Study on Lindley Distribution Accelerated Life Tests: Application and Numerical Simulation
Abstract
:1. Introduction
2. Lindley Distribution and Its Importance
Assumptions and the Test Procedure and the Steps Used
 Let us assume that we have n identical and independent products that follows the Lindley distribution and these were subject to a lifetime examination in a lifetime experiment;
 The examination of the products ends as soon as the ${m}_{th}$ failure happens such that: $(m\le n)$;
 All units run in normaluse conditions and after a prefixed time $\eta $, the stress is increased by a certain value;
 From the physical experiments on products, engineers have stated that the following law controls the connection between the stress on the products S and scale parameter $\sigma $. Thus, the law can be stated as follows: The model of inverse power law (IPL) is given by: $ln\left({\sigma}_{i}\right)=a+b[ln\left({S}_{i}\right)]$, where $b>0$, and voltage is denoted by S, and a is the model parameter;
 We will apply progressive Type II censoring, as discussed above, on the units of this experiment;
 After running the test on the products, the number of units that failed before stress is ${n}_{1}$. In addition, ${n}_{2}$ is the total number of failed items after applying the stress at time $\eta $;
 We used the tampered random variable (TRV) model provided by [20]. This model states that under step stress partially accelerated life test (SSPALT), the lifetime of a unit can be written as:$$Z=\left\{\begin{array}{ccc}z\hfill & & if\phantom{\rule{4.pt}{0ex}}z\phantom{\rule{4.pt}{0ex}}\le \eta ,\hfill \\ \eta +\frac{z\eta}{\zeta}\hfill & & if\phantom{\rule{4.pt}{0ex}}z\phantom{\rule{4.pt}{0ex}}>\eta ,\hfill \end{array}\right.$$
 The PDF is divided as follows:$$f\left(z\right)=\left\{\begin{array}{c}0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}z<0,\hfill \\ {f}_{1}\left(z\right)=\left[{\displaystyle \frac{{\psi}^{2}(1+z){e}^{\psi z}}{1+\psi}}\right],\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}0<z<\eta ,\hfill \\ {f}_{2}\left(z\right)={\displaystyle \frac{{\psi}^{2}\zeta \left(1+\left(\zeta \right(z\eta )+\eta \right)){e}^{\psi \left(\zeta \right(z\eta )+\eta )}}{1+\psi}},\phantom{\rule{1.em}{0ex}}\eta <z<\infty .\hfill \end{array}\right.$$
3. Estimation Using the Maximum Likelihood Function
Point Estimation
4. Bayes Estimation
4.1. Using MCMC Method in Bayesian Estimation
4.2. The Metropolis—Hasting Algorithm
Algorithm 1 MCMC algorithm 
$$E\left(B\right)=\frac{1}{NM}\sum _{i=M+1}^{N}B({\psi}^{\left(i\right)},{\zeta}^{\left(i\right)}),$$
where $M=N/5$ is the burnin period. 
5. Interval Estimation
5.1. Finding Confidence Intervals for the Parameters
5.2. Bootstrap Confidence Intervals
Algorithm 2 Bootstrap algorithm 
Thus, we can get $100\phantom{\rule{0.277778em}{0ex}}(1\alpha )\%$ bootstrap CIs for ${\theta}_{i}$, given by:
$$\left(at{{\theta}_{i}}_{L}^{*},\phantom{\rule{0.277778em}{0ex}}at{{\theta}_{i}}_{U}^{*}\right)=\left(at{{\theta}_{i}}^{*[\zeta B/2]},\phantom{\rule{0.277778em}{0ex}}at{{\theta}_{i}}^{*\left[\right(1\zeta /2\left)B\right]}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,2,$$

5.3. Credible Confidence Intervals
Algorithm 3 Credible interval 
Thus, we can get $100\phantom{\rule{0.277778em}{0ex}}(1\alpha )\%$ credible CIs for ${\theta}_{i}$ is given by:
$$\left(at{{\theta}_{i}}_{SE}^{[\alpha N/2]},\phantom{\rule{0.277778em}{0ex}}at{{\theta}_{i}}_{SE}^{\left[\right(1\alpha /2\left)N\right]}\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,2.$$

6. Application on Real Data Set for Lindley Distribution
6.1. Example
6.2. Comparison with Competitive Distribution
6.3. Important Results Conducted from Real Data
 In the experiment with real data, we used a modified KS method to ensure that our data was a good fit for our distribution;
 According to the pvalues in Table 1, we deduced that our distribution made a good fit for the failure times of the experiment. After that, we first estimated the parameters using this real data, and then we concluded the CIs;
 By using the estimated parameters and the acceleration model estimates $ata,atb$, we deduced ${\theta}_{0}$, where ${\theta}_{0}$ is the scale parameter under normal use. From Equation (27), we can evaluate the MLE of the scale parameter under normal conditions $at{\theta}_{0}={e}^{ata+atbln\left({S}_{0}\right)}=0.0000214702$. Which is the scale parameter under normal use;
 By estimating the parameter under normal use we can use it to find the following:
 The mean time to failure (MTTF) under normal conditions is$$MTTF=\frac{2+{\theta}_{0}}{{\theta}_{0}(1+{\theta}_{0})}=93,151.3\phantom{\rule{0.277778em}{0ex}}\mathrm{h}.$$
 The failure rates (hazard rate function ) under normal conditions is:$$h\left(y\right)=\frac{\left((1+z){\theta}_{0}^{2}\right)}{{\theta}_{0}((1+{\theta}_{0}+z{\theta}_{0})},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}z>0,$$
 The reliability function under normal conditions is:$$R\left(y\right)=\left(1+{\displaystyle \frac{{\theta}_{0}z}{1+{\theta}_{0}}}\right){e}^{{\theta}_{0}z},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}z>0,$$
 By graphing the reliability function we deduced the following: Reliability function under normal use, at time, equals zero the reliability function equal one, see Figure 1. Under stress conditions, we concluded that the reliability function decreases, as time increases, see Figure 2. As the stress increase once more it approaches zero see Figure 3.
7. Simulation Studies
Important Results Conducted from Simulated Data
 As the sample size increased, the MSEs of BEs and MLEs estimation for the parameters $\psi $ and $\zeta $ decreased. Sometimes this situation did not occur because of small disturbances in data generation;
 The MSEs for BEs of $\psi $ and $\zeta $ are smaller than the MSEs of MLEs, and this is rational because the BE is the updated method, and more accurate than MLE;
 When the sample size increases, the length of the approximate, Bootstrap, and credible CIs reduced, except in some small iterations, and that is due to the randomization in the generation of data using the Mathematica package;
 The shortest interval is the credible CIs of $\psi $ and $\zeta $ according to the length, and credible CIs had the highest coverage probability;
 The length of Bootstrap CIs is shorter than the approximate CIs in most cases.
 We deduced that the credible CIs was the shortest one and had the highest coverage probability among all intervals.
Algorithm 4 The complete algorithm for all simulation in the paper 

8. Conclusions on Real Data and Simulation Results
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
Probability density function  
CDF  Cumulative distribution function 
SSPALT  Step Stress Partially Accelerated Life Test 
CIs  Credible confidence intervals 
BEs  Bayes estimates 
MLEs  Maximum Likelihood 
TRV  Tampered Random Variable 
KS  Kolmogorov–Smirnov 
ALT  Accelerated Life Test 
IPL  inverse power law 
SE  Square error 
MCMC  Markov chain Monte Carlo 
MTTF  The mean time to failure 
SEL  Square error loss 
CS  The censoring schemes 
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Value of Voltage Stress  2.25 V  2.44 V 

statistic  0.323529  0.945 
pvalue  0.0563701  0.789 
Value of Voltage Stress  2.25 V  2.44 V 

statistic  0.794118  0.6646 
pvalue  5.70644 × 10^{−20}  0.912 
MLEs  BSE  CI  

ACI  Credible Interval  Bootstrap CI  
$at{a}_{ML}$  $at{\psi}_{ML}$  $at{\psi}_{BS}$  $at\psi $  $at\psi $  $at\psi $ 
$at{b}_{ML}$  $at{\zeta}_{ML}$  $at{\zeta}_{BS}$  $at\zeta $  $at\zeta $  $at\zeta $ 
−51.8084  0.0230107  0.633091  0.0103733  0.73061  0.0276981 
59.2364  2.80211  4.29412  2.79169  2.11759  0.7885519 
n  m  C.S  Parameter  ML  SEL  CI  

ACI  Credible  Bootstrap  
20  10  1  $\psi $  0.0285412  4.2 $\times {10}^{6}$  1.03137  0.03463  0.31198 
$\zeta $  0.0188279  0.003221  3.48522  0.03546  0.41693  
50  30  1  $\psi $  0.0410944  6.31019 $\times {10}^{6}$  0.548811  0.03802  0.900282 
$\zeta $  0.437962  0.00227621  2.4142  0.03567  1.17764  
45  30  1  $\psi $  0.0199778  8.26362 $\times {10}^{6}$  0.602594  0.03751  0.352305 
$\zeta $  0.0172304  $1.41265\times {10}^{6}$  2.72445  0.03605  0.409257  
65  45  1  $\psi $  0.0200046  8.26362 $\times {10}^{6}$  0.6025941  0.03751  0.352305 
$\zeta $  0.0146925  $3.27477\times {10}^{6}$  2.53374  0.03745  0.410015  
100  60  1  $\psi $  0.0500046  4.26362 $\times {10}^{6}$  0.3025941  0.07001  0.302305 
$\zeta $  0.0126925  $2.27477\times {10}^{6}$  1.7924  0.03045  0.39105  
120  65  1  $\psi $  0.0127985  0.0000159836  0.402973  0.03755  0.350897 
$\zeta $  0.0204538  $4.19817\times {10}^{6}$  2.44254  0.0377  0.415675  
120  80  1  $\psi $  0.0204759  0.0000440016  0.368116  0.037755  0.297268 
$\zeta $  0.0150823  $4.48929\times {10}^{6}$  1.61679  0.0375  0.407272  
165  120  1  $\psi $  0.0206759  0.0000550016  0.291216  0.03885  0.24556 
$\zeta $  0.03223343  $6.39076\times {10}^{6}$  1.2992  0.03603  0.40672 
n  m  C.S  Parameter  Results Obtained for Each Interval  

ACI  Credible Intervale  Bootstrap CI  
20  10  1  $\psi $  0.60  0.87  0.65 
$\zeta $  0.88  0.871  0.88  
50  30  1  $\psi $  0.65  0.98  0.7 
$\zeta $  1  0.94  0.78  
45  30  1  $\psi $  0.75  0.92  0.88 
$\zeta $  0.9  0.91  0.92  
65  45  1  $\psi $  0.85  0.94  0.96 
$\zeta $  0.93  0.93  0.95  
120  65  1  $\psi $  0.86  0.95  0.96 
$\zeta $  0.94  0.95  0.96  
120  80  1  $\psi $  0.90  0.97  0.98 
$\zeta $  0.95  0.97  0.98  
120  80  1  $\psi $  0.90  0.97  0.98 
$\zeta $  0.95  0.97  0.98  
165  120  1  $\psi $  0.92  0.98  0.99 
$\zeta $  0.96  0.97  0.98 
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Hafez, E.H.; Riad, F.H.; Mubarak, S.A.M.; Mohamed, M.S. Study on Lindley Distribution Accelerated Life Tests: Application and Numerical Simulation. Symmetry 2020, 12, 2080. https://doi.org/10.3390/sym12122080
Hafez EH, Riad FH, Mubarak SAM, Mohamed MS. Study on Lindley Distribution Accelerated Life Tests: Application and Numerical Simulation. Symmetry. 2020; 12(12):2080. https://doi.org/10.3390/sym12122080
Chicago/Turabian StyleHafez, E. H., Fathy H. Riad, Sh. A. M. Mubarak, and M. S. Mohamed. 2020. "Study on Lindley Distribution Accelerated Life Tests: Application and Numerical Simulation" Symmetry 12, no. 12: 2080. https://doi.org/10.3390/sym12122080