Parametric and Semiparametric Models With Applications to Reliability, Survival Analysis, and Quality of Life

, by Nikulin, M. S.; Balakrishnan, N.; Mesbah, M.; Limnios, N.

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ISBN: 9780817632311 | 081763231X
Cover: Hardcover
Copyright: 6/30/2004

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Parametric and semiparametric models are tools with a wide range of applications to reliability, survival analysis, and quality of life. This self-contained volume examines these tools in survey articles written by experts currently working on the development and evaluation of models and methods. While a number of chapters deal with general theory, several explore more specific connections and recent results in "real-world" reliability theory, survival analysis and related fields. Specific topics covered include: * non-parametric estimation of lifetimes of subjects exposed to radiation * statistical analysis of simultaneous degradation-mortality data with covariates of the aged * estimation of maintenance efficiency in semiparametric imperfect repair models * cancer prognosis using survival forests * short-term health problems related to air pollution: analysis using semiparametric generalized additive models * parametric models in accelerated life testing and fuzzy data * semiparametric models in the studies of aging and longevity This book will be of use as a reference text for general statisticians, theoreticians, graduate students, reliability engineers, health researchers, and biostatisticians working in applied probability and statistics.

Preface

xix

Catherine Huber-Carol: Career and Accomplishments

xxi

Contributors

xxix

List of Tables

xxxv

List of Figures

xxxix

PART I: COX MODELS AND ANALYSES

1 Estimation in Partly Parametric Additive Cox Models

(10)

H. Läuter

1.1 Introduction

(2)

1.2 Types of Regression Functions

(1)

1.2.1 Linear models

(1)

1.2.2 Nonlinear models

(1)

1.3 Random Covariates and Partial Likelihood

(3)

1.3.1 Linear case

(1)

1.4 Fix Covariates and Partial Likelihood

(1)

1.5 Least Squares Estimation

(3)

References

(1)

2 Nonparametric Maximum Likelihood Estimation in the Proportional Hazards Model with Covariate Measurement Error

(14)

J.-F. Dupuy

2.1 Introduction

(2)

2.2 Nonparametric Maximum Likelihood Estimation

(2)

2.3 Existence of the NPML Estimators

(3)

2.4 Consistency of the NPML Estimators

(2)

2.5 Discussion

(1)

References

(5)

3 Diagnostics for Cox's Proportional Hazards Model

(14)

C. Caroni

3.1 Introduction

(2)

3.2 Schoenfeld Residuals

(1)

3.3 Martingale Residuals

(2)

3.4 Using Residuals to Examine the PH Assumption

(1)

3.5 Influence

(1)

3.6 Added Variable Plots

(2)

3.7 Conclusions

(1)

References

(5)

PART II: DEGRADATION MODELS AND ANALYSES

4 Semiparametric Analysis of Degradation and Failure Time Data with Covariates

(24)

V. Bagdonavicius and Al. Nikulin

4.1 Introduction

(1)

4.2 Modeling the Degradation-Failu e Time Process

(2)

4.3 Survival and Degradation Characteristics

(2)

4.4 Modeling Covariate Effects on Degradation and Survival

(3)

4.5 Semiparametric Estimation of Degradation and Survival Characteristics: Models Without Covariates

(9)

4.6 Semiparametric Estimation of Degradation and Survival Characteristics: Models with Covariates

(4)

References

(3)

5 On a Degradation-Failure Model for Repairable Items

(16)

A. Lehmann

5.1 Introduction

(1)

5.2 Types of Covariates

(3)

5.3 Degradation-Failure Model for Non-Repairable Items

(4)

5.4 Degradation-Failure Model for Repairable Items

(2)

5.5 Maximum Likelihood Estimation

(2)

5.6 Semiparametric Estimation

(2)

5.7 Concluding Remarks

(1)

References

(2)

6 Comparison of Parametric and Semiparametric Estimates in a Degradation Model with Covariate and Traumatic Censoring

(18)

V. Couallier

6.1 The Degradation Model

(1)

6.2 Some Traumatic Events Censure the Degradation Paths

(1)

6.3 The Covariates

(1)

6.4 The Data and the Estimation

(5)

6.4.1 Estimation in the parametric framework

(1)

6.4.2 Estimation in the semipa.rametric framework

(2)

6.5 Comparison and Analysis by Monte Carlo Simulations

(5)

6.5.1 Computational aspects

(1)

6.5.2 The simulations

(4)

6.6 The Path Model with Noise

(1)

References

(4)

PART III: ACCELERATED FAILURE TIME MODELS AND ANALYSES

7 Accelerated Life Testing, Fuzzy Information and Generalized Probability

(8)

R. Viertl

7.1 Introduction

(1)

7.2 Fuzzy Numbers, Fuzzy Vectors and Fuzzy Valued Functions

100

(2)

7.3 Semiparametric ALT and Bayes' Theorem

102

(1)

7.4 Fuzzy Data and Generalized Bayes' Theorem

102

(2)

7.5 Conclusion

104

(1)

References

104

(3)

8 Asymptotic Theory in Rank Estimation for AFT Model Under Fixed Censorship

107

(14)

Z. Jin and Z. Ying

8.1 Introduction

107

(2)

8.2 Main Results

109

(3)

8.2.1 Notation and assumptions

109

(2)

8.2.2 Main results

111

(1)

8.3 Proof of Main Results

112

(6)

8.3.1 Preliminaries

112

(5)

8.3.2 Proofs of Theorems 8.2.1 and 8.2.2

117

(1)

8.4 Remarks

118

(1)

References

119

(2)

9 An Example of Optimal Design iN Accelerated Experiments

121

L. Gerville-Reache

9.1 Introduction

121

(1)

9.2 Semiparametric Estimation

122

(4)

9.3 Numerical Simulations

126

(3)

9.4 Conclusion

129

(1)

References

130

(5)

PART IV: AGING PROPERTIES AND ANALYSES

10 Aspects of Multivariate Aging in Exchangeable Frailty Models

135

(32)

F. Spizzichino

10.1 Introduction

135

(1)

10.2 Exchangeable Frailty Models

136

(4)

10.3 Multivariate Aging For Exchangeable Lifetimes

140

(3)

10.4 Exchangeable Frailties and Multivariate Aging

143

(4)

References

147

(2)

11 Semiparametric Models in the Studies of Aging and Longevity

149

(9)

A.I. Yashin

11.1 Introduction

149

(2)

11.2 The Shared Gannna-frailty Model

151

(1)

11.3 Identifiability and Confounding

152

(1)

11.4 The Positive-stable-frailty Models

153

(1)

11.5 The Correlated Gamma frailty Model

154

(1)

11.6 Other Correlated Frailty Models

155

(1)

11.7 The Underlying Hazard

156

(1)

11.8 An Advantage for Genetic Studies

157

(1)

11.9 Dependence Between Observed and Hidden Covariates

158

(1)

11.10 The Identifiability of the Dependent Competing Risk Model for Twins

159

(1)

11.11 Discussion

160

(1)

References

161

(6)

PART V: ANALYSES OF CENSORED AND TRUNCATED DATA

12 Semipararnetric Transformation Models for Arbitrarily Censored and Truncated Data

167

(54)

C. Huber-Carol and F. Vonta

12.1 Introduction

167

(2)

12.2 Nonparametric Estimation of the Survival Function

169

(2)

12.3 Semiparametric Transformation Models and the Likelihood Formulation

171

(3)

12.4 Real Data Example

174

(1)

References

175

(2)

13 EM Algorithm for Type-II Right Censored Bivariate Normal Data

177

(1)

N. Balakrishnan and J.-A. Kim

13.1 Introduction

177

(2)

13.2 Conditional Distributions and Expectations of Censored Data

179

(1)

13.3 EM Algorithm

180

(3)

13.4 Asymptotic Variances and Covariances

183

(2)

13.5 Illustrative Example

185

(2)

13.6 Simulation Results

187

(1)

13.7 Probability Coverages

187

(1)

Appendix

188

(2)

References

190

(21)

14 Statistical Estimation Based on Interval Censored Data

211

(1)

M.S. Tikhov

14.1 Introduction

211

(2)

14.2 Results

213

(3)

14.3 Examples

216

(1)

References

217

(4)

PART VI: REGRESSION METHODS AND APPLICATIONS

15 The Covariate Order Method for Nonparametric Exponential Regression and Some Applications in Other Lifetime Models

221

(32)

J.T. Kvaløy and B.H. Lindquist

15.1 Introduction

221

(1)

15.2 The Covariate Order Method for Exponential Regression

222

(6)

15.2.1 Method description and main theoretical results

223

(2)

15.2.2 Smoothing details

225

(1)

15.2.3 Several covariates

226

(1)

15.2.4 Testing for covariate effect

227

(1)

15.2.5 Example: Cardiac arrest versus air temperature

227

(1)

15.3 Applications in Cox Regression

228

(3)

15.3.1 Model checking and model fitting in classical Cox regression

228

(1)

15.3.2 Example: PBC data

229

(2)

15.4 Proofs

231

(5)

15.4.1 Proof of Theorem 15.2.1

231

(3)

15.4.2 Proof of Lemma 15.2.1

234

(1)

15.4.3 Proof of Theorem 15.2.2

234

(2)

15.5 Conclusions

236

(1)

References

236

(3)

16 Effect of Ignoring Heterogeneity in Hazards Regression

239

(1)

H-D. I. Wu,

16.1 Introduction

239

(1)

16.2 Calculation of Bias

240

(2)

16.3 Numerical Study

242

(3)

16.3.1 Examples

242

(2)

16.3.2 Direction of bias

244

(1)

16.4 The Stratified PH Analysis Under the HHR Model

245

(3)

16.5 Discussion

248

(1)

References

249

(4)

PART VII: TIME SERIES ANALYSIS

17 An Introduction to Efficient Estimation for Semiparametric Time Series

253

(20)

P.E. Greenwood, U.U. Müller and W. Wefelmeter

17.1 Introduction

253

(1)

17.2 Nonparametric Efficiency of the Least Squares Estimator

254

(3)

17.3 Linear Autoregression

257

(4)

17.4 Independent Innovations

261

(4)

References

265

(8)

PART VIII: INFERENTIAL ANALYSIS

18 Distance-based Multivariate Two Sample Tests

273

(60)

C.M. Cuadras and J. Fortiana

18.1 Introduction

273

(1)

18.2 Some Geometrical Concepts

274

(3)

18.3 DB Approach in Comparing Two Populations

277

(1)

18.4 The Generalized T2 Test

278

(1)

18.5 The Proximity Function Test

279

(3)

18.5.1 Linear and quadratic discrimination

279

(1)

18.5.2 Marginal parametrization

280

(1)

18.5.3 Non-parametric tests with proximity functions

281

(1)

18.6 Bootstrapping Distances

282

(1)

18.7 Comparing Variabilities

283

(1)

18.8 Real Data Examples

283

(2)

18.9 Conclusions

285

(3)

References

288

(3)

19 Empirical Likelihood in Nonparametric and Semiparametric Models

291

(1)

P. Bertail

19.1 Introduction

291

(1)

19.2 Empirical Likelihood for Hadamard Differentiable Functionals

292

(6)

19.2.1 Definitions

292

(1)

19.2.2 Some convex duality arguments

293

(2)

19.2.3 Extension to Hadamard differentiable functionals

295

(3)

19.3 Extensions to Some Semi-parametric Problems

298

(2)

19.4 Some Extensions to Bias Sampling Problems

300

(4)

References

304

(3)

20 Goodness of Fit Tests of L2-Type

307

(1)

H. Liero

20.1 Introduction

307

(1)

20.2 Tests for Densities

308

(7)

20.2.1 Testing whether a density has a parametric form

308

(4)

20.2.2 Testing independence

312

(3)

20.3 Tests for Sparse Data Sets

315

(5)

20.3.1 Testing cell probabilities in sparse multinomial data

315

(3)

20.3.2 Testing independence in sparse Contingency tables

318

(2)

20.4 Parametric Versus Nonparametric Regression Fit

320

(1)

20.5 Testing Homoscedasticity in. Nonparametric Regression

321

(2)

20.6 Testing the Hazard Function Under Censoring

323

(6)

20.6.1 Survival model without covariates

323

(3)

20.6.2 Survival model with fixed covariates

326

(3)

References

329

(4)

PART IX: MULIICENTRE STUDIES

21 Asymptotic Properties of the CRT Estimators for Multicentre Studies

333

(16)

V.V. Anisimov and V.V. Fedorov

21.1 Introduction

333

(1)

21.2 Estimator of CRT

334

(1)

21.3 Asymptotic Properties of CRT Estimator

335

(7)

21.3.1 Deterministic setting

336

(4)

21.3.2 Random setting

340

(2)

21.4 Approximation of Variance of the Estimator

342

(1)

21.5 Conclusions

343

(1)

References

343

(6)

PART X: QUALITY OF LIFE STUDIES

22 HRQoL and Concomitant Adjusted Mean Residual Life Analysis

349

(36)

P.K. Sen

22.1 Introduction

349

(1)

22.2 DMRL Characterizations

350

(4)

22.3 QLAMRL: Statistical Perspectives

354

(2)

22.4 Matrix-Valued Counting Process in HRQoL

356

(3)

22.5 Stopped Counting Processes

359

(2)

References

361

(2)

23 Semiparametric Approach to the Multivariate Mixed Rasch Model

363

(1)

M.-L. Feddag and M. Mesbah

23.1 Introduction

363

(3)

23.2 Model Description

366

(1)

23.3 An Overview of Estimating Methods in GLMMs

367

(2)

23.4 GEE to the Multivariate Mixed Basch Model

369

(7)

23.4.1 Approximations of marginal likelihood and joint moments

369

(3)

23.4.2 Estimation of the parameters

372

(3)

23.4.3 Asymptotic properties

375

(1)

23.5 Illustrations

376

(3)

23.5.1 Simulations

376

(1)

23.5.2 Example

377

(2)

23.6 Conclusions

379

(1)

References

380

(5)

PART XI: BREAST CANCER STUDIES

24 Breast Cancer Prognosis Using Survival Forests

385

(34)

T.M. Hoáng and V.L. Parsons

24.1 Introduction

385

(1)

24.2 Survival Forests

386

(1)

24.3 Choosing and Timing SF Parameters

387

(1)

24.4 Assessing the Fit by SF

388

(2)

24.5 Sample Runs Through SF

390

(1)

24.6 Using SF for Cancer Prognosis

391

(6)

References

397

(2)

25 Semiparametric Versus Parametric Regression Analysis Based on the Bounded Cumulative Hazard Model: An Application to Breast Cancer Recurrence

399

(1)

K.M. Boucher. B. Asselain, A.D. Tsodikov and A.Y. Yakovlev

25.1 Introduction

399

(2)

25.2 BCH Regression Models

401

(5)

25.2.1 Mixture models and generalizations

401

(3)

25.2.2 An EM-based estimation procedure for the PH mixture model

404

(2)

25.3 Analysis of Data from the Curie Institute

406

(4)

25.4 Discussion

410

(3)

References

413

(6)

PART XII: INFERENCE FOR PROCESSES

26 Estimation of Analytic Spectral Density of Gaussian Stationary Processes

419

(100)

I. Ibragimov

26.1 Introduction

419

(2)

26.2 Proof of Theorem 26.1.1. Upper Bounds

421

(7)

26.3 Lower Bounds

428

(8)

26.4 Processes with Continuous Time

436

(6)

References

442

(3)

27 Sub-optimal Estimation of an Unknown Function from Stationary Noisy Data

445

(1)

V. Solev

27.1 Statistical Problem

445

(2)

27.2 Process with Stationary Increments

447

(1)

27.3 Linear Estimators

448

(1)

27.4 Pseudo-periodic Functions

449

(1)

27.5 Parametric Set L*

450

(2)

27.6 Example

452

(6)

27.6.1 Hyperrectangle

453

(1)

27.6.2 The one-dimensional parametric set

454

(3)

27.6.3 General case

457

(1)

27.7 Main Result

458

(1)

References

459

(2)

28 On Parameter Estimation by Contaminated Observations of Ergodic Diffusion Processes

461

(1)

Yu. A. Kutoyants

28.1 Introduction

461

(2)

28.2 Regular Case

463

(2)

28.3 Nonregular Cases

465

(7)

28.3.1 Change-point estimation

465

(5)

28.3.2 Cusp estimation

470

(2)

References

472

(1)

29 On Parameter Estimation for a Position-Dependent Marking of a Doubly Stochastic Poisson Process

473

(1)

H. Wendt and W. Kahle

29.1 Introduction

473

(1)

29.2 A Cumulative Damage Process

474

(3)

29.3 Maximum Likelihood Estimations

477

(4)

29.4 The Large Sample Case

481

(4)

References

485

(2)

30 Discrete Time Semi-Markov Processes for Reliability and Survival Analysis - A Nonparametric Estimation Approach

487

(1)

V. Barbu and N. Limnios

30.1 Introduction

487

(1)

30.2 Discrete Time Semi-Markov Processes: Definitions and Basic Properties

488

(5)

30.3 Estimation of Survival Function and Asymptotic Properties

493

(3)

30.4 Proof

496

(2)

30.5 Numerical Example

498

(2)

References

500

(3)

31 Non-parametric Estimation on Lifetimes of Subjects Exposed to Radiation from a Semi-Markov Process

503

(1)

A-L. Afchain

31.1 Experimental Context and Its Modelling

503

(2)

31.2 Preliminaries

505

(1)

31.3 Semi-Markov Kernel Estimator

506

(3)

31.4 Survival Function Estimator

509

(2)

31.5 Numerical Application

511

(1)

31.5.1 Raw data. set

511

(1)

31.5.2 Results

512

(1)

31.6 Conclusion

512

(2)

References

514

(5)

PART XIII: PROBABILITY THEORY AND APPLICATIONS

32 An Extension of Levy's Formula to Weighted Wiener Processes

519

(34)

P. Deheuvels

32.1 Introduction and Results

519

(3)

32.2 Proofs

522

(4)

32.3 An Alternative Proof of Theorem 32.1.1 Based upon a Result of Biane and Yor (1987)

526

(2)

References

528

(5)

33 Sur l'Inégalité de Concentration de Doeblin-Lévy, Rogozin-Kesten

533

(1)

J. Bretagnolle

33.1 Introduction et résultat principal

533

(3)

33.2 Notations et premieres réductions

536

(2)

33.3 Présentation de la preuve

538

(1)

33.4 Preuve de l'assertion (33.7)

539

(11)

33.5 Appendice 544 References

550

(3)

Index

553

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Parametric and Semiparametric Models With Applications to Reliability, Survival Analysis, and Quality of Life

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