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

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

• ISBN13: 9780817632311
• ISBN10: 081763231X
• Format: Hardcover
• Copyright: 2004-06-30
• Publisher: Birkhauser

Summary

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.

Table of Contents

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	3	(10)
H. Läuter
1.1 Introduction	3	(2)
1.2 Types of Regression Functions	5	(1)
1.2.1 Linear models	5	(1)
1.2.2 Nonlinear models	5	(1)
1.3 Random Covariates and Partial Likelihood	6	(3)
1.3.1 Linear case	8	(1)
1.4 Fix Covariates and Partial Likelihood	9	(1)
1.5 Least Squares Estimation	9	(3)
References	12	(1)
2 Nonparametric Maximum Likelihood Estimation in the Proportional Hazards Model with Covariate Measurement Error	13	(14)
J.-F. Dupuy
2.1 Introduction	13	(2)
2.2 Nonparametric Maximum Likelihood Estimation	15	(2)
2.3 Existence of the NPML Estimators	17	(3)
2.4 Consistency of the NPML Estimators	20	(2)
2.5 Discussion	22	(1)
References	22	(5)
3 Diagnostics for Cox's Proportional Hazards Model	27	(14)
C. Caroni
3.1 Introduction	27	(2)
3.2 Schoenfeld Residuals	29	(1)
3.3 Martingale Residuals	30	(2)
3.4 Using Residuals to Examine the PH Assumption	32	(1)
3.5 Influence	33	(1)
3.6 Added Variable Plots	34	(2)
3.7 Conclusions	36	(1)
References	36	(5)
PART II: DEGRADATION MODELS AND ANALYSES
4 Semiparametric Analysis of Degradation and Failure Time Data with Covariates	41	(24)
V. Bagdonavicius and Al. Nikulin
4.1 Introduction	41	(1)
4.2 Modeling the Degradation-Failu e Time Process	42	(2)
4.3 Survival and Degradation Characteristics	44	(2)
4.4 Modeling Covariate Effects on Degradation and Survival	46	(3)
4.5 Semiparametric Estimation of Degradation and Survival Characteristics: Models Without Covariates	49	(9)
4.6 Semiparametric Estimation of Degradation and Survival Characteristics: Models with Covariates	58	(4)
References	62	(3)
5 On a Degradation-Failure Model for Repairable Items	65	(16)
A. Lehmann
5.1 Introduction	65	(1)
5.2 Types of Covariates	66	(3)
5.3 Degradation-Failure Model for Non-Repairable Items	69	(4)
5.4 Degradation-Failure Model for Repairable Items	73	(2)
5.5 Maximum Likelihood Estimation	75	(2)
5.6 Semiparametric Estimation	77	(2)
5.7 Concluding Remarks	79	(1)
References	79	(2)
6 Comparison of Parametric and Semiparametric Estimates in a Degradation Model with Covariate and Traumatic Censoring	81	(18)
V. Couallier
6.1 The Degradation Model	82	(1)
6.2 Some Traumatic Events Censure the Degradation Paths	83	(1)
6.3 The Covariates	84	(1)
6.4 The Data and the Estimation	84	(5)
6.4.1 Estimation in the parametric framework	86	(1)
6.4.2 Estimation in the semipa.rametric framework	87	(2)
6.5 Comparison and Analysis by Monte Carlo Simulations	89	(5)
6.5.1 Computational aspects	89	(1)
6.5.2 The simulations	90	(4)
6.6 The Path Model with Noise	94	(1)
References	95	(4)
PART III: ACCELERATED FAILURE TIME MODELS AND ANALYSES
7 Accelerated Life Testing, Fuzzy Information and Generalized Probability	99	(8)
R. Viertl
7.1 Introduction	99	(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