Ill-posed Problems in Probability and Stability of Random Sums PDF Download
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Author: Lev Borisovich Klebanov
Publisher: Nova Publishers
ISBN: 9781600212628
Category : Mathematics
Languages : en
Pages : 454
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Book Description
This volume is concerned with the problems in probability and statistics. Ill-posed problems are usually understood as those results where small changes in the assumptions lead to arbitrarily large changes in the conclusions. Such results are not very useful for practical applications where the presumptions usually hold only approximately (because even a slightest departure from the assumed model may produce an uncontrollable shift in the outcome). Often, the ill-posedness of certain practical problems is due to the lack of their precise mathematical formulation. Consequently, one can deal with such problems by replacing a given ill-posed problem with another, well-posed problem, which in some sense is 'close' to the original one. The goal in this book is to show that ill-posed problems are not just a mere curiosity in the contemporary theory of mathematical statistics and probability. On the contrary, such problems are quite common, and majority of classical results fall into this class. The objective of this book is to identify problems of this type, and re-formulate them more correctly. Thus, alternative (more precise in the above sense) versions are proposed of numerous classical theorems in the theory of probability and mathematical statistics. In addition, some non-standard problems are considered from this point of view.
Author: Lev Borisovich Klebanov
Publisher: Nova Publishers
ISBN: 9781600212628
Category : Mathematics
Languages : en
Pages : 454
Get Book Here
Book Description
This volume is concerned with the problems in probability and statistics. Ill-posed problems are usually understood as those results where small changes in the assumptions lead to arbitrarily large changes in the conclusions. Such results are not very useful for practical applications where the presumptions usually hold only approximately (because even a slightest departure from the assumed model may produce an uncontrollable shift in the outcome). Often, the ill-posedness of certain practical problems is due to the lack of their precise mathematical formulation. Consequently, one can deal with such problems by replacing a given ill-posed problem with another, well-posed problem, which in some sense is 'close' to the original one. The goal in this book is to show that ill-posed problems are not just a mere curiosity in the contemporary theory of mathematical statistics and probability. On the contrary, such problems are quite common, and majority of classical results fall into this class. The objective of this book is to identify problems of this type, and re-formulate them more correctly. Thus, alternative (more precise in the above sense) versions are proposed of numerous classical theorems in the theory of probability and mathematical statistics. In addition, some non-standard problems are considered from this point of view.
Author: Svetlozar T. Rachev
Publisher: John Wiley & Sons
ISBN: 1444392700
Category : Business & Economics
Languages : en
Pages : 264
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Book Description
A Probability Metrics Approach to Financial Risk Measures relates the field of probability metrics and risk measures to one another and applies them to finance for the first time. Helps to answer the question: which risk measure is best for a given problem? Finds new relations between existing classes of risk measures Describes applications in finance and extends them where possible Presents the theory of probability metrics in a more accessible form which would be appropriate for non-specialists in the field Applications include optimal portfolio choice, risk theory, and numerical methods in finance Topics requiring more mathematical rigor and detail are included in technical appendices to chapters
Author: Ashis Sengupta
Publisher: World Scientific
ISBN: 9814468835
Category : Mathematics
Languages : en
Pages : 492
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Book Description
This volume contains a collection of research articles on multivariate statistical methods, encompassing both theoretical advances and emerging applications in a variety of scientific disciplines. It serves as a tribute to Professor S N Roy, an eminent statistician who has made seminal contributions to the area of multivariate statistical methods, on his birth centenary. In the area of emerging applications, the topics include bioinformatics, categorical data and clinical trials, econometrics, longitudinal data analysis, microarray data analysis, sample surveys, statistical process control, etc.Researchers, professionals and advanced graduates will find the book an essential resource for modern developments in theory as well as for innovative and emerging important applications in the area of multivariate statistical methods.
Author: Roman Vershynin
Publisher: Cambridge University Press
ISBN: 1108415199
Category : Business & Economics
Languages : en
Pages : 299
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Book Description
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
Author: Aram Aruti?u?novich Sveshnikov
Publisher: Courier Corporation
ISBN: 9780486637174
Category : Mathematics
Languages : en
Pages : 516
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Book Description
Approximately 1,000 problems — with answers and solutions included at the back of the book — illustrate such topics as random events, random variables, limit theorems, Markov processes, and much more.
Author: Sandra L. Newsome
Publisher: Springer Science & Business Media
ISBN: 1461236460
Category : Technology & Engineering
Languages : en
Pages : 349
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Book Description
In 1984, Nam Sub, who was then the Assistant Director for Engineering at the National Science Foundation (NSF), created the Design Theory and Methodology Program. Among his goals in creating this program were to develop a science of engineering design and to establish design as an accepted field of engineering research. From 1984 to 1986 this program was directed by Susan Finger; from 1986 to the present Jack Dixon has been the director. The program itself has covered a broad range of disciplines, from chemical engineering to architecture, and a broad range of research paradigms, from psychological experiments to mathematical models. The present volume is based on the second NSF Grantee Workshop on Design Theory and Methodology, called Design Theory '88, which was held June 2-5, 1988 at Rensselaer Polytechnic Institute in Troy, NY, USA. It is, however, not strictly a proceedings since it includes some material that was not presented at a the Workshop and since it omits some papers and discussions that were presented at the Workshop. At the Workshop, invited speakers presented overviews of six different research areas based on summaries submitted in advance by the grantees of the Design Theory and Methodology Program. Since most of the speakers were not supported under the NSF program they brought fresh views to it. The other papers in this book were submitted directly to this volume and were not presented at the Workshop.
Author: Boris V. Gnedenko
Publisher: CRC Press
ISBN: 1000141179
Category : Mathematics
Languages : en
Pages : 282
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Book Description
This book provides an introduction to the asymptotic theory of random summation, combining a strict exposition of the foundations of this theory and recent results. It also includes a description of its applications to solving practical problems in hardware and software reliability, insurance, finance, and more. The authors show how practice interacts with theory, and how new mathematical formulations of problems appear and develop. Attention is mainly focused on transfer theorems, description of the classes of limit laws, and criteria for convergence of distributions of sums for a random number of random variables. Theoretical background is given for the choice of approximations for the distribution of stock prices or surplus processes. General mathematical theory of reliability growth of modified systems, including software, is presented. Special sections deal with doubling with repair, rarefaction of renewal processes, limit theorems for supercritical Galton-Watson processes, information properties of probability distributions, and asymptotic behavior of doubly stochastic Poisson processes. Random Summation: Limit Theorems and Applications will be of use to specialists and students in probability theory, mathematical statistics, and stochastic processes, as well as to financial mathematicians, actuaries, and to engineers desiring to improve probability models for solving practical problems and for finding new approaches to the construction of mathematical models.
Author: Gennadij M. Fel'dman
Publisher: American Mathematical Soc.
ISBN: 9780821897447
Category : Mathematics
Languages : en
Pages : 236
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Book Description
This book studies the problem of the decomposition of a given random variable introduction a sum of independent random variables (components). The central feature of the book is Feldman's use of powerful analytical techniques.
Author: Gennadiĭ Mikhaĭlovich Felʹdman
Publisher: American Mathematical Soc.
ISBN: 9780821845936
Category : Mathematics
Languages : en
Pages : 236
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Book Description
This book studies the problem of the decomposition of a given random variable introduction a sum of independent random variables (components). The central feature of the book is Feldman's use of powerful analytical techniques.
Author: James W. Demmel
Publisher: Forgotten Books
ISBN: 9781330257166
Category : Mathematics
Languages : en
Pages : 37
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Book Description
Excerpt from The Probability That a Numerical, Analysis Problem Is Difficult Numerous problems in numerical analysis, including matrix inversion, eigenvalue calculations and polynomial zero finding, share the following property: the difficulty of solving a given problem is large when the distance from that problem to the nearest "ill-posed" one is small. For example, the closer a matrix is to the set of noninvertible matrices, the larger its condition number with respect to inversion. We show that the sets of ill-posed problems for matrix inversion, eigenproblems, and polynomial zero finding all have a common algebraic and geometric structure which lets us compute the probability distribution of the distance from a "random" problem to the set. From this probability distribution we derive, for example, the distribution of the condition number of a random matrix. We examine the relevance of this theory to the analysis and construction of numerical algorithms destined to be run in finite precision arithmetic. To investigate the probability that a numerical analysis problem is difficult, we need to do three things: 1) Choose a measure of difficulty, 2) Choose a probability distribution on the set of problems, 3) Compute the distribution of the measure of difficulty induced by the distribution on the set of problems. The measure of difficulty we shall use in this paper is the condition number, which measures the sensitivity of the solution to small changes in the problem. For the problems we consider in this paper (matrix inversion, polynomial zero finding and eigenvalue calculation), there are well known condition numbers in the literature of which we shall use slightly modified versions to be discussed more fully later. The condition number is an appropriate measure of difficulty because it can be used to measure the expected loss of accuracy in the computed solution, or even the number of iterations required for an iterative algorithm to converge to a solution. The probability distribution on the set of problems for which we will attain most of our results will be the "uniform distribution" which we define as follows. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
