Journal of Systems Engineering and Electronics ›› 2018, Vol. 29 ›› Issue (5): 1089-1100.doi: 10.21629/JSEE.2018.05.19
• Reliability • Previous Articles Next Articles
Xiaolin SHI1,*(), Pu LU2(
), Yimin SHI2(
)
Received:
2017-06-17
Online:
2018-10-26
Published:
2018-11-14
Contact:
Xiaolin SHI
E-mail:linda20016@163.com;1464293801@163.com;ymshi@nwpu.edu.cn
About author:
SHI Xiaolin was born in 1980. She received her B.S., M.S., and Ph.D. degrees in electrical engineering from Northwestern Polytechnical University. She is currently working as an associate professor of School of Electronics Engineering at Xi’an University of Posts & Telecommunications. Her research interests include system reliability analysis and statistical inference for masked data. E-mail: Supported by:
Xiaolin SHI, Pu LU, Yimin SHI. Inference and optimal design on step-stress partially accelerated life test for hybrid system with masked data[J]. Journal of Systems Engineering and Electronics, 2018, 29(5): 1089-1100.
Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks
Table 1
A-MLE of parameter and reliability index and MSE for different $\mathit{\boldsymbol{(n, R)}}$ and CS $\mathit{\boldsymbol{(p = 0.3)}}$ "
n | CS | $T_0 = 0.6, p = 0.3$ | |||||
$\widehat {\alpha }$ (MSE) | $\widehat {\beta }$ (MSE) | $\widehat {\lambda }$ (MSE) | $\widehat {b}$ (MSE) | $\widehat {R}_S $ (MSE) | $\widehat {h}_S $ (MSE) | ||
(40, 20) | 1 | 0.342 3 | 0.733 7 | 0.480 5 | 2.357 2 | 0.838 0 | 0.599 3 |
(0.044 8) | (0.080 1) | (0.569 3) | (1.972 7) | (0.007 2) | (0.059 2) | ||
2 | 0.259 7 | 0.691 5 | 0.397 6 | 1.782 3 | 0.909 4 | 0.457 2 | |
(0.035 1) | (0.219 0) | (1.904 3) | (1.752 6) | (0.003 4) | (0.067 2) | ||
3 | 0.268 8 | 0.718 8 | 0.574 5 | 2.202 1 | 0.825 8 | 0.403 9 | |
(0.028 9) | (0.086 8) | (1.326 3) | (1.802 2) | (0.002 6) | (0.041 1) | ||
(40, 25) | 1 | 0.312 1 | 0.835 9 | 0.485 4 | 1.977 0 | 0.883 9 | 0.490 8 |
(0.019 4) | (0.070 1) | (1.021 4) | (1.650 6) | (0.006 8) | (0.023 7) | ||
2 | 0.264 5 | 0.698 3 | 0.414 3 | 1.839 8 | 0.900 9 | 0.487 4 | |
(0.023 9) | (0.195 1) | (1.846 0) | (1.500 2) | (0.003 1) | (0.066 9) | ||
3 | 0.288 2 | 0.720 7 | 0.557 4 | 2.211 8 | 0.824 0 | 0.424 9 | |
(0.023 1) | (0.076 9) | (1.087 4) | (1.394 9) | (0.002 6) | (0.025 0) | ||
(50, 35) | 1 | 0.296 1 | 0.837 1 | 0.456 2 | 2.001 9 | 0.881 8 | 0.450 4 |
(0.012 6) | (0.028 6) | (0.516 5) | (1.411 1) | (0.002 6) | (0.017 9) | ||
2 | 0.261 9 | 0.754 7 | 0.511 6 | 1.861 9 | 0.895 8 | 0.484 5 | |
(0.014 2) | (0.049 3) | (1.128 4) | (1.211 6) | (0.001 5) | (0.052 3) | ||
3 | 0.285 2 | 0.723 1 | 0.557 6 | 1.908 1 | 0.855 1 | 0.418 6 | |
(0.014 6) | (0.030 5) | (0.572 0) | (1.303 4) | (0.002 3) | (0.023 5) |
Table 2
A-MLE of parameter and reliability index and MSE for different $\mathit{\boldsymbol{(n, R)}}$ and CS $\mathit{\boldsymbol{(p = 0.8)}}$ "
n | CS | $T_0 = 0.6, p = 0.3$ | |||||
$\widehat {\alpha }$ (MSE) | $\widehat {\beta }$ (MSE) | $\widehat {\lambda }$ (MSE) | $\widehat {b}$ (MSE) | $\widehat {R}_S $ (MSE) | $\widehat {h}_S $ (MSE) | ||
(40, 20) | 1 | 0.444 6 | 0.875 2 | 0.450 1 | 2.162 1 | 0.892 4 | 0.394 0 |
(0.082 0) | (0.084 1) | (1.043 2) | (2.571 3) | (0.008 1) | (0.033 2) | ||
2 | 0.239 7 | 0.745 6 | 0.572 0 | 1.849 7 | 0.912 7 | 0.308 7 | |
(0.038 2) | (0.241 0) | (1.921 6) | (1.827 6) | (0.003 8) | (0.066 7) | ||
3 | 0.269 8 | 0.759 6 | 0.588 7 | 2.227 5 | 0.845 6 | 0.396 1 | |
(0.032 4) | (0.087 1) | (1.674 6) | (1.857 8) | (0.002 8) | (0.051 4) | ||
(40, 25) | 1 | 0.283 7 | 0.848 1 | 0.512 1 | 2.039 5 | 0.895 0 | 0.400 4 |
(0.029 0) | (0.072 2) | (1.019 2) | (2.145 4) | (0.001 7) | (0.030 9) | ||
2 | 0.248 0 | 0.753 3 | 0.500 3 | 1.883 9 | 0.909 6 | 0.323 0 | |
(0.024 2) | (0.224 5) | (1.903 9) | (1.792 1) | (0.003 4) | (0.051 1) | ||
3 | 0.226 0 | 0.755 9 | 0.541 1 | 2.076 7 | 0.869 3 | 0.414 6 | |
(0.030 8) | (0.082 0) | (1.601 8) | (2.542 2) | (0.002 7) | (0.045 1) | ||
(50, 35) | 1 | 0.266 5 | 0.856 8 | 0.454 1 | 1.980 0 | 0.895 0 | 0.414 2 |
(0.012 7) | (0.030 1) | (0.589 4) | (1.515 5) | (0.003 6) | (0.023 2) | ||
2 | 0.241 5 | 0.832 9 | 0.542 6 | 1.970 6 | 0.901 6 | 0.369 8 | |
(0.022 4) | (0.051 2) | (1.280 9) | (1.296 2) | (0.001 9) | (0.028 9) | ||
3 | 0.263 9 | 0.762 0 | 0.611 2 | 0.902 5 | 0.866 7 | 0.409 5 | |
(0.015 7) | (0.037 5) | (0.785 3) | (2.298 6) | (0.003 4) | (0.038 6) |
Table 3
Average value of 95% CIs of parameters and corresponding CIL, CP $\mathit{\boldsymbol{(T_0 = 1.2, (n, R) = (50, 5))}}$ "
MP | CS | Parameter | ACIs\CIL CP | Boot-p CIs \ CIL CP | Boot-t CIs \ CIL CP |
p=0.3 | 1 | α | (0.112 1, 0.556 3)\0.444 1 95.0% | (0.111 0, 0.518 8)\0.407 7 95.6% | (0.191 4, 0.456 4)\0.264 9 95.4% |
β | (0.250 9, 1.151 8)\0.900 8 95.3% | (0.610 4, 0.975 3)\0.364 9 95.7% | (0.760 9, 1.080 2)\0.319 2 95.4% | ||
λ | ( – 0.282 5, 1.072 6)\1.355 1 95.0% | (0.145 9, 0.864 5)\0.718 6 96.4% | ( – 0.037 1, 0.819 1)\0.856 2 96.1% | ||
b | ( – 0.308 7, 2.432 3)\2.741 8 95.0% | (1.183 1, 2.619 6)\1.436 5 95.6% | (1.371 0, 2.472 3)\1.101 3 95% | ||
2 | α | (0.051 5, 0.525 9)\0.474 3 94.9% | (0.106 4, 0.548 9)\0.442 5 95.0% | (0.254 5, 0.408 4)\0.153 8 95.0% | |
β | (0.432 2, 0.926 2)\0.494 0 94.7% | (0.564 2, 0.945 6)\0.381 4 95.5% | (0.695 2, 0.962 0)\0.266 8 95.3% | ||
λ | ( – 0.044 6, 1.201 3)\1.245 9 94.8% | (0.114 1, 0.854 9)\0.740 8 95.4% | (0.250 2, 0.830 7)\0.580 5 95.0% | ||
b | (0.341 9, 2.793 7)\2.451 7 94.7% | (1.788 9, 2.475 3)\0.686 3 95.5% | (1.757 8, 2.399 2)\0.641 4 95.3% | ||
3 | α | (0.222 6, 0.570 4)\0.347 7 94.5% | (0.125 6, 0.547 8)\0.422 2 95.0% | (0.282 5, 0.416 4)\0.133 9 94.9% | |
β | ( – 0.286 8, 1.722 1)\2.009 1 94.5% | (0.611 8, 0.985 9)\0.374 1 95.2% | (0.511 3, 0.891 6)\0.380 2 95.1% | ||
λ | ( – 0.684 6, 1.798 4)\2.483 0 94.3% | (0.270 4, 0.849 1)\0.578 6 95.2% | (0.382 8, 0.636 5)\0.253 6 95.1% | ||
b | (1.225 9, 2.720 5)\1.494 6 94.0% | (1.458 6, 2.275 7)\0.817 1 95.5% | (1.602 1, 2.349 7)\0.747 6 95.2% | ||
p=0.8 | 1 | α | ( – 0.006 5, 0.857 7)\0.864 2 93.8% | (0.143 9, 0.639 9)\0.495 9 94.1% | (0.210 1, 0.357 1)\0.146 9 93.9% |
β | (0.072 2, 2.713 6)\2.641 3 94.2% | (0.552 3, 1.065 1)\0.512 7 94.3% | (0.600 4, 0.940 6)\0.340 2 93.7% | ||
λ | ( – 0.310 6, 1.362 4)\1.673 0 93.6% | ( – 0.009 9, 1.648 7)\1.658 6 96.1 % | ( – 0.468 3, 0.647 9)\1.116 3 94.1% | ||
b | (0.302 1, 3.282 3)\2.980 2 92.2% | (1.380 4, 3.029 4)\1.648 9 92.7% | (1.561 5, 3.007 3)\1.445 8 92.3% | ||
2 | α | ( – 0.000 7, 0.566 1)\0.566 8 92.2% | (0.072 4, 0.606 1)\0.533 7 93.4% | (0.175 4, 0.364 7)\0.189 2 93.2% | |
β | ( – 0.594 8, 1.400 8)\1.995 6 91.8% | (0.712 7, 1.777 3)\1.064 6 94.7% | (0.710 2, 1.015 9)\0.305 7 94.6% | ||
λ | ( – 0.085 7, 1.765 1)\1.850 8 92.7% | (0.075 9, 1.588 5)\1.512 6 93.9% | (0.173 3, 0.779 1)\0.605 8 92.8% | ||
b | ( – 0.301 2, 2.442 9)\2.744 1 92.6% | (1.092 5, 2.468 2)\1.375 7 92.8% | (1.616 3, 3.088 5)\1.472 2 92.6% | ||
3 | α | (0.222 9, 0.594 9)\0.372 0 93.3% | (0.119 8, 0.601 1)\0.481 2 93.9% | (0.209 4, 0.454 8)\0.245 3 93.4% | |
β | ( – 0.582 7, 1.478 9)\2.061 6 93.1% | (0.388 1, 0.925 9)\0.537 8 94.7% | (0.505 6, 0.988 4)\0.482 9 94.1% | ||
λ | ( – 0.585 6, 2.292 3)\2.877 9 93.7% | (0.127 3, 1.153 6)\1.026 3 94.6% | (0.301 1, 0.908 0)\0.606 8 94.3% | ||
b | (0.937 1, 2.871 0)\1.933 9 93.1% | (1.208 2, 2.550 0)\1.341 7 94.2% | (1.158 7, 2.405 6)\1.246 8 94.0% |
Table 4
Optimal time of changing stress levels under the two optimalcriteria"
CS | $p = 0.3$ | $p = 0.8$ | |||||
$\tau _D^\ast $ | $\tau _A^\ast $ | $\tau _D^\ast $ | $\tau _A^\ast $ | ||||
T0 = 0.6 | (40, 20) | 1 | 0.315 1 | 0.300 6 | 0.382 5 | 0.306 1 | |
2 | 0.373 3 | 0.293 1 | 0.340 1 | 0.362 0 | |||
3 | 0.385 5 | 0.308 6 | 0.381 8 | 0.299 3 | |||
(40, 25) | 1 | 0.349 4 | 0.302 9 | 0.393 9 | 0.308 1 | ||
2 | 0.374 2 | 0.313 6 | 0.369 8 | 0.363 5 | |||
3 | 0.393 8 | 0.312 7 | 0.382 7 | 0.300 8 | |||
(50, 35) | 1 | 0.388 2 | 0.311 6 | 0.411 5 | 0.318 3 | ||
2 | 0.392 3 | 0.357 4 | 0.377 6 | 0.370 3 | |||
3 | 0.391 3 | 0.317 0 | 0.383 9 | 0.304 9 | |||
T0 = 1.2 | (40, 20) | 1 | 0.305 9 | 0.355 0 | 0.337 7 | 0.342 6 | |
2 | 0.325 8 | 0.342 3 | 0.338 1 | 0.375 4 | |||
3 | 0.306 3 | 0.342 5 | 0.353 9 | 0.330 2 | |||
(40, 25) | 1 | 0.316 3 | 0.356 5 | 0.339 8 | 0.345 9 | ||
2 | 0.366 9 | 0.345 8 | 0.350 9 | 0.376 1 | |||
3 | 0.311 0 | 0.344 8 | 0.372 8 | 0.332 7 | |||
(50, 35) | 1 | 0.302 9 | 0.362 6 | 0.344 6 | 0.351 8 | ||
2 | 0.375 1 | 0.358 3 | 0.365 3 | 0.377 1 | |||
3 | 0.334 8 | 0.347 4 | 0.379 7 | 0.332 5 |
1 |
USHER J, HODGSON T. Maximum likelihood analysis of component reliability using masked system life-test data. IEEE Trans. on Reliability, 1988, 37 (5): 550- 555.
doi: 10.1109/24.9880 |
2 |
SARHAN A, GUESS F, USHER J. Estimators for reliability measures in geometric distribution model using dependent masked system life test data. IEEE Trans. on Reliability, 2007, 56 (2): 312- 320.
doi: 10.1109/TR.2007.895300 |
3 | BASU S, SEN A, BANERJEE M. Bayesian analysis of competing risks with partially masked cause of failure. Journal of the Royal Statistical Society, 2003, 52 (1): 77- 93. |
4 | SARHAN A M, EL-BASSIOUNY A H. Estimation of components reliability in a parallel system using masked system life data. Applied Mathematics & Computation, 2003, 138 (1): 61- 75. |
5 | FAN T H, WANG W L. Accelerated life tests for independent Weibull series systems with masked Type-I censored data. IEEE Trans. on Reliability, 2011, 60 (30): 557- 569. |
6 |
XU A, BASU S, TANG Y. A full Bayesian approach for masked data in step-stress accelerated life testing. IEEE Trans. on Reliability, 2014, 63 (3): 798- 806.
doi: 10.1109/TR.2014.2315940 |
7 |
FAN T H, HSU T M. Accelerated life tests of a series system with masked interval data under exponential lifetime distributions. IEEE Trans. on Reliability, 2012, 61 (3): 798- 808.
doi: 10.1109/TR.2012.2209259 |
8 | FAN T H, HSU T M, PENG K J. Bayesian inference of a series system on Weibull step-stress accelerated life tests with dependent masking. Quality Technology & Quantitative Management, 2013, 10 (3): 291- 303. |
9 | CAI J, SHI Y, LIU B, et al. Bayesian analysis for Burr-XII masked system in step-stress partially accelerated life test under Type-I progressive hybrid censoring. International Journal of Reliability Quality & Safety Engineering, 2015, 22 (4): 524- 526. |
10 |
LIU B, SHI Y, ZHANG F, et al. Reliability nonparametric Bayesian estimation for the masked data of parallel systems in step-stress accelerated life tests. Journal of Computational and Applied Mathematics, 2017, 311, 375- 386.
doi: 10.1016/j.cam.2016.07.029 |
11 | ZHANG Y, YUAN Y, ZHANG Q. Analysis on the reliability of the dual electric firing blasting network and application of the network in chamber blasting. Mining Research and Development, 2001, 21 (5): 41- 43. |
12 |
WANG R H, SHA N J, GU B Q, et al. Parameter inference in a hybrid system with masked data. IEEE Trans. on Reliability, 2015, 64 (2): 636- 644.
doi: 10.1109/TR.2015.2412537 |
13 | SHA N, WANG R, HU P, et al. Statistical inference in dependent component hybrid systems with masked data. Advances in Statistics, 2015, 2015, 525136. |
14 |
LAI C, XIE M, MURTHY D. A modified Weibull distribution. IEEE Trans. on Reliability, 2003, 52 (1): 33- 37.
doi: 10.1109/TR.2002.805788 |
15 | SOLIMAN A AHMED H, NASER A, et al. Modified Weibull model: a Bayes study using MCMC approach based on progressive censoring data. Reliability Engineering & System Safety, 2012, 100 (6): 48- 57. |
16 | NELSON W. Accelerated life testing: statistical models, data analysis and test plans. New York: John Wiley and Sons, 1990. |
17 | ISMAIL A, SARHAN A. Optimal design of step-stress life test with progressively Type-II censored exponential data. Proc. of the International Mathematical Forum, 2009, 4 (40): 1963- 1976. |
18 |
ABD-ELFATTAH A M, HASSAN A S, NASSR S G. Estimation in step-stress partially accelerated life tests for the Burr Type-XII distribution using Type-I censoring. Statistical Methodology, 2008, 5 (6): 502- 514.
doi: 10.1016/j.stamet.2007.12.001 |
19 | ABDEL-HAMID A, AL-HUSSAINI E. Inference and optimal design based on step-partially accelerated life tests for the generalized Pareto distribution under progressive Type-I censoring. Communication in Statistics-Simulation and Computation, 2014, 44 (7): 1750- 1769. |
20 | ISMAIL A. Inference in the generalized exponential distribution under partially accelerated tests with progressive Type-II censoring. Theoretical & Applied Fracture Mechanics, 2012, 59 (59): 49- 56. |
21 | ABDEL-GHANI M. The estimation problem of the loglogistic parameters in step partially accelerated life tests using Type-I censored data. The National Review of Social Sciences, 2004, 41 (2): 1- 19. |
22 | ISMAIL A. Inference for a step-stress partially accelerated life test model with an adaptive Type-II progressively hybrid censored data from Weibull distribution. Journal of Computational and Applied Mathematics, 2014, 260 (11): 533- 542. |
23 |
ABDEL-HAMID A, AL-HUSSAINI E. Inference and optimal design based on step partially accelerated life tests for the generalized pareto distribution under progressive Type-I censoring. Communications in Statistics-Simulation and Computation, 2015, 44 (7): 1750- 1769.
doi: 10.1080/03610918.2013.826363 |
24 | KUNDU D, JOARDER A. Analysis of Type-II progressively hybrid censored data. Computational Statistics & Data Analysis, 2006, 50 (10): 2509- 2528. |
25 | CHILDS A, CHANDRASEKAR B, BALAKRISHNAN N. Exact likelihood inference for an exponential parameter under progressive hybrid censoring schemes. VON T A F, NLIKULIN M, LIMNIOS N, et al., ed. Statistical models and methods for biomedical and technical systems, Boston, MA: Birkhäuser, 2008, 319-330. |
26 | TOMER S, GUPTA V, KUMAR J. Reliability estimation for Burr Type-XII distribution under Type-I progressive hybrid censoring scheme. Mathematics in Engineering, Science & Aerospace, 2015, 2 (6): 123- 134. |
27 | TIAN Y, ZHU Q, TIAN M. Estimation for mixed exponential distributions under Type-II progressively hybrid censored samples. Computational Statistics & Data Analysis, 2015, 89, 85- 96. |
28 |
PING S, NG H, FENG S. Exact likelihood inference for the two-parameter exponential distribution under Type-II progressively hybrid censoring. Metrika, 2015, 78 (6): 747- 770.
doi: 10.1007/s00184-014-0525-5 |
29 | SUN T, SHI Y, WEI W. Reliability analysis for B-S components under progressive hybrid Type-I censoring test. Journal of Systems Engineering and Electronics, 2014, 36 (11): 2326- 2332. |
30 |
CAI J, SHI Y, YUE H. Accelerated life tests for log-normal series system with dependent masked data under Type-I progressive hybrid censoring. Communications in Statistics- Simulation and Computation, 2017, 46 (2): 1628- 1646.
doi: 10.1080/03610918.2015.1045078 |
31 |
DEGROOT M, GOEL P. Bayesian estimation and optimal designs in partially accelerated life testing. Naval Research Logistics Quarterly, 1979, 26 (2): 223- 235.
doi: 10.1002/nav.3800260204 |
32 | EFRON B. The Jackknife, the Bootstrap and other resampling plans. Philadelphia: Society for Industrial and Applied Mathematics, 1982. |
33 |
HALL P. Theoretical comparison of bootstrap confidence intervals. Annals of Statistics, 1988, 16 (3): 927- 953.
doi: 10.1214/aos/1176350933 |
34 | BALAKRISHNAN N, AGGARWALA R. Progressive censoring: theory, methods, and applications. Boston, MA: Birkhäuser, 2000. |
[1] | Shunyi Zhao and Fei Liu. Bayesian estimation for nonlinear stochastic hybrid systems with state dependent transitions [J]. Journal of Systems Engineering and Electronics, 2012, 23(2): 242-249. |
[2] | He You, Xiong Wei & Ma Qiang. Composite filtering with feedback information [J]. Journal of Systems Engineering and Electronics, 2007, 18(1): 54-56. |
Viewed | ||||||
Full text |
|
|||||
Abstract |
|
|||||