Author: Dinghai XuPublisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
Continuous Empirical Characteristic Function Estimation of Mixtures of Normal Parameters
Author: Dinghai XuAn Empirical Characteristic Function Approach to VaR Under a Mixture-of-Normal Distribution with Time-Varying Volatility
Author: Dinghai XuPublisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
This article considers risk measures constructed under a discrete mixture-of-normal distribution on the innovations of a GARCH model with time-varying volatility. The authors use an approach based on a continuous empirical characteristic function to estimate the parameters of the model using several daily foreign exchange rates' return data. This approach, compared to the likelihood-based approach, has several important advantages as a method for estimating this type of models; in particular, the characteristic function, unlike its likelihood function counterpart, is always uniformly bounded over the model's parameter space due to the Fourier transformation. To evaluate VaR and expected shortfall measures obtained from alternative specifications, the authors construct several evaluation criteria, such as the number of violations and the sum square of violations. Based on these criteria, the authors find that both the VaR and expected shortfall measures obtained from the proposed model outperform those obtained from other competing models.
Bias in Mixtures of Normal Distributions and Joint Modeling of Longitudinal and Time-to-event Data with Monotonic Change Curves
Author: Spencer LourensPublisher:
ISBN:
Category : Biometric identification
Languages : en
Pages : 248
Book Description
Estimating parameters in a mixture of normal distributions dates back to the 19th century when Pearson originally considered data of crabs from the Bay of Naples. Since then, many real world applications of mixtures have led to various proposed methods for studying similar problems. Among them, maximum likelihood estimation (MLE) and the continuous empirical characteristic function (CECF) methods have drawn the most attention. However, the performance of these competing estimation methods has not been thoroughly studied in the literature and conclusions have not been consistent in published research. In this article, we review this classical problem with a focus on estimation bias. An extensive simulation study is conducted to compare the estimation bias between the MLE and CECF methods over a wide range of disparity values. We use the overlapping coefficient (OVL) to measure the amount of disparity, and provide a practical guideline for estimation quality in mixtures of normal distributions. Application to an ongoing multi-site Huntington disease study is illustrated for ascertaining cognitive biomarkers of disease progression. We also study joint modeling of longitudinal and time-to-event data and discuss pattern-mixture and selection models, but focus on shared parameter models, which utilize unobserved random effects in order to "join" a marginal longitudinal data model and marginal survival model in order to assess an internal time-dependent covariate's effect on time-to-event. The marginal models used in the analysis are the Cox Proportional Hazards model and the Linear Mixed model, and both of these models are covered in some detail before defining joints models and describing the estimation process. Joint modeling provides a modeling framework which accounts for correlation between the longitudinal data and the time-to-event data, while also accounting for measurement error in the longitudinal process, which previous methods failed to do. Since it has been shown that bias is incurred, and this bias is proportional to the amount of measurement error, utilizing a joint modeling approach is preferred. Our setting is also complicated by monotone degeneration of the internal covariate considered, and so a joint model which utilizes monotone B-Splines to recover the longitudinal trajectory and a Cox Proportional Hazards (CPH) model for the time-to-event data is proposed. The monotonicity constraints are satisfied via the Projected Newton Raphson Algorithm as described by Cheng et al., 2012, with the baseline hazard profiled out of the $Q$ function in each M-step of the Expectation Maximization (EM) algorithm used for optimizing the observed likelihood. This method is applied to assess Total Motor Score's (TMS) ability to predict Huntington Disease motor diagnosis in the Biological Predictors of Huntington's Disease study (PREDICT-HD) data.
Estimation of the Parameters of Mixtures Via Distance Between Densities Or Characteristic Functions
Author: J. L. BryantPublisher:
ISBN:
Category :
Languages : en
Pages : 98
Book Description
The integrated weighted distance between the sample characteristic function and the assumed characteristic function, or equivalently, the integrated distance between the smoothed assumed density and its kernel-estimate, is shown to be affective procedure for estimation of mixing proportions and for estimating all parameters of a modified compound Poisson distribution. These procedures are compared against their competitors in terms of efficiency, mean square error, and computational time. The characteristic function-based procedures are generally superior in terms of computation time for each of two types of procedures. The procedure introduced for the modified compound distribution is widely applicable since it is basically nonlinear modified x squared minimum. THe role of the sampling interval in estimating the parameters of the modified compound distribution is discussed and recommendations are made. Information matrices associated with this distribution are given for a spectrum of parameter values.
Finite Mixture Models
Author: Geoffrey McLachlanPublisher: John Wiley & Sons
ISBN: 047165406X
Category : Mathematics
Languages : en
Pages : 419
Book Description
An up-to-date, comprehensive account of major issues in finitemixture modeling This volume provides an up-to-date account of the theory andapplications of modeling via finite mixture distributions. With anemphasis on the applications of mixture models in both mainstreamanalysis and other areas such as unsupervised pattern recognition,speech recognition, and medical imaging, the book describes theformulations of the finite mixture approach, details itsmethodology, discusses aspects of its implementation, andillustrates its application in many common statisticalcontexts. Major issues discussed in this book include identifiabilityproblems, actual fitting of finite mixtures through use of the EMalgorithm, properties of the maximum likelihood estimators soobtained, assessment of the number of components to be used in themixture, and the applicability of asymptotic theory in providing abasis for the solutions to some of these problems. The author alsoconsiders how the EM algorithm can be scaled to handle the fittingof mixture models to very large databases, as in data miningapplications. This comprehensive, practical guide: * Provides more than 800 references-40% published since 1995 * Includes an appendix listing available mixture software * Links statistical literature with machine learning and patternrecognition literature * Contains more than 100 helpful graphs, charts, and tables Finite Mixture Models is an important resource for both applied andtheoretical statisticians as well as for researchers in the manyareas in which finite mixture models can be used to analyze data.
Applications of Characteristic Functions
Author: Eugene LukacsPublisher:
ISBN:
Category : Characteristic functions
Languages : en
Pages : 208
Book Description
Empirical Bayes Methods with Applications
Author: J.S. MaritzPublisher: CRC Press
ISBN: 1351080113
Category : Mathematics
Languages : en
Pages : 296
Book Description
The second edition of Empirical Bayes Methods details are provided of the derivation and the performance of empirical Bayes rules for a variety of special models. Attention is given to the problem of assessing the goodness of an empirical Bayes estimator for a given set of prior data. A chapter is devoted to a discussion of alternatives to the empirical Bayes approach and there is also a chapter giving details of several actual applications of empirical Bayes method.
Empirical Bayes Methods
Author: J. S. MaritzPublisher: Routledge
ISBN: 1351140639
Category : Business & Economics
Languages : en
Pages : 299
Book Description
Originally published in 1970; with a second edition in 1989. Empirical Bayes methods use some of the apparatus of the pure Bayes approach, but an actual prior distribution is assumed to generate the data sequence. It can be estimated thus producing empirical Bayes estimates or decision rules. In this second edition, details are provided of the derivation and the performance of empirical Bayes rules for a variety of special models. Attention is given to the problem of assessing the goodness of an empirical Bayes estimator for a given set of prior data. Chapters also focus on alternatives to the empirical Bayes approach and actual applications of empirical Bayes methods.
The Normal Distribution
Author: Wlodzimierz BrycPublisher: Springer Science & Business Media
ISBN: 1461225604
Category : Mathematics
Languages : en
Pages : 142
Book Description
This book is a concise presentation of the normal distribution on the real line and its counterparts on more abstract spaces, which we shall call the Gaussian distributions. The material is selected towards presenting characteristic properties, or characterizations, of the normal distribution. There are many such properties and there are numerous rel evant works in the literature. In this book special attention is given to characterizations generated by the so called Maxwell's Theorem of statistical mechanics, which is stated in the introduction as Theorem 0.0.1. These characterizations are of interest both intrin sically, and as techniques that are worth being aware of. The book may also serve as a good introduction to diverse analytic methods of probability theory. We use characteristic functions, tail estimates, and occasionally dive into complex analysis. In the book we also show how the characteristic properties can be used to prove important results about the Gaussian processes and the abstract Gaussian vectors. For instance, in Section 5.4 we present Fernique's beautiful proofs of the zero-one law and of the integrability of abstract Gaussian vectors. The central limit theorem is obtained via characterizations in Section 7.3.
Estimating Functions
Author: V. P. GodambePublisher: Oxford University Press on Demand
ISBN: 9780198522287
Category : History
Languages : en
Pages : 344
Book Description
This volume comprises a comprehensive collection of original papers on the subject of estimating functions. It is intended to provide statisticians with an overview of both the theory and the applications of estimating functions in biostatistics, stochastic processes, and survey sampling. From the early 1960s when the concept of optimality criterion was first formulated, together with the later work on optimal estimating functions, this subject has become both an active research area in its own right and also a cornerstone of the modern theory of statistics. Individual chapters have been written by experts in their respective fields and as a result this volume will be an invaluable reference guide to this topic as well as providing an introduction to the area for non-experts.