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E-raamat: Advances on Theoretical and Methodological Aspects of Probability and Statistics

Edited by N. Balakrishnan (McMaster University, Hamilton, Ontario, Canada)

Formaat: 560 pages
Ilmumisaeg: 30-Apr-2019
Kirjastus: CRC Press Inc
Keel: eng
ISBN-13: 9781135465100

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Formaat: 560 pages
Ilmumisaeg: 30-Apr-2019
Kirjastus: CRC Press Inc
Keel: eng
ISBN-13: 9781135465100

Teised raamatud teemal:

Business & management

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At the International Indian Statistical Association Conference, held at McMaster University in Ontario, Canada, participants focused on advancements in theory and methodology of probability and statistics. This is one of two volumes containing invited papers from the meeting. The 32 chapters deal with different topics of interest, including stochastic processes and inference, distributions and characterizations, inference, Bayesian inference, selection methods, regression methods, and methods in health research. The text is ideal for applied mathematicians, statisticians, and researchers in the field.

Preface

xix

List of Contributors

xxi

List of Tables

xxvii

List of Figures

xxix

Part I Stochastic Processes and Inference

Nonlinear Filtering with Stochastic Delay Equations

(34)

G. Kallianpur

Pranab Kumar Mandal

Introduction

(2)

Preliminaries

(2)

Stochastic Delay Differential Equations

(13)

The Filtering Problem

(11)

Zakai Equation and Uniqueness

(6)

References

(2)

Sigma Oscillatory Processes

(12)

Randall J. Swift

Some Classes of Nonstationary Processes

(4)

Sigma Oscillatory Processes

(3)

Determination of the Evolutionary Spectra

(5)

References

(3)

Some Properties of Harmonizable Processes

(8)

Marc H. Mehlman

Introduction

(2)

Incremental Processes

(1)

Moments of Harmonizable Processes

(2)

Virile Representations

(3)

References

(1)

Inference for Branching Processes

(18)

V. Basawa

Introduction

(1)

Galton-Watson Branching Process: Background

(1)

Locally Asymptotic Mixed Normal (Lamn) Family

(1)

G-W Branching Process as a Proto-Type Example of a Lamn Model

(1)

Estimation Efficiency

(1)

Test Efficiency

(2)

Confidence Bounds

(1)

Conditional Inference

(1)

Prediction and Test of Fit

(1)

Quasilikelihood Estimation

(1)

Bayes and Empirical Bayes Estimation

(2)

Concluding Remarks

(5)

References

(5)

Part II Distributions and Characterizations

The Conditional Distribution of X Given X = Y Can Be Almost Anything!

(8)

B. C. Arnold

C. A. Robertson

Introduction

(1)

The Distribution of X Given X = Y Can be Almost Anything

(2)

Dependent Variables

(1)

Related Examples

(4)

References

(2)

An Application of Record Range and Some Characterization Results

(14)

P. Basak

Introduction

(2)

The Stopping Time N

(6)

The Mean and the Variance of N

(4)

Behavior for Large c: Almost Sure Limits

(2)

Characterization Results

(6)

References

(2)

Contents of Random Simplices and Random Parallelotopes

(20)

A. M. Mathai

Introduction

(10)

Some Basic Results from Linear Algebra

(4)

Some Basic Results on Jacobians of Matrix Transformations

102

(2)

Some Practical Situations

104

(3)

Distribution of the Volume or Content of a Random Parallelotope in Rn

107

(4)

Matrix-Variate Type-1 Beta Distribution

109

(1)

Matrix-Variate Type-2 Beta Density

110

(1)

Spherically Symmetric Distributions

111

(2)

Arrival of Points by a Poisson Process

113

(4)

References

114

(3)

The Distribution of Functions of Elliptically Contoured Vectors in Terms of Their Gaussian Counterparts

117

(14)

Young-Ho Cheong

Serge B. Provost

Introduction and Notation

117

(2)

A Representation of the Density Function of Elliptical Vectors

119

(1)

The Exact Distribution of Quadratic Forms

120

(3)

Moments and Approximate Distribution

123

(2)

A Numerical Example

125

(6)

References

126

(5)

Inverse Normalizing Transformations and an Extended Normalizing Transformation

131

(16)

Haim Shore

Introduction

132

(1)

Derivation of the Transformations

133

(3)

Numerical Assessment

136

(1)

Estimation

137

(2)

Conclusions

139

(8)

References

140

(7)

Curvature: Gaussian or Riemann

147

(16)

William Chen

Definition of the Gaussian Curvature

147

(3)

Examples

150

(3)

Some Basic Properties of Gaussian Curvature

153

(4)

Applications of the Gauss Equations

157

(6)

References

158

(5)

Part III Inference

Convex Geometry, Asymptotic Minimaxity and Estimating Functions

163

(10)

Schultz Chan

Malay Ghosh

Introduction

163

(1)

A Convexity Result

164

(2)

Asymptotic Minimaxity

166

(7)

Appendix

170

(1)

References

171

(2)

Nonnormal Filtering Via Estimating Functions

173

(12)

A. Thavaneswaran

M. E. Thompson

Introduction

173

(2)

Linear and Nonlinear Filters

175

(4)

Optimal Combination Extension

177

(2)

Applications to State Space Models

179

(6)

Linear State Space Models

179

(1)

Generalized Nonnormal Filtering

180

(1)

Robust Estimation Filtering Equations

180

(1)

Censored Autocorrelated Data

181

(1)

References

182

(3)

Recent Developments in Conditional-Frequentist Sequential Testing

185

(14)

B. Boukai

Introduction

185

(2)

The Setup

187

(1)

The `Conventional' Approaches

188

(3)

The New Conditional Sequential Test

191

(2)

An Application

193

(6)

References

196

(3)

Some Remarks on Generalizations of the Likelihood Function and the Likelihood Principle

199

(14)

Tapan K. Nayak

Subrata Kundu

Introduction

199

(3)

A General Framework

202

(2)

Sufficiency and Weak Conditionality

204

(3)

The Sufficiency Principle

204

(2)

Weak Conditionality

206

(1)

The Likelihood Principle

207

(3)

Discussion

210

(3)

References

211

(2)

Cusum Procedures for Detecting Changes in the Tail Probability of A Normal Distribution

213

(12)

Rasul A. Khan

Introduction

213

(1)

A Shewhart Chart and a Cusum Scheme

214

(2)

Noncentral t-Statistics Based Cusum Procedures

216

(5)

Simulations

221

(4)

References

223

(2)

Detecting Changes in the Von Mises Distribution

225

(14)

Kaushik Ghosh

Introduction

225

(2)

The Tests

227

(3)

Change in κ,μ Fixed and Known

227

(1)

Change in κ,μ Fixed but Unknown

228

(1)

Change in μ or κ or Both

229

(1)

Simulation Results

230

(1)

Power Comparisons

231

(1)

An Example

232

(7)

References

233

(6)

One-Way Random Effects Model with A Covariate: Nonnegative Estimators

239

(8)

Poduri S. R. S. Rao

Introduction

239

(1)

Ancova Estimator and its Modification

240

(2)

Ancova Estimator

240

(1)

Adjustment for Nonnegativeness

241

(1)

The Minqe and a Modification

242

(1)

An Estimator Derived from the Mivque Procedure

242

(1)

Special Cases of the Estimator

243

(1)

Comparison of the Estimators

243

(4)

References

245

(2)

On a Two-Stage Procedure with Higher Than Second-Order Approximations

247

(32)

N. Mukhopadhyay

Introduction

247

(2)

General Formulation and Main Results

249

(6)

Proofs of the Main Results

255

(10)

Proof of Theorem 18.2.1

256

(1)

Auxiliary Lemmas

257

(7)

Proof of Theorem 18.2.2

264

(1)

Proof of Theorem 18.2.3

264

(1)

Applications of the Main Results

265

(9)

Negative Exponential Location Estimation

266

(2)

Multivariate Normal Mean Vector Estimation

268

(2)

Linear Regression Parameters Estimation

270

(1)

Multiple Decision Theory

271

(3)

Concluding Thoughts

274

(5)

References

275

(4)

Bounded Risk Point Estimation of a Linear Function of K Multinormal Mean Vectors when Covariance Matrices are Unknown

279

(10)

M. Aoshima

Y. Takada

Introduction

279

(2)

Two-Stage Procedure

281

(1)

Asymptotic Properties

282

(7)

References

286

(3)

The Elusive and Illusory Multivariate Normality

289

(16)

G. S. Mudholkar

D. K. Srivastava

Introduction

290

(1)

Tests of Multivariate Normality

291

(3)

Dubious Normality of Some Well Known Data

294

(4)

Conclusions

298

(7)

References

298

(7)

Part IV Bayesian Inference

Characterizations of Tailfree and Neutral to the Right Priors

305

(12)

R. V. Ramamoorthi

L. Draghici

J. Dey

Introduction

305

(1)

Tailfree Priors

306

(4)

Neutral to Right Priors

310

(3)

Nr Priors From Censored Observations

313

(4)

References

315

(2)

Empirical Bayes Estimation and Testing For a Location Parameter Family of Gamma Distributions

317

(14)

N. Balakrishnan

Yimin Ma

Introduction

317

(1)

Bayes Estimator and Bayes Testing Rule

318

(2)

Bayes Estimation

318

(1)

Bayes Testing

319

(1)

Empirical Bayes Estimator and Empirical Bayes Testing

320

(1)

Empirical Bayes Estimator

320

(1)

Empirical Bayes Testing Rule

321

(1)

Asymptotic Optimality of the Empirical Bayes Estimator

321

(4)

Asymptotic Optimality of the Empirical Bayes Testing Rule

325

(6)

References

328

(3)

Rate of Convergence for Empirical Bayes Estimation of a Distribution Function

331

(14)

T. C. Liang

Introduction

331

(2)

The Empirical Bayes Estimators

333

(2)

Asymptotic Optimality

335

(10)

References

341

(4)

Part V Selection Methods

On a Selection Procedure for Selecting the Best Logistic Population Compared with a Control

345

(26)

S. S. Gupta

Z. Lin

X. Lin

Introduction

346

(1)

Formulation of the Selection Problem with the Selection Rule

347

(5)

Asymptotic Optimality of the Proposed Selection Procedure

352

(10)

Simulations

362

(9)

References

363

(8)

On Selection From Normal Populations in Terms of the Absolute Values of their Means

371

(20)

Khaled Hussein

S. Panchapakesan

Introduction

371

(2)

Some Preliminary Results

373

(1)

Indifference Zone Formulation: Known Common Variance

373

(2)

Subset Selection Formulation: Known Common Variance

375

(1)

Indifference Zone Formulation: Unknown Common Variance

376

(2)

Subset Selection Formulation: Unknown Common Variance

378

(1)

An Integrated Formulation

379

(1)

Simultaneous Selection of the Extreme Populations: Indifference Zone Formulation and Known Common Variance

380

(4)

Simultaneous Selection of the Extreme Populations: Subset Selection Formulation Known Common Variance

384

(2)

Concluding Remarks

386

(5)

References

387

(4)

A Selection Procedure Prior to Signal Detection

391

(16)

Pinyin Chen

Introduction

391

(1)

The Selection Procedure

392

(4)

Table, Simulation Study and An Example

396

(11)

References

398

(9)

Part VI Regression Methods

Tolerance Intervals and Calibration in Linear Regression

407

(20)

YI-TZU Lee

Thomas Mathew

Introduction

407

(3)

Tolerance Intervals, Simultaneous Tolerance Intervals and a Marginal Property

410

(4)

Numerical Results

414

(7)

The Simulation of (27.2.17) and (27.2.18)

416

(4)

An Example

420

(1)

Calibration

421

(1)

Conclusions

422

(5)

Appendix A: Some Fitted Functions k(c)

423

(2)

References

425

(2)

An Overview of Sequential and Multistage Methods in Regression Models

427

(24)

Sugary Datta

Introduction

427

(1)

The Models and the Methodologies---A General Discussion

428

(5)

Linear Regression and Related Models

429

(1)

Sequential and Multistage Methodologies

430

(2)

Sequential Inference in Regression: A Motivating Example

432

(1)

Fixed-Precision Inference in Deterministic Regression Models

433

(5)

Confidence Set Estimation

434

(2)

Point Estimation

436

(2)

Hypotheses Testing

438

(1)

Sequential Shrinkage Estimation In Regression

438

(1)

Bayes Sequential Inference in Regression

439

(1)

Sequential Inference in Stochastic Regression Models

440

(1)

Sequential Inference in Inverse Linear Regression and Errors-in-Variables Models

441

(1)

Some Miscellaneous Topics

442

(9)

References

443

(8)

Bayesian Inference for a Change-Point in Nonlinear Modeling

451

(22)

V. K. Jandhyala

J. A. Alsaleh

Introduction

451

(2)

Gibbs Sampler

453

(3)

Bayesian Preliminaries and the Nonlinear Change-Point Model

456

(1)

Bayesian Inferential Methods

457

(3)

Implementation and the Results

460

(13)

Appendix

463

(2)

References

465

(8)

Convergence to Tweedie Models and Related Topics

473

(20)

Bent Jørgensen

Vladimir Vinogradov

Introduction

474

(4)

Special Cases: Inverse Gaussian and Generalized Inverse Gaussian Distributions

478

(2)

Critical Points in the Formation of Large Deviations

480

(3)

Different Mechanisms of Ruin in Non-Life Insurance

483

(10)

References

486

(7)

Part VII Methods in Health Research

Estimation of Stage Occupation Probabilities in Multistage Models

493

(14)

Somnath Datta

Glen A. Satten

Susmita Datta

Introduction

494

(1)

The Fractional Risk Set Estimators

495

(5)

Variance Estimation

500

(2)

Extension to Multistage Models

502

(5)

References

503

(4)

Statistical Methods in the Validation Process of a Health Related Quality of Life Questionnaire: Classical and Modern Theory

507

(1)

Mounir Mesbah

Agnes Hamon

Introduction

507

(2)

Classical Psychometric Theory

509

(1)

The Strict Parallel Model

509

(1)

Reliability of an Instrument

510

(4)

Modern Psychometric Theory

514

(1)

The Rasch Model

515

(8)

Conclusion

523

(1)

References

523

(4)

Annex 1: Communication Dimension of The Sip (9 Items)

527

(1)

Annex 2: Social Interaction Dimension of the Sip (20 Items)

528

N. Balaskrishnan, McMaster University, Hamilton, Ontario, Canada

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