Saddlepoint Approximations

Saddlepoint Approximations
Author: Jens Ledet Jensen
Publisher: Oxford University Press
Total Pages: 348
Release: 1995
Genre: Mathematics
ISBN: 9780198522959

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This book explains the ideas behind the saddlepoint approximations as well as giving a detailed mathematical description of the subject and many worked out examples.

Saddlepoint Approximations with Applications

Saddlepoint Approximations with Applications
Author: Ronald W. Butler
Publisher: Cambridge University Press
Total Pages: 548
Release: 2007-08-16
Genre: Mathematics
ISBN: 1139466518

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Modern statistical methods use complex, sophisticated models that can lead to intractable computations. Saddlepoint approximations can be the answer. Written from the user's point of view, this book explains in clear language how such approximate probability computations are made, taking readers from the very beginnings to current applications. The core material is presented in chapters 1-6 at an elementary mathematical level. Chapters 7-9 then give a highly readable account of higher-order asymptotic inference. Later chapters address areas where saddlepoint methods have had substantial impact: multivariate testing, stochastic systems and applied probability, bootstrap implementation in the transform domain, and Bayesian computation and inference. No previous background in the area is required. Data examples from real applications demonstrate the practical value of the methods. Ideal for graduate students and researchers in statistics, biostatistics, electrical engineering, econometrics, and applied mathematics, this is both an entry-level text and a valuable reference.

Saddlepoint Approximation Methods in Financial Engineering

Saddlepoint Approximation Methods in Financial Engineering
Author: Yue Kuen Kwok
Publisher: Springer
Total Pages: 134
Release: 2018-02-16
Genre: Mathematics
ISBN: 3319741012

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This book summarizes recent advances in applying saddlepoint approximation methods to financial engineering. It addresses pricing exotic financial derivatives and calculating risk contributions to Value-at-Risk and Expected Shortfall in credit portfolios under various default correlation models. These standard problems involve the computation of tail probabilities and tail expectations of the corresponding underlying state variables. The text offers in a single source most of the saddlepoint approximation results in financial engineering, with different sets of ready-to-use approximation formulas. Much of this material may otherwise only be found in original research publications. The exposition and style are made rigorous by providing formal proofs of most of the results. Starting with a presentation of the derivation of a variety of saddlepoint approximation formulas in different contexts, this book will help new researchers to learn the fine technicalities of the topic. It will also be valuable to quantitative analysts in financial institutions who strive for effective valuation of prices of exotic financial derivatives and risk positions of portfolios of risky instruments.

Mathematical Statistics Theory and Applications

Mathematical Statistics Theory and Applications
Author:
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 871
Release: 2020-05-26
Genre: Technology & Engineering
ISBN: 3112319087

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Saddlepoint Approximations to Tail Expectations Under Non-Gaussian Base Distributions

Saddlepoint Approximations to Tail Expectations Under Non-Gaussian Base Distributions
Author: Yuantao Zhang
Publisher:
Total Pages: 20
Release: 2019
Genre:
ISBN:

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The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatilitymodel.

Small Sample Asymptotics

Small Sample Asymptotics
Author: Christopher A. Field
Publisher: IMS
Total Pages: 166
Release: 1990
Genre: Mathematical statistics
ISBN: 9780940600188

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Statistical Foundations of Actuarial Learning and its Applications

Statistical Foundations of Actuarial Learning and its Applications
Author: Mario V. Wüthrich
Publisher: Springer Nature
Total Pages: 611
Release: 2022-11-22
Genre: Mathematics
ISBN: 303112409X

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This open access book discusses the statistical modeling of insurance problems, a process which comprises data collection, data analysis and statistical model building to forecast insured events that may happen in the future. It presents the mathematical foundations behind these fundamental statistical concepts and how they can be applied in daily actuarial practice. Statistical modeling has a wide range of applications, and, depending on the application, the theoretical aspects may be weighted differently: here the main focus is on prediction rather than explanation. Starting with a presentation of state-of-the-art actuarial models, such as generalized linear models, the book then dives into modern machine learning tools such as neural networks and text recognition to improve predictive modeling with complex features. Providing practitioners with detailed guidance on how to apply machine learning methods to real-world data sets, and how to interpret the results without losing sight of the mathematical assumptions on which these methods are based, the book can serve as a modern basis for an actuarial education syllabus.

Intermediate Probability

Intermediate Probability
Author: Marc S. Paolella
Publisher: John Wiley & Sons
Total Pages: 430
Release: 2007-09-27
Genre: Mathematics
ISBN: 9780470035054

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Intermediate Probability is the natural extension of the author's Fundamental Probability. It details several highly important topics, from standard ones such as order statistics, multivariate normal, and convergence concepts, to more advanced ones which are usually not addressed at this mathematical level, or have never previously appeared in textbook form. The author adopts a computational approach throughout, allowing the reader to directly implement the methods, thus greatly enhancing the learning experience and clearly illustrating the applicability, strengths, and weaknesses of the theory. The book: Places great emphasis on the numeric computation of convolutions of random variables, via numeric integration, inversion theorems, fast Fourier transforms, saddlepoint approximations, and simulation. Provides introductory material to required mathematical topics such as complex numbers, Laplace and Fourier transforms, matrix algebra, confluent hypergeometric functions, digamma functions, and Bessel functions. Presents full derivation and numerous computational methods of the stable Paretian and the singly and doubly non-central distributions. A whole chapter is dedicated to mean-variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions. A whole chapter is dedicated to nesting, generalizing, and asymmetric extensions of popular distributions, as have become popular in empirical finance and other applications. Provides all essential programming code in Matlab and R. The user-friendly style of writing and attention to detail means that self-study is easily possible, making the book ideal for senior undergraduate and graduate students of mathematics, statistics, econometrics, finance, insurance, and computer science, as well as researchers and professional statisticians working in these fields.

Survival Analysis

Survival Analysis
Author: John O'Quigley
Publisher: Springer Nature
Total Pages: 475
Release: 2021-04-27
Genre: Mathematics
ISBN: 3030334392

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This book provides an extensive coverage of the methodology of survival analysis, ranging from introductory level material to deeper more advanced topics. The framework is that of proportional and non-proportional hazards models; a structure that is broad enough to enable the recovery of a large number of established results as well as to open the way to many new developments. The emphasis is on concepts and guiding principles, logical and graphical. Formal proofs of theorems, propositions and lemmas are gathered together at the end of each chapter separate from the main presentation. The intended audience includes academic statisticians, biostatisticians, epidemiologists and also researchers in these fields whose focus may be more on the applications than on the theory. The text could provide the basis for a two semester course on survival analysis and, with this goal in mind, each chapter includes a section with a range of exercises as a teaching aid for instructors.