Reliability analysis using an exponential power model with bathtubshaped failure rate function: a Bayes study
 Romana Shehla^{1}Email author and
 Athar Ali Khan^{1}
Received: 12 February 2016
Accepted: 30 June 2016
Published: 13 July 2016
Abstract
Models with bathtubshaped hazard function have been widely accepted in the field of reliability and medicine and are particularly useful in reliability related decision making and cost analysis. In this paper, the exponential power model capable of assuming increasing as well as bathtubshape, is studied. This article makes a Bayesian study of the same model and simultaneously shows how posterior simulations based on Markov chain Monte Carlo algorithms can be straightforward and routine in R. The study is carried out for complete as well as censored data, under the assumption of weaklyinformative priors for the parameters. In addition to this, inference interest focuses on the posterior distribution of nonlinear functions of the parameters. Also, the model has been extended to include continuous explanatory variables and Rcodes are well illustrated. Two real data sets are considered for illustrative purposes.
Keywords
Background
In reliability analysis, hazard rate plays an indispensable role to characterize life phenomena. Technically, failure or hazard rate represents the propensity of a device of age t to fail in the small interval of time t to \(t + \mathrm {d}t\). The parametric models, such as gamma, Weibull, and truncated normal distributions, which are commonly used lifetime distributions display monotone failure rates. However, many physical phenomena exhibit failure rates which are nonmonotonic. For example, the failure pattern of many mechanical and electronic components comprise of three stages: initial stage (or burnin) where failure is high at the beginning of the product life cycle due to design and manufacturing problems, and decreases towards a constant level, the middle stage with an approximately constant failure rate, which is followed by a final stage (or wearout phase), from where the failure rate starts to increase. Such failure rates are usually termed as bathtub (BT) or U shaped. The aforementioned models which allow only monotone failure rates are unable to produce bathtub curves and thus cannot adequately interpret data with this character. Bathtub models are possibly more realistic models than monotone failure rate models. Several models have been proposed one by one to model the real data with bathtubshaped failure rate since 1980s (see Aarset 1987; Xie et al. 2002; Gupta et al. 2008 for detailed discussion). There are number of papers discussing several flexible distributions with more than two parameters, which can accommodate increasing, decreasing, unimodal and bathtubshaped hazard functions (see, for examples Mudholkar et al. 1996; Pham and Lai 2007; Carrasco et al. 2008). Nevertheless, from the practical point of view, it is always important to consider parsimonious models with as few parameters as possible.
An interesting twoparameter lifetime model capable of producing increasing as well as bathtub hazard curve is exponential powerdistribution introduced by Smith and Bain (1975). This model has been discussed by several authors not only in the context of reliability literature (for examples, Rajarshi and Rajarshi 1988 and Leemis 1986) but also within asymmetric distributions; see, for example, Delicado and Goria (2008). This model may be useful in certain cases where the product may be quite reliable and possibly even improve for some period of time, and then fail rather quickly after it begins to wearout. A notable amount of work has been done on this model from the frequentists perspective. For example, Smith and Bain (1975) considered least squarestype estimators for the model parameters and performed Monte Carlo simulation to obtain their distributions in order to get inference results for the reliability. Koh and Leemis (1989) developed statistical procedures for maximum likelihood and least squares estimation of the parameters for the complete as well as TypeII censored data. Chen (1999) proposed an exact statistical test for the shape parameter of the model and found an exact confidence interval for the same parameter. Srivastava and Kumar (2011) presented the exponential power distribution as a software reliability model and carried out the Bayesian analysis in OpenBUGS using informative priors (gamma priors) for the parameters but didn’t consider censoring mechanism. To the best of our knowledge, regression modeling of the exponential power distribution has not yet been discussed. This article analyzes the model in the Bayesian framework assuming weaklyinformative priors for the model parameters for both the complete and censored reliability data. A distinguishing feature of this paper is that the whole analysis is done in R (R Core Team 2015) and codes developed are well illustrated. Both the analytic and simulationbased Bayesian studies are conducted.
Exponential power distribution
Characterization of failure rate function
The role of the parameter \(\gamma \) in determining different shapes of the failure rate function can be studied under two situations:
Case 1:
 i
For any \(t>0, h'(t)>0\), thus, h(t) is an increasing function.
 ii
\(h(t)\rightarrow +\infty \) as \(t \rightarrow +\infty \).
Case 2:
 i
Letting \(h'(t_0)=0\), we obtain \(t_0=\alpha \bigg (\frac{1\gamma }{\gamma }\bigg )^{\frac{1}{\gamma }}\). It is evident that when \(0<\gamma <1, t_0\) exists and is finite. For \(t<t_0, h(t)\) is decreasing while it is increasing for \(t>t_0\) showing a bathtub shaped property.
 ii
\(h(t) \rightarrow \infty \) for \(t \rightarrow 0\) and \(t \rightarrow +\infty \).
Model formulation
Independence Metropolis algorithm
 1
Select a starting value of the chain \(\theta ^{(0)}\).
 2For \(s=1, \ldots , S\), repeat the following steps

set \(\theta =\theta ^{(s1)}\)

generate a new parameter value, i.e. a proposal \(\theta ^*\), from a proposal distribution \(q(\theta ^*)\).

calculate acceptance probability as the ratio$$\begin{aligned} \alpha =\text {min}\big (1, \frac{{p(\theta ^*y)}{q(\theta )}}{{p(\theta y)}{q(\theta ^*)}}\big ) \end{aligned}$$

update \(\theta ^{(s)} = \theta ^{*}\) with probability \(\alpha \); otherwise set \(\theta ^{(s)} = \theta \).

Laplace approximation
The influence of prior distribution on posterior inferences decreases as the sample size n increases. These ideas are sometimes referred to as asymptotic theory. The large sample results are not actually necessary for performing Bayesian data analysis but are often useful for quick references and as starting points for iterative simulation algorithms (Gelman et al. 2004). A remarkable method of asymptotic approximation is the Laplace approximation (Tierney and Kadane 1986; Tierney et al. 1989) which accurately approximates the unimodal posterior moments and marginal posterior densities in many cases. A brief and informal description of Laplace approximation method is as follows:
Bayesian computation with R
There are significant number of packages contributing to the Comprehensive R Archive Network (CRAN) such as MCMCpack (Martin et al. 2013), arm (Gelman et al. 2015), LearnBayes (Albert 2014) that provide tools for Bayesian inference. But, these packages are not flexible enough to handle highdimensional problems and at the same time, the censoring mechanism which is the most important feature of reliability data. This paper presents the contributed R package LaplacesDemon that facilitates multidimensional Bayesian inference and is freely available at http://www.bayesianinference.com/software. The MCMC algorithms in LaplacesDemon are generalizable and robust to correlation between variables or parameters.
The package LaplacesDemon (Statisticat LLC 2015) not only facilitates the implementation of simple as well as more advanced simulation algorithms though the function LaplacesDemon, but also provides a function LaplaceApproximation for Laplace approximation, that evaluates objective function many times, per iteration thus, making MCMC algorithms faster per iteration (see, for example, Shehla and Khan 2013). Although, this package is not specifically meant for reliability data analysis, we have developed codes in it to deal with uncensored and censored reliability data problems.
The function LaplaceApproximation
The function LaplaceApproximation deterministically maximizes the logarithm of the unnormalized joint posterior density using one of the several optimization techniques. The aim of LaplaceApproximation is to estimate posterior mode and variance of each parameter. Currently, this function offers 19 optimization algorithms. The function LaplaceApproximation is also typically faster because it is seeking pointestimates, rather than attempting to represent the target distribution with enough simulation draws. Another striking feature of this function is that it carries the possibility of drawing independent samples through sampling importance resampling technique via one of its arguments sir. A short length discussion of its arguments are as follows:
LaplaceApproximation(Model, parm, Data, Interval=1.0E6,
Iterations=100, Method="SPG", Samples=1000, CovEst="Hessian",
sir=TRUE, Stop.Tolerance=1.0E5, CPUs=1, Type="PSOCK")
where Model receives the model from a userdefined function. The argument parm requires a vector of initial values for the parameters for optimization. The argument Data accepts a listed data object on which the model is to be fitted. The argument sir takes a logical value to specify whether sampling importance resampling is to be implemented or not. It is implemented via SIR function of this package which draws independent posterior samples.
The function LaplacesDemon
Given data, a model specification, and initial values, LaplacesDemon maximizes the logarithm of the unnormalized joint posterior density with Markov chain Monte Carlo (MCMC) algorithms, also called samplers, and provides samples of the marginal posterior distributions, deviance and other monitored variables. The function LaplacesDemon offers 41 MCMC algorithms for numerical approximation in Bayesian inference. The default algorithm is “MetropoliswithinGibbs (MWG)”. The arguments of this function are as follows:
LaplacesDemon(Model, Data, Initial.Values, Covar=NULL,
Iterations=10000, Status=100, Thinning=10, Algorithm="MWG",
Specs=NULL, LogFile="", ...)
where Model receives the same userdefined model, Data stands for the listed data object. The argument Initial.Values requires a vector of initial values equal in length to the number of parameters. However, if Laplace approximation has been performed, the results obtained are input as initial values in this function. The argument Covar receives a \(d\times d\) proposal covariance matrix (where d is the number of parameters) as returned by the function LaplaceApproximation. If NULL, it indicates that variance vector or covariance matrix has not been specified, so the algorithm will begin with its own estimates. The argument Iterations specifies the number of iterations that LaplacesDemon will update the parameters searching for target distribution and Status is reported after every 100 iterations. Thinning is performed via the argument Thinning to reduce autocorrelation and the number of marginal posterior samples. The argument Specs=NULL is default argument, and accepts a list of specifications for the MCMC algorithm declared in the Algorithm argument.
Bayesian analysis of exponential power model
Simulated dataset from exponential power model
8.068  11.464  18.465  36.609  6.094  35.695 
40.787  21.987  20.672  1.854  6.226  5.325 
23.125  11.855  27.114 
We now proceed for the posterior analysis of the model in R, which essentially requires the creation of data, model building and choosing initial values for the parameters. Before applying the independenceMetropolis algorithm to approximate the posterior density, an attempt is made to approximate it using Laplace approximation. For that, we progress by the following steps:
Creation of data
Model specification
Initial values
Approximation by Laplace’s method
Summarizing output
Asymptotic posterior summaries along with 0.025, 0.5, 0.975 quantiles based on Laplace’s approximation
Parameter  Mode  SD  LB  UB 

log.gamma  0.15  0.23  −0.31  0.60 
log.alpha  3.38  0.13  3.11  3.65 
Posterior means and standard deviations of the parameters along with the quantiles, Monte Carlo standard errors and effective sample sizes as obtained by sampling importance resampling technique
Parameter  Mean  SD  MCSE  ESS  LB  Median  UB 

log.gamma  0.07  0.23  0.01  1000.00  −0.43  0.08  0.50 
log.alpha  3.40  0.14  0.00  1000.00  3.13  3.40  3.70 
Deviance  115.37  1.95  0.06  1000.00  113.47  114.75  120.65 
It may be noted that the results obtained are in log scale and must be exponentiated to get the values in original metric.
Posterior analysis using simulation technique
Summarizes marginal posterior distributions of the parameters, deviance based on the MCMC samples
Parameter  Mean  SD  MCSE  ESS  LB  Median  UB 

log.gamma  0.14  0.14  0.00  1149.03  −0.13  0.14  0.41 
log.alpha  3.38  0.08  0.00  1243.82  3.23  3.38  3.53 
Deviance  114.16  0.75  0.03  1001.04  113.44  113.94  116.17 
Analysis of censored data with R
A distinctive aspect of the statistical analysis of reliability data is regarding the natural occurrence of censored observations. In Bayesian set up, censoring mechanisms are easily handled as Bayesian methods take into account only the observed lifetimes and does not bother about the cause or type of censoring. Thus, for a Bayesian analyst, Type I, Type II, Type III, Type IV and random rightcensoring, all correspond to the right censored data. Hence, Bayesian approach provides a common framework to analyze all censored data types.
Exponential power regression analysis
In the previous section, we considered the use of exponential power model to describe responses with no covariates (or explanatory variables). In practice, many situations involve heterogeneous populations, and to represent that heterogeneity, it is important to consider the relationship of failure time to other factors (or explanatory variables). In the present section, we focus our attention to the models containing explanatory variables namely, failure time regression models in the context of reliability.
A model with explanatory variables (or regressors) can sometimes best describe the heterogeneity in a population. It explains or predicts why some units survive a long time whereas others fail quickly. The main objective behind regression modeling is to explore the relationship between failuretime and the explanatory variables. This involves specifying a model for the distribution of \(\underline{t}\), given \(\underline{x}\), where \(\underline{t}\) represents lifetime and \(\underline{x}\) is a vector of regressor variables. It is an important class of regression models which allows one or more elements of the model parameter vector \(\theta =(\theta _1, \ldots , \theta _k)\) to be a function of the regressor variables. In the present section, we develop Bayesian analysis for the nonlinear regression model with random errors distributed according to the exponential power distribution. More specifically, we shall demonstrate the regression modeling of a data set in R with an underlying exponential power distribution using the LaplacesDemon package.
Formulation of the model
Initial values
Posterior analysis by Laplace’s method
Summarizing output
Posterior summaries using the function LaplaceApproximation
Parameter  Mode  SD  LB  UB 

beta[1]  1.13  0.25  0.63  1.62 
beta[2]  1.69  0.45  0.79  2.59 
log.gamma  0.17  0.23  −0.28  0.62 
Posterior means and standard deviations of the exponential power parameters based on the samples drawn by sampling importance resampling algorithm
Parameter  Mean  SD  MCSE  ESS  LB  Median  UB 

beta[1]  1.20  0.27  0.01  1000.00  0.68  1.18  1.78 
beta[2]  1.57  0.51  0.02  1000.00  0.62  1.58  2.55 
log.gamma  0.03  0.24  0.01  1000.00  −0.46  0.05  0.48 
Deviance  71.93  3.36  0.11  1000.00  68.75  71.04  83.87 
It follows from Table 5 that the posterior modes of the regression parameters \(\beta _0\) and \(\beta _1\) for the concerned model is \(1.13\pm 0.25\) and \(1.69\pm 0.45\), respectively. For more accurate summary, we resort to simulation technique.
Simulationbased study
Marginal posterior summaries based on the MCMC samples using independenceMetropolis algorithm
Parameter  Mean  SD  MCSE  ESS  LB  Median  UB 

beta[1]  1.13  0.15  0.01  890.36  0.85  1.13  1.43 
beta[2]  1.68  0.26  0.01  779.53  1.18  1.68  2.21 
log.gamma  0.15  0.13  0.01  815.56  −0.10  0.15  0.41 
Deviance  69.52  0.86  0.05  574.48  68.58  69.30  71.68 
Determination of burnin and replacement time
A bathtub curve are useful in reliability related decision making. Reducing the burnin time of a new product with too high initial failure rate results in improved reliability of the product. Similarly, during the wearout phase of the product, the failure increases rapidly and replacement is needed to reduce the risk of immediate failure. The problem of determining burnin time and replacement time can easily be tackled by the failurerate criteria (Xie and Lai 1996).
Real data modeling
The R codes are used to model two real data sets and the most relevant results are reported.
Electronic device failure time data
Transistor data
Discussion and conclusion
This paper develops the Bayesian inference procedures for the exponential power model assuming weaklyinformative priors for the model parameters. This parsimonious model with just twoparameters is fairly applicable to various reallife failuretime data capable of producing increasing as well as bathtubshaped failure rate. These two properties along with the availability of invertible cumulative distribution function makes the exponential power model, a useful alternative to the conventional Weibull distribution. A distinguishing feature of this paper is that both the analytic and simulationbased Bayesian studies are conducted in R language using the package LaplacesDemon. The main body of the manuscript contains the complete description of R codes both for the null and regression models with random errors distributed according to the exponential power distribution. Illustrations have been made using simulated data sets which is finally concluded on realworld reliability problems. The posterior means, modes and 95 % credible intervals for the parameters are obtained. The exact posterior densities of the parameters together with that of hazard and reliability functions are plotted. It is seen that, the two functions LaplaceApproximation and LaplacesDemon exploited throughout the paper, allow fast and precise posterior analysis. However, since, LaplaceApproximation is asymptotic in nature, it should be noted that the sample size is at least 5 times the number of parameters, in order to observe its good performance. Simulation tools are free from such restrictions. Furthermore, it has been observed throughout that the simulation technique, particularly, independenceMetropolis algorithm summarizes the posterior more pecisely, in terms of the lower standard deviations of the parameters. However, it is to be noted that, IM algorithm performs well if the proposal is a good approximation of the posterior. Therefore, the posterior approximation using Laplace approximation can always be improved with independenceMetropolis algorithm.
Limitations of the study
The Bayesian study has been carried out only for complete and right censored data. The case of leftcensored and intervalcensored data are yet to be considered.
Declarations
Authors' contributions
The first author RS studied the model and carried out the Bayesian reliability analysis. Both authors read and approved the final manuscript.
Acknowledgements
The authors appreciate the anonymous referees and editor for their valuable comments and suggestions towards improving the quality of the manuscript. The first author would like to thank University Grants Commission (UGC), New Delhi for financial assistance.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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