Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality

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ISBN: 9781584884316 | 1584884312
Cover: Hardcover
Copyright: 7/27/2004

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This book reviews the state-of-the-art advances in skew-elliptical distributions and provides many new developments in a single volume, collecting theoretical results and applications previously scattered throughout the literature. The main goal of this research area is to develop flexible parametric classes of distributions beyond the classical normal distribution. The book is divided into two parts. The first part discusses theory and inference for skew-elliptical distribution. The second part examines applications and case studies, including areas such as economics, finance, oceanography, climatology, environmetrics, engineering, image processing, astronomy, and biomedical science.

I Theory and Inference

(172)

The Skew-Normal Distribution

(22)

Alessandra Dalla Valle

Introduction

(1)

The Univariate Skew-Normal Distribution

(5)

Moments

(1)

Cumulative Distribution Function

(1)

Estimation and Inference

(4)

Checking the Hypothesis of Skew-Normality

(1)

The Multivariate Skew-Normal Distribution

(5)

Genesis

(1)

The Cumulative Distribution Function

(1)

The Moment Generating Function

(2)

Extensions of Properties Holding in the Scalar Case

(1)

Quadratic Forms

(2)

The Conditional Distribution

(1)

Miscellanea

(1)

Future Research Problems

(1)

The Closed Skew-Normal Distribution

(18)

Graciela Gonzalez-Farias

J. Armando Dominguez-Molina

Arjun K. Gupta

Introduction

(1)

Basic Results on the Multivariate CSN Distribution

(3)

Linear Transformations

(4)

Joint Distribution of Independent CSN Random Vectors

(2)

Sums of Independent CSN Random Vectors

(2)

Examples of Sums of Skew-Normal Random Vectors

(3)

Azzalini and Dalla Valle (1996)

(1)

Arnold and Beaver (2002)

(1)

Liseo and Loperfido (2003a)

(1)

A Multivariate Extended Skew-Elliptical Distribution

(1)

Concluding Remarks

(2)

Skew-Elliptical Distributions

(22)

Junfeng Liu

Dipak K. Dey

Introduction

(1)

General Multivariate Skew-Elliptical Distributions

(6)

The Branco and Dey Approach

(1)

The Arnold and Beaver Approach

(1)

The Wang-Boyer-Genton Approach

(1)

The Dey and Liu Approach

(1)

Linear Constraint and Linear Combination of Type-1 (LCLC1)

(1)

Linear Constraint and Linear Combination of Type-2 (LCLC2)

(1)

Examples of Skew-Elliptical Distributions

(6)

Skew-Scale Mixture of Normal Distribution

(1)

Skew-Finite Mixture of Normal

(1)

Skew-Logistic Distribution

(1)

Skew-Stable Distribution

(1)

Skew-Exponential Power Distribution

(1)

Skew-t Distribution

(1)

Skew-Pearson Type II Distribution

(1)

Different Types of Multivariate Skew-Normal Distributions

(1)

Type A-SN(μ, Ω, α)

(1)

Type B-SN(μ, Ω, D)

(1)

The Liseo and Loperfido Class of SN

(1)

The Dominguez-Molina-Gonzalez-Farias-Gupta Class of SN

(1)

Skew-Elliptical Distribution SEk(μ, Ω, δ, g (k+1)

(1)

Skew-Elliptical Distribution SEm(μ, Σ, D, g(m)

(1)

Some Properties of Skew-Elliptical Distributions

(6)

Moment Generating Functions

(1)

Skew-Scale Mixture of Normal Distributions

(2)

LCLC1

(1)

LCLC2

(1)

Marginal and Conditional Closure Property

(1)

Distribution of Quadratic Forms

(1)

LCLC1

(1)

LCLC2

(1)

Quadratic Forms from LCLC1/LCLC2

(2)

Other Distributional Properties

(1)

Properties of the Branco-Dey Skew-Elliptical Models

(1)

Density Shape of Univariate Skew-Elliptical Distributions

(1)

Concluding Remarks

(3)

Generalized Skew-Normal Distributions

(16)

Nicola M. R. Loperfido

Introduction

(1)

Definition and Characterization

(3)

Transformations and Moments

(1)

Diagnostics

(3)

Parametric Inference

(2)

Australian Athletes Data

(2)

Concluding Remarks

(1)

Appendix: Proofs

(4)

Skew-Symmetric and Generalized Skew-Elliptical Distributions

(20)

Marc G. Genton

Introduction

(1)

Skew-Symmetric Distributions

(6)

Invariance

(3)

Stochastic Representation and Simulations

(1)

Skew-Symmetric Representation of Multivariate Distributions

(1)

Example: Intensive Care Unit Data

(2)

Generalized Skew-Elliptical Distributions

(2)

Flexible Skew-Symmetric Distributions

(10)

Flexibility and Multimodality

(2)

Example: Australian Athletes Data

(2)

Locally Efficient Semiparametric Estimators

(5)

Concluding Remarks

100

(1)

Elliptical Models Subject to Hidden Truncation or Selective Sampling

101

(12)

Barry C. Arnold

Robert J. Beaver

Introduction

101

(1)

Univariate Skew-Normal Models

101

(2)

Estimation for the Skew-Normal Distribution

103

(1)

Other Univariate Skewed Distributions

104

(1)

Multivariate Skewed Distributions

105

(2)

General Multivariate Skewed Distributions

107

(2)

Hidden Truncation Paradigm

107

(1)

Hidden Truncation, More General

107

(1)

Additive Component Paradigm

108

(1)

Additive Component, More General

108

(1)

The Jones Construction

108

(1)

Skew-Elliptical Distributions

109

(3)

Discussion

112

(1)

From Symmetric to Asymmetric Distributions: A Unified Approach

113

(18)

Reinaldo B. Arellano-Valle

Guido E. del Pino

Introduction

113

(2)

Signs, Absolute Values, and Skewed Distributions

115

(2)

Latent Variables, Selection Models, Skewed Distributions

117

(1)

Symmetry, Invariance, and Skewness

118

(3)

Glossary and Basic Facts

118

(1)

Three Groups of Transformations

119

(1)

Conditional Representations

119

(1)

Density or Probability Functions for the Maximal Invariant

120

(1)

The SI Class of Sign Invariant Distributions

121

(3)

Examples and Main Results

121

(2)

Application to the Density Formula for a Skewed Distribution

123

(1)

A Stochastic Representation Associated with the SI Class

124

(2)

Moments of a Multivariate Skewed Distribution

125

(1)

Distribution of the Square of a Skewed Random Variable

126

(1)

Application to the Multivariate Skew-Normal Distribution

126

(2)

A Canonical Form for Skew-Elliptical Distributions

128

(3)

Skewed Link Models for Categorical Response Data

131

(22)

Ming-Hui Chen

Introduction

131

(1)

Preliminaries

132

(2)

Importance of Links in Fitting Categorical Response Data

134

(3)

Relationship between Regression Coefficients under Different Links

134

(1)

Prediction under Different Links

135

(2)

General Skewed Link Models

137

(6)

Independent Binary and/or Ordinal Regression Models

137

(3)

Correlated Binary and/or Ordinal Regression Models

140

(2)

Discrete Choice Models

142

(1)

Bayesian Inference

143

(1)

Bayesian Model Assessment

144

(4)

Weighted L Measure

144

(3)

Conditional Predictive Ordinate

147

(1)

Deviance Information Criterion

148

(1)

Bayesian Model Diagnostics and Outlier Detection

148

(3)

Bayesian Latent Residuals

149

(1)

Bayesian CPO-Based Residuals

149

(1)

Observationwise Weighted L Measure

150

(1)

Concluding Remarks

151

(2)

Skew-Elliptical Distributions in Bayesian Inference

153

(20)

Brunero Liseo

Introduction

153

(1)

Skewed Prior Distributions for Location Parameters

154

(4)

Hierarchical Models with Linear Constraints

154

(2)

Efficiency of Linear Bayes Rules with Skewed Priors

156

(1)

Heavy Tail Priors

157

(1)

Skew-Elliptical Likelihood

158

(5)

Inferential Problems

159

(1)

Regression Models with SE Errors

160

(2)

Some Applications

162

(1)

Objective Bayesian Analysis of the Skew-Normal Model

163

(6)

The Scalar Case

164

(2)

Some Multivariate Results

166

(3)

Appendix

169

(4)

II Applications and Case Studies

173

(186)

Bayesian Multivariate Skewed Regression Modeling with an Application to Firm Size

175

(16)

Jose T. A. S. Ferreira

Mark F. J. Steel

Introduction

175

(1)

Multivariate Skewed Distributions

176

(3)

FS Skewed Distributions

177

(1)

SDB Skewed Distributions

178

(1)

Regression Models

179

(2)

Prior Distributions

179

(1)

Numerical Implementation

180

(1)

Application to Firm Size

181

(8)

Distribution of Firm Size

183

(3)

Analysis of Firm Growth

186

(3)

Discussion

189

(2)

Capital Asset Pricing for UK Stocks under the Multivariate Skew-Normal Distribution

191

(14)

Chris Adcock

Introduction

191

(3)

The Multivariate Skew-Normal Model

194

(2)

The Market Model

196

(2)

Estimation Methodology and Data

198

(1)

Empirical Study

199

(5)

Summary and Conclusions

204

(1)

A Skew-in-Mean GARCH Model

205

(18)

Giovanni De Luca

Nicola M. R. Loperfido

Introduction

205

(2)

Assumptions

207

(1)

News

208

(2)

Returns

210

(1)

Data

211

(2)

Estimation

213

(2)

Conclusions

215

(1)

Appendix

216

(7)

Skew-Normality in Stochastic Frontier Analysis

223

(20)

J. Armando Dominguez-Molina

Graciela Gonzalez-Farias

Rogelio Ramos-Quiroga

Introduction

223

(2)

Estimation

225

(5)

Model Assumptions

226

(1)

Model I: Homoscedastic and Uncorrelated Errors

227

(1)

Model II: Heteroscedastic and Uncorrelated Errors

227

(1)

Model III: Correlated Errors

227

(1)

Likelihood

228

(1)

Estimation of Inefficiencies/Efficiencies

229

(1)

A Correlated Structure for the Compound Error

230

(4)

Simulated Example with Correlated Compound Errors

231

(3)

SFA with Skew-Elliptical Components

234

(1)

Conclusions

235

(1)

Appendix

236

(7)

Distributional Properties of Multivariate Compound Errors

236

(2)

Expectation of the Truncated Multivariate Normal Distribution

238

(1)

Efficiencies for Individual Errors of Model III

239

(4)

Coastal Flooding and the Multivariate Skew-t Distribution

243

(16)

Keith R. Thompson

Yingshuo Shen

Introduction

243

(1)

A Seasonally Varying Skew-t Distribution

244

(3)

Observations of Coastal Sea Level

247

(3)

Fitting the Skew-t Distribution

250

(3)

Applications of the Skew-t Distribution

253

(5)

Quality Control of Sea Level Observations

253

(1)

Detecting Secular Changes in the Sea Level Distribution

254

(1)

Estimating Flooding Risk

255

(3)

Discussion

258

(1)

Time Series Analysis with a Skewed Kalman Filter

259

(20)

Philippe Naveau

Marc G. Genton

Caspar Ammann

Introduction

259

(1)

The Classical Kalman Filter

260

(2)

State-Space Model

260

(1)

The Kalman Filter Procedure in the Gaussian Case

261

(1)

A Skewed Kalman Filter

262

(6)

The Closed Skew-Normal Distribution

262

(1)

Extension of the Linear Gaussian State-Space Model

263

(2)

The Steps of our Skewed Kalman Filter

265

(3)

Applications to Paleoclimate Time Series

268

(6)

Multi-Process Linear Models

269

(1)

The Smoothing Spline Model for Trends

269

(1)

The Skewed Component

270

(3)

The State-Space Model

273

(1)

Simulations and Results

273

(1)

Conclusions

274

(1)

Appendix

275

(4)

Spatial Prediction of Rainfall Using Skew-Normal Processes

279

(12)

Hyoung-Moon Kim

Eunho Ha

Bani K. Mallick

Introduction

279

(1)

Data and Model

280

(4)

Automatic Weather Stations and Their Sensors

280

(1)

Model Description

280

(2)

Bayesian Analysis

282

(2)

Data Analyses

284

(4)

Discussion

288

(3)

Shape Representation with Flexible Skew-Symmetric Distributions

291

(18)

Sajjad H. Baloch

Hamid Krim

Marc G. Genton

Introduction

291

(1)

Problem Statement

292

(2)

Shape Analysis

294

(8)

Posterior Learning

296

(3)

Selection of a Distribution for the Angle

299

(1)

Overall Shape Distribution

300

(1)

Performance Assessment of the Learning Process

301

(1)

Experimental Results

302

(5)

Case Study 1

303

(1)

Case Study 2

304

(1)

Case Study 3

304

(1)

Case Study 4

305

(1)

Case Study 5

305

(1)

Sampling from Models

306

(1)

Conclusions

307

(1)

Appendix

308

(1)

An Astronomical Distance Determination Method Using Regression with Skew-Normal Errors

309

(12)

Laurent Eyer

Marc G. Genton

Introduction

309

(1)

The Trigonometric Parallax

310

(2)

Astrometric Satellites

311

(1)

Some Statistical Aspects

311

(1)

Parallax Is a Positive Quantity

311

(1)

Parent Distribution

312

(1)

Formation of Samples

312

(1)

Famous Standard Candles: the Cepheids

312

(2)

Calibration of the Period-Luminosity Relation

314

(7)

Regression with Skew-Normal Errors

315

(1)

Discussion

316

(1)

Fixed Slope

316

(1)

Free Slope

317

(4)

On a Bayesian Multivariate Survival Model with a Skewed Frailty

321

(18)

Sujit K. Sahu

Dipak K. Dey

Introduction

321

(2)

Frailty Models

323

(3)

Frailty Distributions

323

(2)

Comparison of Frailty Distributions

325

(1)

Baseline Hazard Function

326

(1)

Model Specification

327

(3)

Likelihood Specification

327

(1)

Prior Specification

328

(1)

Hyper-Parameter Values and Prior Sensitivity

329

(1)

Reversible Jump Steps

330

(2)

Examples

332

(4)

Kidney Infection Data Example

332

(2)

Litters Example

334

(2)

Conclusion

336

(3)

Linear Mixed Effects Models with Flexible Generalized Skew-Elliptical Random Effects

339

(20)

Yanyuan Ma

Marc G. Genton

Marie Davidian

Introduction

339

(1)

FGSE Distributions and the Linear Mixed Effects Model

340

(3)

Implementation and Inference

343

(7)

Maximum Likelihood via the EM Algorithm

343

(3)

Bayesian Inference via Markov Chain Monte Carlo Simulation

346

(4)

Simulation Results

350

(6)

Application to Cholesterol Data

356

(2)

Discussion

358

(1)

References

359

(18)

Index

377

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Skew-Elliptical Distributions and Their Applications: A Journey Beyond Normality

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