Anthony F. Desmond

Stochastic models of failure in random environments

Stochastic models of failure modes of frequent occurrence in the engineering sciences are considered. The failure-producing stress environment is modelled as a stationary stochastic process. Using theoretical properties of the sample paths of these processes, failure-time distributions which belong to the Birnbaum-Saunders family are obtained. Several examples of particular engineering relevance are treated. Grǎce aux processus stochastiques, il est possible de representer de maniere assez juste de nombreux types de defaillances ďorigine repandue en ingenierie. Un processus stochastique en regime permanent peut notamment servir de modele pour certaines causes exogenes de defaillance. En faisant appel aux proprietes des chemins echantillonnaux de tels processus, il est possible de calculer les distributions des temps de bris. On constate que celles-ci appartiennent a la famille de lois dite de Birnbaum-Saunders. On presente aussi de nombreux exemples ďinterět pratique.

Review and implementation of cure models based on first hitting times for Wiener processes

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501–513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.

Modified censored moment estimation for the two-parameter Birnbaum-Saunders distribution

The maximum likelihood estimators (MLEs) and the moment estimators of a two-parameter Birnbaum-Saunders (BISA) distribution are studied by various authors when data are either complete or subject to Type-I or Type-II censoring. But there is not much research on parameter estimation for the BISA distribution under random censoring. A simple method of modified censored moment estimation is proposed to estimate parameters of the BISA distribution under random censoring. Bias-reduced versions of these estimators are constructed as well. Asymptotic theory for the estimators is established. The performance of these estimators is compared with that of the MLEs through Monte Carlo simulations for small, moderate, and large proportions of censoring and different sample sizes. An analysis of real data is used to illustrate the proposed method.

Optimal estimating functions, quasi-likelihood and statistical modelling

Abstract The value of studying quasi-likelihood and recent extensions within the framework of estimating function theory is explored. The notion of a quasi-score function as an optimal surrogate for the true score function in semi-parametric models is outlined. It is shown that such quasi-score functions may be identified with optimal estimating functions within certain classes of unbiased estimating functions, defined in terms of an underlying semi-parametric model. Wedderburns quasi-score function is seen to be a special case, for a particular semi-parametric model and a particular class of estimating functions. Applications of the estimating function approach to problems with overdispersed data and longitudinal data are discussed. A recent formulation of the analysis of case-control data via estimating functions is also discussed as well as the role of orthogonality and Bayesian inference in estimating function approaches.

Estimation of parameters for a Birnbaum–Saunders regression model with censored data

Little work has been published on the analysis of censored data for the Birnbaum–Saunders distribution (BISA). In this article, we implement the EM algorithm to fit a regression model with censored data when the failure times follow the BISA. Three approaches to implement the E-Step of the EM algorithm are considered. In two of these implementations, the M-Step is attained by an iterative least-squares procedure. The algorithm is exemplified with a single explanatory variable in the model.

Shortest prediction intervals for the birnbaum-saunders distribution

Shortest prediction intervals for a future observation from the Birnbaum-Saunders distribution are obtained from both frequentist and Bayesian perspectives. Comparisons are made with alternative intervals obtained via inversion. Monte Carlo simulations are performed to assess the approximate intervals.

Bayesian and likelihood inference for cure rates based on defective inverse Gaussian regression models

Failure time models are considered when there is a subpopulation of individuals that is immune, or not susceptible, to an event of interest. Such models are of considerable interest in biostatistics. The most common approach is to postulate a proportion p of immunes or long-term survivors and to use a mixture model [5]. This paper introduces the defective inverse Gaussian model as a cure model and examines the use of the Gibbs sampler together with a data augmentation algorithm to study Bayesian inferences both for the cured fraction and the regression parameters. The results of the Bayesian and likelihood approaches are illustrated on two real data sets.

A competing risks model for correlated data based on the subdistribution hazard

Family-based follow-up study designs are important in epidemiology as they enable investigations of disease aggregation within families. Such studies are subject to methodological complications since data may include multiple endpoints as well as intra-family correlation. The methods herein are developed for the analysis of age of onset with multiple disease types for family-based follow-up studies. The proposed model expresses the marginalized frailty model in terms of the subdistribution hazards (SDH). As with Pipper and Martinussen’s (Scand J Stat 30:509–521, 2003) model, the proposed multivariate SDH model yields marginal interpretations of the regression coefficients while allowing the correlation structure to be specified by a frailty term. Further, the proposed model allows for a direct investigation of the covariate effects on the cumulative incidence function since the SDH is modeled rather than the cause specific hazard. A simulation study suggests that the proposed model generally offers improved performance in terms of bias and efficiency when a sufficient number of events is observed. The proposed model also offers type I error rates close to nominal. The method is applied to a family-based study of breast cancer when death in absence of breast cancer is considered a competing risk.

A mixed effects log-linear model based on the Birnbaum-Saunders distribution

In lifetime data analysis and particularly in engineering reliability contexts, the Birnbaum-Saunders (BISA) density is often suggested as a suitable model; see Birnbaum and Saunders (1969), Mann et al. (1974), and Desmond (1985). A linear regression model, obtained from a logarithmic transformation of the response variable, is useful in studying the effect of covariates on the response variable; see Rieck and Nedelman (1991), Tsionas (2001) and Galea et al. (2004). In this paper, an extension of the log-linear regression model of Rieck and Nedelman (1991), which considers random effects, is introduced. From a Monte Carlo simulation study, the performance of various estimation and prediction methods are studied. The usefulness of the mixed log-linear model is stressed and compared to the pure fixed effects log-linear regression BISA model. The new model is used to analyze a real data set, for which a fixed effects model is inappropriate.

A kernel smoothed semiparametric survival model

Abstract In a semiparametric survival model with a flexible covariate effect, we suppose the baseline hazard function is parameterized, while the risk function associated with covariates is modeled in a semiparametric way. A maximum generalized profile likelihood estimator for the parameters is considered. We show that the resulting estimator is root- n consistent, asymptotically normal and efficient. An estimator for the nonparametric risk function is also given and its asymptotic properties are derived. Its rate of convergence is shown to be comparable to that in nonparametric regression. An application to mouse leukemia data is presented to illustrate the proposed method.