Weibull model selection for reliability modelling

A large number of models have been derived from the two-parameter Weibull distribution and are referred to as Weibull models. They exhibit a wide range of shapes for the density and hazard functions, which makes them suitable for modelling complex failure data sets. The WPP and IWPP plot allows one to determine in a systematic manner if one or more of these models are suitable for modelling a given data set. This paper deals with this topic.

Models involving two inverse Weibull distributions

In this paper, we look at three models (mixture, competing risk and multiplicative) involving two inverse Weibull distributions. We study the shapes of the density and failure-rate functions and discuss graphical methods to determine if a given data set can be modelled by one of these models. (C) 2001 Elsevier Science Ltd. All rights reserved.

n-fold Weibull multiplicative model

In this paper we study the n-fold multiplicative model involving Weibull distributions and examine some properties of the model. These include the shapes for the density and failure rate functions and the WPP plot. These allow one to decide if a given data set can be adequately modelled by the model. We also discuss the estimation of model parameters based on the WPP plot. (C) 2001 Elsevier Science Ltd. All rights reserved.

A modified Weibull distribution

A new lifetime distribution capable of modeling a bathtub-shaped hazard-rate function is proposed. The proposed model is derived as a limiting case of the Beta Integrated Model and has both the Weibull distribution and Type I extreme value distribution as special cases. The model can be considered as another useful 3-parameter generalization of the Weibull distribution. An advantage of the model is that the model parameters can be estimated easily based on a Weibull probability paper (WPP) plot that serves as a tool for model identification. Model characterization based on the WPP plot is studied. A numerical example is provided and comparison with another Weibull extension, the exponentiated Weibull, is also discussed. The proposed model compares well with other competing models to fit data that exhibits a bathtub-shaped hazard-rate function.

Study of n-fold Weibull competing risk model

This paper deals with an n-fold Weibull competing risk model. A characterisation of the WPP plot is given along with estimation of model parameters when modelling a given data set. These are illustrated through two examples. A study of the different possible shapes for the density and failure rate functions is also presented. (C) 2003 Elsevier Ltd. All rights reserved.

Modelling the Distribution of Ischaemic Stroke-Specific Survival Time Using an EM-based Mixture Approach with Random Effects Adjustment

A two-component survival mixture model is proposed to analyse a set of ischaemic stroke-specific mortality data. The survival experience of stroke patients after index stroke may be described by a subpopulation of patients in the acute condition and another subpopulation of patients in the chronic phase. To adjust for the inherent correlation of observations due to random hospital effects, a mixture model of two survival functions with random effects is formulated. Assuming a Weibull hazard in both components, an EM algorithm is developed for the estimation of fixed effect parameters and variance components. A simulation study is conducted to assess the performance of the two-component survival mixture model estimators. Simulation results confirm the applicability of the proposed model in a small sample setting. Copyright (C) 2004 John Wiley Sons, Ltd.

Development and survival of the diamondback moth (Lepidoptera : Plutellidae) at constant and alternating temperatures

Survival and development time from egg to adult emergence of the diamondback moth, Plutella xylostella (L.), were determined at 19 constant and 14 alternating temperature regimes from 4 to 40degreesC. Plutella xylostella developed successfully front egg to adult emergence at constant temperatures from 8 to 32degreesC. At temperatures from 4 to 6degreesC or from 34 to 40degreesC, partial or complete development of individual stages or instars was possible, with third and fourth instars having the widest temperature limits. The insect developed successfully from egg to adult emergence under alternating regimes including temperatures as low as 4degreesC or as high as 38degreesC. The degree-day model, the logistic equation, and the Wang model were used to describe the relationships between temperature and development rate at both constant and alternating temperatures. The degree-day model described the relationships well from 10 to 30degreesC. The logistic equation and the Wang model fit the data well at temperatures 32degreesC. Under alternating regimes, all three models gave good simulations of development in the mid-temperature range, but only the logistic equation gave close simulations in the low temperature range, and none gave close or consistent simulations in the high temperature range. The distribution of development time was described satisfactorily by a Weibull function. These rate and time distribution functions provide tools for simulating population development of P. xylostella over a wide range of temperature conditions.

Damage by eutectic particle cracking in aluminum casting alloys A356/357

The strain dependence of particle cracking in aluminum alloys A356/357 in the T6 temper has been studied in a range of microstructures produced by varying solidification rate and Mg content, and by chemical (Sr) modification of the eutectic silicon. The damage accumulates linearly with the applied strain for all microstructures, but the rate depends on the secondary dendrite arm spacing and modification state. Large and elongated eutectic silicon particles in the unmodified alloys and large pi-phase (Al9FeMg3Si5) particles in alloy A357 show the greatest tendency to cracking. In alloy A356, cracking of eutectic silicon particles dominates the accumulation of damage while cracking of Fe-rich particles is relatively unimportant. However, in alloy A357, especially with Sr modification, cracking of the large pi-phase intermetallics accounts for the majority of damage at low and intermediate strains but becomes comparable with silicon particle cracking at large strains. Fracture occurs when the volume fraction of cracked particles (eutectic silicon and Fe-rich intermetallics combined) approximates 45 pct of the total particle volume fraction or when the number fraction of cracked particles is about 20 pct. The results are discussed in terms of Weibull statistics and existing models for dispersion hardening.

Analysis of Melanoma Onset: Assessing Familial Aggregation by Using Estimating Equations and Fitting Variance Components via Bayesian Random Effects Models

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.

Heavy tails, importance sampling and cross-entropy

We consider the problem of estimating P(Yi + (...) + Y-n > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a toot for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y-1 + (...) + Y-n > x), n - 1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/l and GI/G/l queues are also discussed.

Improved algorithms for rare event simulation with heavy tails

The estimation of P(S-n > u) by simulation, where S, is the sum of independent. identically distributed random varibles Y-1,..., Y-n, is of importance in many applications. We propose two simulation estimators based upon the identity P(S-n > u) = nP(S, > u, M-n = Y-n), where M-n = max(Y-1,..., Y-n). One estimator uses importance sampling (for Y-n only), and the other uses conditional Monte Carlo conditioning upon Y1,..., Yn-1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.