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E-raamat: Elliptically Contoured Models in Statistics and Portfolio Theory

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 07-Sep-2013
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781461481546
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Elliptically Contoured Models in Statistics and Portfolio TheorySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 07-Sep-2013
  • Kirjastus: Springer-Verlag New York Inc.
  • Keel: eng
  • ISBN-13: 9781461481546
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Elliptically Contoured Models in Statistics and Portfolio Theory fully revises the first detailed introduction to the theory of matrix variate elliptically contoured distributions. There are two additional chapters, and all the original chapters of this classic text have been updated. Resources in this book will be valuable for researchers, practitioners, and graduate students in statistics and related fields of finance and engineering. Those interested in multivariate statistical analysis and its application to portfolio theory will find this text immediately useful. In multivariate statistical analysis, elliptical distributions have recently provided an alternative to the normal model. Elliptical distributions have also increased their popularity in finance because of the ability to model heavy tails usually observed in real data. Most of the work, however, is spread out in journals throughout the world and is not easily accessible to the investigators. A noteworthy function of this book is the collection of the most important results on the theory of matrix variate elliptically contoured distributions that were previously only available in the journal-based literature. The content is organized in a unified manner that can serve an a valuable introduction to the subject.

 

Arvustused

From the book reviews:

The book is a thorough presentation of the results from the literature and some new ones and is a product of a careful work of its authors. the book under review is a useful collection of numerous facts concerning elliptically contoured distributions for random matrices. It may find its place in research libraries as a valuable reference source and will surely appeal to the researches working in the theory of multivariate distributions. (Ilya S. Molchanov, zbMATH 1306.62028, 2015)

Part I Preliminaries
1 Preliminaries
3(12)
1.1 Introduction and Literature Review
3(1)
1.2 Some Results from Matrix Algebra
4(7)
1.3 A Functional Equation
11(4)
Part II Definition and Distributional Properties
2 Basic Properties
15(44)
2.1 Definition
15(5)
2.2 Probability Density Function
20(3)
2.3 Marginal Distributions
23(1)
2.4 Expected Value and Covariance
24(2)
2.5 Stochastic Representation
26(8)
2.6 Conditional Distributions
34(15)
2.7 Examples
49(10)
2.7.1 One-Dimensional Case
49(1)
2.7.2 Vector Variate Case
50(3)
2.7.3 General Matrix Variate Case
53(3)
2.7.4 Generating Elliptically Contoured Distributions
56(3)
3 Probability Density Function and Expected Values
59(44)
3.1 Probability Density Function
59(7)
3.2 More on Expected Values
66(37)
4 Mixtures of Normal Distributions
103(22)
4.1 Mixture by Distribution Function
103(13)
4.2 Mixture by Weighting Function
116(9)
5 Quadratic Forms and Other Functions of Elliptically Contoured Matrices
125(20)
5.1 Extension of Cochran's Theorem to Multivariate Elliptically Contoured Distributions
125(12)
5.2 Rank of Quadratic Forms
137(2)
5.3 Distributions of Invariant Matrix Variate Functions
139(6)
6 Characterization Results
145(28)
6.1 Characterization Based on Invariance
145(3)
6.2 Characterization of Normality
148(25)
Part III Estimation and Hypothesis Testing
7 Estimation
173(20)
7.1 Maximum Likelihood Estimators of the Parameters
173(10)
7.2 Properties of Estimators
183(10)
8 Hypothesis Testing
193(26)
8.1 General Results
193(5)
8.2 Two Models
198(3)
8.2.1 Model I
198(1)
8.2.2 Model II
199(2)
8.3 Testing Criteria
201(18)
8.3.1 Testing That a Mean Vector Is Equal to a Given Vector
201(1)
8.3.2 Testing That a Covariance Matrix Is Equal to a Given Matrix
202(1)
8.3.3 Testing That a Covariance Matrix Is Proportional to a Given Matrix
203(1)
8.3.4 Testing That a Mean Vector and Covariance Matrix Are Equal to a Given Vector and Matrix
204(2)
8.3.5 Testing That a Mean Vector Is Equal to a Given Vector and a Covariance Matrix Is Proportional to a Given Matrix
206(2)
8.3.6 Testing Lack of Correlation Between Sets of Variates
208(1)
8.3.7 Testing That a Correlation Coefficient Is Equal to a Given Number
209(2)
8.3.8 Testing That a Partial Correlation Coefficient Is Equal to a Given Number
211(1)
8.3.9 Testing That a Multiple Correlation Coefficient Is Equal to a Given Number
212(1)
8.3.10 Testing Equality of Means
213(1)
8.3.11 Testing Equality of Covariance Matrices
214(1)
8.3.12 Testing Equality of Means and Covariance Matrices
215(4)
Part IV Applications
9 Linear Models
219(18)
9.1 Estimation of the Parameters in the Multivariate Linear Regression Model
219(9)
9.2 Hypothesis Testing in the Multivariate Linear Regression Model
228(3)
9.3 Inference in the Random Effects Model
231(6)
10 Application in Portfolio Theory
237(36)
10.1 Elliptically Contoured Distributions in Portfolio Theory
237(2)
10.2 Estimation of the Global Minimum Variance
239(10)
10.2.1 Distribution of the Global Minimum Variance
240(3)
10.2.2 Examples
243(2)
10.2.3 Determination of the Elliptical Model
245(2)
10.2.4 Tests for the Global Minimum Variance
247(2)
10.3 Test for the Weights of the Global Minimum Variance Portfolio
249(9)
10.3.1 Distribution of the Estimated Weights of the Global Minimum Variance Portfolio
250(1)
10.3.2 The General Linear Hypothesis for the Global Minimum Variance Portfolio Weights
251(5)
10.3.3 The General Linear Hypothesis for the Global Minimum Variance Portfolio Weights in an Elliptical Model
256(2)
10.3.4 International Global Minimum Variance Portfolio
258(1)
10.4 Inference for the Markowitz Efficient Frontier
258(15)
10.4.1 Derivation of the Efficient Frontier
259(2)
10.4.2 Sample Efficient Frontier in Elliptical Models
261(1)
10.4.3 Confidence Region for the Efficient Frontier
262(4)
10.4.4 Unbiased Estimator of the Efficient Frontier
266(2)
10.4.5 Overall F-Test
268(1)
10.4.6 Empirical Illustration
269(4)
11 Skew Elliptically Contoured Distributions
273(30)
11.1 Skew Normal Distribution
273(4)
11.2 Matrix Variate Skew Normal Distribution
277(4)
11.2.1 Basic Properties
279(2)
11.3 Quadratic Forms of the Matrix Variate Skew Normal Distributions
281(1)
11.4 A Multivariate Stochastic Frontier Model
282(2)
11.5 Global Minimum Variance Portfolio Under the Skew Normality
284(13)
11.5.1 Model Estimation
287(3)
11.5.2 Goodness-of-Fit Test
290(4)
11.5.3 Application to Some Stocks in the Dow Jones Index
294(3)
11.6 General Class of the Skew Elliptically Contoured Distributions
297(6)
11.6.1 Multivariate Skew t-Distribution
298(3)
11.6.2 Multivariate Skew Cauchy Distribution
301(2)
References 303(10)
Author Index 313(4)
Subject Index 317
Arjun Gupta is affiliated with the Department of Statistics at Bowling Green State University. Tamas Varga is a statistical researcher in Hungary. Taras Bodnar is affiliated with the Department of Statistics at Stiftung European University Viadrina.
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