Abstract
In reliability engineering and lifetime analysis, many units of the product fail with different causes of failure, and some tests require stress higher than normal stress. Also, we need to design the life experiments which present methodology for formulating scientific and engineering problems using statistical models. So, in this paper, we adopted a partially constant stress accelerated life test model to present times to failure in a small period of time for Gompertz life products. Also, considering that, units are failing with the only two independent causes of failure and tested under type-I generalized hybrid censoring scheme the data built. Obtained data are analyzed with two methods of estimations, maximum likelihood and Bayes methods. These two methods are used to construct the point and interval estimators with the help of the MCMC method. The developed results are measured and compared under Monte Carlo studying. Also, a data set is analyzed for illustration purposes. Finally, some comments are presented to describe the numerical results.
Highlights
In the plan of type-I hybrid censoring scheme, n units are randomly selected from the product. e ideal test time and a suitable number of failure units that need statistical inference are proposed to be η∗ and m, respectively. e experimenter terminates the test at the min (η∗, Xm)
The ideal test time and a suitable number of failure units that need statistical inference are proposed to be η∗ and m, respectively. e experimenter terminates the test at the max (η∗, Xm); for more details, see [4]. e types of censoring, type-I HCS and type-II HCS satisfy the property that a smaller number of failures may be zero and there is a large test time, respectively; see [5]. en, this problem has been treated with a generalized form of two types of censoring schemes known as a generalized hybrid censoring scheme (GHCS)
Different forms of accelerated life tests (ALTs) are available. e first is known as constant stress ALTs, in which the experiment is loaded under constant stress until the final point of the experiments. e second type is called step stress ALTs, in which the experiment is running at different stress levels and changing at a prefixed time or number [11]. e last type is progressive stress ALTs, in which the stress is kept with a continuous increase at all experiment steps [12]
Summary
GD : Gompertz distribution MLE : maximum likelihood estimation ME : mean PC : probability coverage CDF : cumulative distribution function HRF : hazard failure rate function xij: failure time under stress i and cause j MH : Metropolis–Hastings algorithm CI : credible intervals MCMC : Markov chain Monte Carlo MSE : mean squared error ML : mean interval length PDF : probability density function SF : survival function SEL : squared error loss δj: the indicator value expressed to cause. E joint likelihood function of observed data {(x1j;nj, δ1j) < (x2j;nj, δ2j) < · · · < (x]jj;nj, δ]jj)} with the CDF and PDF of random variables given byFlj(x) and flj(x), l 1, 2, denotes cause 1 and cause 2, respectively; it is given by 2 ⎧⎨ ]j δij 1− δij⎫⎬. Considering that, tested units with CDF given by (4) for used condition and for independent two causes of failure that are reduced to the distribution have PDFs given by f1l(x) θl expexp(αx). 3. Estimation under ML Method e results of the point and asymptotic confidence intervals of model parameters are discussed with MLE for two independent causes of failure. E Fisher information matrix is defined as the minus expectation of second derivatives from the log-likelihood function with respect to model parameters. Where zc presents the standard normal values with probability tailed diagonal of c and the the matrix vΦal−0u1e(sα,σθ111,, θσ22, 2β,).σ 33 , and σ44 are the
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