consistent estimators and inference procedures for it with desirable properties are the main goals of partial identification analysis. This review introduces the fundamental tools of the theory of random sets, which brings together elements of topology, convex geometry, and probability theory to develop a coherent mathematical framework to analyze random elements whose realizations are sets. It then elucidates how these tools have been fruitfully applied in econometrics to reach the goals of partial identification analysis.
Author: Van-Nam Huynh Publisher: Springer ISBN: 3319134493 Category : Technology & Engineering Languages : en Pages : 486
Book Description
This edited book contains several state-of-the-art papers devoted to econometrics of risk. Some papers provide theoretical analysis of the corresponding mathematical, statistical, computational, and economical models. Other papers describe applications of the novel risk-related econometric techniques to real-life economic situations. The book presents new methods developed just recently, in particular, methods using non-Gaussian heavy-tailed distributions, methods using non-Gaussian copulas to properly take into account dependence between different quantities, methods taking into account imprecise ("fuzzy") expert knowledge, and many other innovative techniques. This versatile volume helps practitioners to learn how to apply new techniques of econometrics of risk, and researchers to further improve the existing models and to come up with new ideas on how to best take into account economic risks.
Author: Ilya Molchanov
Author: Ilya Molchanov
Author: Ilya Molchanov
Author: Ilya Molchanov
Author: Hung T. Nguyen
Author: Ilya Molchanov
Author: 
Author: James Charles Hickman
Author: Inés Couso
Author: Ilya Molchanov
Author: Van-Nam Huynh