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Estimation And Goodness Of Fit For Multivariate Survival Models Based On Copulas
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Book Synopsis Estimation and Goodness of Fit for Multivariate Survival Models Based on Copulas by : Yildiz Elif Yilmaz
Download or read book Estimation and Goodness of Fit for Multivariate Survival Models Based on Copulas written by Yildiz Elif Yilmaz and published by . This book was released on 2009 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: We provide ways to test the fit of a parametric copula family for bivariate censored data with or without covariates. The proposed copula family is tested by embedding it in an expanded parametric family of copulas. When parameters in the proposed and the expanded copula models are estimated by maximum likelihood, a likelihood ratio test can be used. However, when they are estimated by two-stage pseudolikelihood estimation, the corresponding test is a pseudolikelihood ratio test. The two-stage procedures offer less computation, which is especially attractive when the marginal lifetime distributions are specified nonparametrically or semiparametrically. It is shown that the likelihood ratio test is consistent even when the expanded model is misspecified. Power comparisons of the likelihood ratio and the pseudolikelihood ratio tests with some other goodness-of-fit tests are performed both when the expanded family is correct and when it is misspecified. They indicate that model expansion provides a convenient, powerful and robust approach. We introduce a semiparametric maximum likelihood estimation method in which the copula parameter is estimated without assumptions on the marginal distributions. This method and the two-stage semiparametric estimation method suggested by Shih and Louis (1995) are generalized to regression models with Cox proportional hazards margins. The two-stage semiparametric estimator of the copula parameter is found to be about as good as the semiparametric maximum likelihood estimator. Semiparametric likelihood ratio and pseudolikelihood ratio tests are considered to provide goodness of fit tests for a copula model without making parametric assumptions for the marginal distributions. Both when the expanded family is correct and when it is misspecified, the semiparametric pseudolikelihood ratio test is almost as powerful as the parametric likelihood ratio and pseudolikelihood ratio tests while achieving robustness to the form of the marginal distributions. The methods are illustrated on applications in medicine and insurance. Sequentially observed survival times are of interest in many studies but there are difficulties in modeling and analyzing such data. First, when the duration of followup is limited and the times for a given individual are not independent, the problem of induced dependent censoring arises for the second and subsequent survival times. Non-identifiability of the marginal survival distributions for second and later times is another issue, since they are observable only if preceding survival times for an individual are uncensored. In addition, in some studies, a significant proportion of individuals may never have the first event. Fully parametric models can deal with these features, but lack of robustness is a concern, and methods of assessing fit are lacking. We introduce an approach to address these issues. We model the joint distribution of the successive survival times by using copula functions, and provide semiparametric estimation procedures in which copula parameters are estimated without parametric assumptions on the marginal distributions. The performance of semiparametric estimation methods is compared with some other estimation methods in simulation studies and shown to be good. The methodology is applied to a motivating example involving relapse and survival following colon cancer treatment.
Book Synopsis Survival Analysis with Correlated Endpoints by : Takeshi Emura
Download or read book Survival Analysis with Correlated Endpoints written by Takeshi Emura and published by Springer. This book was released on 2019-03-25 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces readers to advanced statistical methods for analyzing survival data involving correlated endpoints. In particular, it describes statistical methods for applying Cox regression to two correlated endpoints by accounting for dependence between the endpoints with the aid of copulas. The practical advantages of employing copula-based models in medical research are explained on the basis of case studies. In addition, the book focuses on clustered survival data, especially data arising from meta-analysis and multicenter analysis. Consequently, the statistical approaches presented here employ a frailty term for heterogeneity modeling. This brings the joint frailty-copula model, which incorporates a frailty term and a copula, into a statistical model. The book also discusses advanced techniques for dealing with high-dimensional gene expressions and developing personalized dynamic prediction tools under the joint frailty-copula model. To help readers apply the statistical methods to real-world data, the book provides case studies using the authors’ original R software package (freely available in CRAN). The emphasis is on clinical survival data, involving time-to-tumor progression and overall survival, collected on cancer patients. Hence, the book offers an essential reference guide for medical statisticians and provides researchers with advanced, innovative statistical tools. The book also provides a concise introduction to basic multivariate survival models.
Book Synopsis Analysis of Multivariate Survival Data by : Philip Hougaard
Download or read book Analysis of Multivariate Survival Data written by Philip Hougaard and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 559 pages. Available in PDF, EPUB and Kindle. Book excerpt: Survival data or more general time-to-event data occur in many areas, including medicine, biology, engineering, economics, and demography, but previously standard methods have requested that all time variables are univariate and independent. This book extends the field by allowing for multivariate times. As the field is rather new, the concepts and the possible types of data are described in detail. Four different approaches to the analysis of such data are presented from an applied point of view.
Book Synopsis A Goodness-of-fit Test for Semi-parametric Copula Models of Right-censored Bivariate Survival Times by : Moyan Mei
Download or read book A Goodness-of-fit Test for Semi-parametric Copula Models of Right-censored Bivariate Survival Times written by Moyan Mei and published by . This book was released on 2016 with total page 36 pages. Available in PDF, EPUB and Kindle. Book excerpt: In multivariate survival analyses, understanding and quantifying the association between survival times is of importance. Copulas, such as Archimedean copulas and Gaussian copulas, provide a flexible approach of modeling and estimating the dependence structure among survival times separately from the marginal distributions (Sklar, 1959). However, misspecification in the parametric form of the copula function will directly lead to incor- rect estimation of the joint distribution of the bivariate survival times and other model-based quantities.The objectives of this project are two-folded. First, I reviewed the basic definitions and properties of commonly used survival copula models. In this project, I focused on semi- parametric copula models where the marginal distributions are unspecified but the copula function belongs to a parametric copula family. Various estimation procedures of the de- pendence parameter associated with the copula function were also reviewed. Secondly, I extended the pseudo in-and-out-of-sample (PIOS) likelihood ratio test proposed in Zhang et al. (2016) to testing the semi-parametric copula models for right-censored bivariate sur- vival times. The PIOS test is constructed by comparing two forms of pseudo likelihoods, one is the "in-sample" pseudo likelihood, which is the full pseudo likelihood, and the other is the "out-of-sample" pseudo likelihood, which is a cross-validated pseudo likelihood by the means of jacknife. The finite sample performance of the PIOS test was investigated via a simulation study. In addition, two real data examples were analyzed for illustrative purpose.
Book Synopsis Survival Analysis with Correlated Endpoints by : Takeshi Emura
Download or read book Survival Analysis with Correlated Endpoints written by Takeshi Emura and published by . This book was released on 2019 with total page 118 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces readers to advanced statistical methods for analyzing survival data involving correlated endpoints. In particular, it describes statistical methods for applying Cox regression to two correlated endpoints by accounting for dependence between the endpoints with the aid of copulas. The practical advantages of employing copula-based models in medical research are explained on the basis of case studies. In addition, the book focuses on clustered survival data, especially data arising from meta-analysis and multicenter analysis. Consequently, the statistical approaches presented here employ a frailty term for heterogeneity modeling. This brings the joint frailty-copula model, which incorporates a frailty term and a copula, into a statistical model. The book also discusses advanced techniques for dealing with high-dimensional gene expressions and developing personalized dynamic prediction tools under the joint frailty-copula model. To help readers apply the statistical methods to real-world data, the book provides case studies using the authors' original R software package (freely available in CRAN). The emphasis is on clinical survival data, involving time-to-tumor progression and overall survival, collected on cancer patients. Hence, the book offers an essential reference guide for medical statisticians and provides researchers with advanced, innovative statistical tools. The book also provides a concise introduction to basic multivariate survival models.
Book Synopsis Estimation and Model Selection of Semiparametric Multivariate Survival Functions under General Censorship by : Xiaohong Chen
Download or read book Estimation and Model Selection of Semiparametric Multivariate Survival Functions under General Censorship written by Xiaohong Chen and published by . This book was released on 2008 with total page 39 pages. Available in PDF, EPUB and Kindle. Book excerpt: Many models of semiparametric multivariate survival functions are characterized by nonparametric marginal survival functions and parametric copula functions, where different copulas imply different dependence structures. This paper considers estimation and model selection for these semiparametric multivariate survival functions, allowing for misspecified parametric copulas and data subject to general censoring. We first establish convergence of the two-step estimator of the copula parameter to the pseudo-true value defined as the value of the parameter that minimizes the KLIC between the parametric copula induced multivariate density and the unknown true density. We then derive its root-n asymptotically normal distribution and provide a simple consistent asymptotic variance estimator by accounting for the impact of the nonparametric estimation of the marginal survival functions. These results are used to establish the asymptotic distribution of the penalized pseudo-likelihood ratio statistic for comparing multiple semiparametric multivariate survival functions subject to copula misspecification and general censorship. An empirical application of the model selection test to the Loss-ALAE insurance data set is provided.
Book Synopsis Survival Analysis Using Bivariate Archimedean Copulas by : Krishnendu Chandra
Download or read book Survival Analysis Using Bivariate Archimedean Copulas written by Krishnendu Chandra and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: A suitable bootstrap procedure is yet to be suggested for our method. We further propose a new parameter estimator and a simple goodness-of-fit test for Archimedean copula models when the bivariate data is under fixed left truncation. Our simulation results suggest that our procedure needs to be improved so that it can be more powerful, reliable and efficient. In our strategy, to obtain estimates for the unknown parameters, we heavily exploit the concept of truncated tau (a measure of association established by [Manatunga and Oakes, 1996] for left truncated data). The idea of our goodness of fit test originates from the goodness-of-fit test for Archimedean copula models proposed by [Wang, 2010] for right censored bivariate data.
Book Synopsis Multivariate Survival Modelling by : Pierre Georges
Download or read book Multivariate Survival Modelling written by Pierre Georges and published by . This book was released on 2007 with total page 72 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this paper, we review the use of copulas for multivariate survival modelling. In particular, we study properties of survival copulas and discuss the dependence measures associated to this construction. Then, we consider the problem of competing risks. We derive the distribution of the failure time and order statistics. After having presented statistical inference, we finally provide financial applications which concern the life time value (attrition models), the link between default, prepayment and credit life, the measure of risk for a credit portfolio and the pricing of credit derivatives.
Book Synopsis Analysis of Survival Data with Dependent Censoring by : Takeshi Emura
Download or read book Analysis of Survival Data with Dependent Censoring written by Takeshi Emura and published by Springer. This book was released on 2018-04-13 with total page 84 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces readers to copula-based statistical methods for analyzing survival data involving dependent censoring. Primarily focusing on likelihood-based methods performed under copula models, it is the first book solely devoted to the problem of dependent censoring. The book demonstrates the advantages of the copula-based methods in the context of medical research, especially with regard to cancer patients’ survival data. Needless to say, the statistical methods presented here can also be applied to many other branches of science, especially in reliability, where survival analysis plays an important role. The book can be used as a textbook for graduate coursework or a short course aimed at (bio-) statisticians. To deepen readers’ understanding of copula-based approaches, the book provides an accessible introduction to basic survival analysis and explains the mathematical foundations of copula-based survival models.
Book Synopsis Univariate and Multivariate Survival Models with Flexible Hazard Functions by : Dooti Roy
Download or read book Univariate and Multivariate Survival Models with Flexible Hazard Functions written by Dooti Roy and published by . This book was released on 2017 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Our research focuses on exploring and developing flexible Bayesian methodologies to model both univariate and multivariate survival data. When developing a Bayesian survival model, the most desirable properties are often flexibility of hazard functions, a proper posterior distribution and efficient implementation. The novelty of our work can be classified into three sections: first, we introduce a new distribution to model univariate and bivariate survival data. Although extreme value theory and subsequently the Generalized Extreme Value (GEV) distribution have been explored in the past to model rare events, our work is the first of its kind to extend GEV framework into the foray of survival analysis. We develop a cure rate model and apply it to various types of univariate cancer survival data. Second, we provide a novel method of estimating the copula association parameter for bivariate survival data using an empirical Bayes approach. Lastly we propose a novel Bayesian R-Vine approach to model multivariate survival data. The thesis consists of five chapters. Chapter 1 introduces the problem of survival data analysis and provides a brief overview of both the frequentist and Bayesian methods developed over the past few decades. Chapter 2 briefly introduces the univariate extreme value analysis. In Chapter 3, we use both forms of the GEV distribution, the Maxima and the Minima to develop a Bayesian modeling technique to analyze right-censored log survival data for populations with a surviving fraction. Next in Chapter 4, we consider bivariate survival data and use copula structures to model the association between the survival times. A novel empirical Bayesian method for estimating the copula function has been proposed. Using our model, we enable the user to use different copula functions to model the same data and hence introduce the concept of copula choice using the Bayesian model selection approach. We demonstrate through extensive simulations that the empirical Bayesian approach provides tighter HPD intervals for the copula parameter of association as compared to full Bayesian and two-stage estimation procedures. Lastly, chapter 5 introduces a novel approach to model multivariate survival data using a Bayesian R-vine copula approach. We show that this method provides flexibility and easy computation even for dimensions 3 and higher as compared to direct extension of bivariate copula families to multivariate dimensions.
Book Synopsis Simulation and Goodness-of-Fit Tests of Copulas by : Yiran Chen
Download or read book Simulation and Goodness-of-Fit Tests of Copulas written by Yiran Chen and published by . This book was released on 2022 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multivariate models with dependent variables are popular in financial industry, with applications to option pricing, portfolio optimization, risk management, and modeling credit portfolios. Other applications include geostatistic, hydrology, insurance mathematics,medicine, and reliability theory. Multivariate distributions can be separated into univariate marginals and the dependence structure. Usually, we are more interested in the latter part, which is represented by copulas. In this report, we provide a review of two popular simulation methods. For each method, different random number sequences (MC vs RQMC) are used for comparison. We first compare these simulation methods by a goodness-of-fit test for copulas from literature.Then we propose a new goodness-of-fit test for copulas, investigate the accuracy of the test numerically, and compare it with a well-regarded test in the literature. In addition, we established error bounds for numerical integration and option pricing problems.
Book Synopsis The Frailty Model by : Luc Duchateau
Download or read book The Frailty Model written by Luc Duchateau and published by Springer Science & Business Media. This book was released on 2007-10-23 with total page 329 pages. Available in PDF, EPUB and Kindle. Book excerpt: Readers will find in the pages of this book a treatment of the statistical analysis of clustered survival data. Such data are encountered in many scientific disciplines including human and veterinary medicine, biology, epidemiology, public health and demography. A typical example is the time to death in cancer patients, with patients clustered in hospitals. Frailty models provide a powerful tool to analyze clustered survival data. In this book different methods based on the frailty model are described and it is demonstrated how they can be used to analyze clustered survival data. All programs used for these examples are available on the Springer website.
Book Synopsis Dependence Modeling with Copulas by : Harry Joe
Download or read book Dependence Modeling with Copulas written by Harry Joe and published by CRC Press. This book was released on 2014-06-26 with total page 483 pages. Available in PDF, EPUB and Kindle. Book excerpt: Dependence Modeling with Copulas covers the substantial advances that have taken place in the field during the last 15 years, including vine copula modeling of high-dimensional data. Vine copula models are constructed from a sequence of bivariate copulas. The book develops generalizations of vine copula models, including common and structured factor models that extend from the Gaussian assumption to copulas. It also discusses other multivariate constructions and parametric copula families that have different tail properties and presents extensive material on dependence and tail properties to assist in copula model selection. The author shows how numerical methods and algorithms for inference and simulation are important in high-dimensional copula applications. He presents the algorithms as pseudocode, illustrating their implementation for high-dimensional copula models. He also incorporates results to determine dependence and tail properties of multivariate distributions for future constructions of copula models.
Book Synopsis Copula Theory and Its Applications by : Piotr Jaworski
Download or read book Copula Theory and Its Applications written by Piotr Jaworski and published by Springer Science & Business Media. This book was released on 2010-07-16 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: Copulas are mathematical objects that fully capture the dependence structure among random variables and hence offer great flexibility in building multivariate stochastic models. Since their introduction in the early 50's, copulas have gained considerable popularity in several fields of applied mathematics, such as finance, insurance and reliability theory. Today, they represent a well-recognized tool for market and credit models, aggregation of risks, portfolio selection, etc. This book is divided into two main parts: Part I - "Surveys" contains 11 chapters that provide an up-to-date account of essential aspects of copula models. Part II - "Contributions" collects the extended versions of 6 talks selected from papers presented at the workshop in Warsaw.
Book Synopsis Copulae and Multivariate Probability Distributions in Finance by : Alexandra Dias
Download or read book Copulae and Multivariate Probability Distributions in Finance written by Alexandra Dias and published by Routledge. This book was released on 2013-08-21 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: Portfolio theory and much of asset pricing, as well as many empirical applications, depend on the use of multivariate probability distributions to describe asset returns. Traditionally, this has meant the multivariate normal (or Gaussian) distribution. More recently, theoretical and empirical work in financial economics has employed the multivariate Student (and other) distributions which are members of the elliptically symmetric class. There is also a growing body of work which is based on skew-elliptical distributions. These probability models all exhibit the property that the marginal distributions differ only by location and scale parameters or are restrictive in other respects. Very often, such models are not supported by the empirical evidence that the marginal distributions of asset returns can differ markedly. Copula theory is a branch of statistics which provides powerful methods to overcome these shortcomings. This book provides a synthesis of the latest research in the area of copulae as applied to finance and related subjects such as insurance. Multivariate non-Gaussian dependence is a fact of life for many problems in financial econometrics. This book describes the state of the art in tools required to deal with these observed features of financial data. This book was originally published as a special issue of the European Journal of Finance.
Book Synopsis Economic Time Series by : William R. Bell
Download or read book Economic Time Series written by William R. Bell and published by CRC Press. This book was released on 2018-11-14 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: Economic Time Series: Modeling and Seasonality is a focused resource on analysis of economic time series as pertains to modeling and seasonality, presenting cutting-edge research that would otherwise be scattered throughout diverse peer-reviewed journals. This compilation of 21 chapters showcases the cross-fertilization between the fields of time s
Book Synopsis Copula-Based Markov Models for Time Series by : Li-Hsien Sun
Download or read book Copula-Based Markov Models for Time Series written by Li-Hsien Sun and published by Springer Nature. This book was released on 2020-07-01 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides statistical methodologies for time series data, focusing on copula-based Markov chain models for serially correlated time series. It also includes data examples from economics, engineering, finance, sport and other disciplines to illustrate the methods presented. An accessible textbook for students in the fields of economics, management, mathematics, statistics, and related fields wanting to gain insights into the statistical analysis of time series data using copulas, the book also features stand-alone chapters to appeal to researchers. As the subtitle suggests, the book highlights parametric models based on normal distribution, t-distribution, normal mixture distribution, Poisson distribution, and others. Presenting likelihood-based methods as the main statistical tools for fitting the models, the book details the development of computing techniques to find the maximum likelihood estimator. It also addresses statistical process control, as well as Bayesian and regression methods. Lastly, to help readers analyze their data, it provides computer codes (R codes) for most of the statistical methods.
