Mathematical Methods in Survival Analysis, Reliability and Quality of Life

Catherine Huber, Nikolaos Limnios, Mounir Mesbah, Mikhail S. Nikulin, Catherine Huber, Nikolaos Limnios, Mounir Mesbah, Mikhail S. Nikulin

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848210103
• eBook ISBN: 9781118624111
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Reliability and survival analysis are important applications of stochastic mathematics (probability, statistics and stochastic processes) that are usually covered separately in spite of the similarity of the involved mathematical theory. This title aims to redress this situation: it includes 21 chapters divided into four parts: Survival analysis, Reliability, Quality of life, and Related topics. Many of these chapters were presented at the European Seminar on Mathematical Methods for Survival Analysis, Reliability and Quality of Life in 2006.

PART I

Survival Analysis

Chapter 1

Model Selection for Additive Regression in the Presence of Right-Censoring 1

1.1. Introduction

Statistical tools for handling regression problems when the response is censored have been developed in the last two decades. The response was often assumed to be a linear function of the covariates, but non-parametric regression models provide a very flexible method when a general relationship between covariates and response is first to be explored. In recent years, a vast literature has been devoted to non-parametric regression estimators for completely observed data. However, few methods exist under random censoring. First, Buckley and James [BUC 79] and Koul, Susarla and Van Ryzin [KOU 81] among others introduced the original idea of transforming the data to take the censoring into account, for linear regression curves. Then, Zheng [ZHE 88] proposed various classes of unbiased transformations. Dabrowska [DAB 87] and Zheng [ZHE 88] applied non-parametric methods for estimating the univariate regression curve. Later, Fan and Gijbels [FAN 94] considered a local linear approximation for the data transformed in the same way by using a variable bandwidth adaptive to the sparsity of the design points. Györfi et al. [GYÖ 02] also studied the consistency of generalized Stone’s regression estimators in the censored case. Heuchenne and Van Keilegom [HEU 05] considered a nonlinear semi-parametric regression model with censored data. Park [PAR 04] extended a procedure suggested in Gross and Lai [GRO 96] to a general non-parametric model in the presence of left-truncation and right-censoring, by using B-spline developments. Recently, Kohler et al. [KOH 03] proposed an adaptive mean-square estimator built with polynomial splines.

However, for modeling the relationship between a response and a multivariate re-gressor, new methodologies have to be found to solve the problem of practical implementation in higher dimension. The main objective of the article is to propose a multivariate method of model selection for an additive regression function of a low-dimensional covariate vector. In fact, the particular case of additive models seems to be more realistic in practice and may constitute a way to make the dimension of the covariate greater than 1. Suppose that

is a d-dimensional covariate in a compact set, without loss of generality we assume that

is a [0, 1]d-valued vector. Let

,

be independent identically distributed random variables. Let T > 0 be a fixed time for collecting the data. Therefore, the response variables before censoring are denoted by Yi, T = Yi ∧ T, where a ∧ b denotes the infim...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• PART I
• PART II
• PART III
• PART IV