An Introduction to Random Sets

ISBN-13: 9780367390990 / Angielski / Miękka / 2019 / 272 str.

Hung T. Nguyen

Opis

Random sets as models for set-valued observations, are a new type of data proving useful in areas such as survey sampling, biostatistics, and intelligent systems. This is the first text to explore the topic in depth, using extended probability theory to provide a framework and tools for statistical analysis of random sets. With an abundance of examples, it highlights the basic role random sets play in a variety of statistical settings, links their study to fuzzy logic, fully develops the theory, and concludes with a variety of applications. Written by an author of the best-selling A First Course in Fuzzy Logic, this book is rigorous yet readable and fills a significant need for a textbook treatment of the subject.

Spis treści

GENERALITIES ON PROBABILITY
Survey Sampling Revisited
Mathematical Models for Random Phenomena
Random Elements
Distribution Functions of Random Variables
Distribution Functions of Random Vectors
Exercises

SOME RANDOM SETS IN STATISTICS
Probability Sampling Designs
Confidence Regions
Robust Bayesian Statistics
Probability Density Estimation
Coarse Data Analysis
Perception-Based Information
Stochastic Point Processes
Exercises

FINITE RANDOM SETS
Random Sets and Their Distributions
Set-Valued Observations
Imprecise Probabilities
Decision Making with Random Sets
Exercises

RANDOM SETS AND RELATED UNCERTAINTY MEASURES
Some Set Functions
Incidence Algebras
Cores of Capacity Functionals
Exercises

RANDOM CLOSED SETS
Introduction
The Hit-or-Miss Topology
Capacity Functionals
Notes on the Choquet Theorem on Polish Spaces
Exercises

THE CHOQUET INTEGRAL
Some Motivations
The Choquet Integral
Radon-Nikodym Derivatives
Exercises

CHOQUET WEAK CONVERGENCE
Stochastic Convergence of Random Sets
Convergence in Distribution
Weak Convergence of Capacity Functionals
Exercises

SOME ASPECTS OF STATISTICAL INFERENCE WITH COARSE DATA
Expectations and Limit Theorems
A Statistical Inference Framework for Coarse Data
A Related Statistical Setting
A Variational Calculus of Set Functions
Exercises

APPENDIX: BASIC CONCEPTS AND RESULTS OF PROBABILITY THEORY

References
Index

O autorze

Nguyen, Hung T.