Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution

Sadia Anwar; Sana Shahab; Arif Ul Islam

doi:doi:10.11648/j.ajtas.20150403.22

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Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution

Sadia Anwar, Sana Shahab, Arif Ul Islam

Published in American Journal of Theoretical and Applied Statistics (Volume 4, Issue 3)

Received: 17 April 2015 Accepted: 28 April 2015 Published: 11 May 2015

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Abstract

This article contains the optimum 3 step stress accelerated life test under cumulative exposure model. The lifetimes of test units are assumed to follow a generalized Pareto distribution. The scale parameter of the used failure time distribution at the constant stress level is assumed to have a log-linear and quadratic relationship with the stress. A comparison between linear plan and quadratic plan by maximum likelihood estimators for the different sample sizes is shown in the table. The optimum test plans is obtained by minimizing the asymptotic variance of the maximum likelihood estimator of the percentile of the lifetime distribution at normal stress condition for the model parameters. Tables of optimum times of changing stress level for both plans are also obtained.

Published in	American Journal of Theoretical and Applied Statistics (Volume 4, Issue 3)
DOI	10.11648/j.ajtas.20150403.22
Page(s)	163-169
Creative Commons	This is an Open Access article, 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 or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Accelerated Life Testing, Generalized Pareto Distribution, Asymptotic Variance, Maximum Likelihood Estimate, Life Stress Relationship (Linear and Quadratic), Cumulative Exposure Model

References

[1]	Alhadeed, A. A. and Yang, S. S. (2005). Optimal Simple Step-Stress Plan for Cumulative Exposure Model Using Log-Normal Distribution. IEEE Transaction on Reliability, 54(1), 64-68.
[2]	Bai, D. S., Kim, M.S. and Lee, S. H. (1989). Optimum Simple Step-Stress Accelerated Life Tests with Censoring. IEEE Transactions on Reliability, 38(5), 528-532.
[3]	Bander-Al Zahrani (2012). Maximum Likelihood Estimation for Generalized Pareto Distribution under Progressive Censoring with Binomial Removals. Open Journal of Statistics, Scientific Research, 2, 420-423. http://dx.doi.org/10.4236/ojs.2012.24051
[4]	Fard, N. and Li, C. (2009). Optimal Simple Step Stress Accelerated Life Test Design for Reliability Prediction, Journal of Statistical Planning and Inference, 139(5), 1799-1808.
[5]	Hunt, S. and Xu, X. (2012). Optimum Design for Accelerated Life Testing with Simple Step- Stress Plans. International Journal of per formability Engineering, 8(5), 575- 579.
[6]	Khamis, I. H. and Higgins J. J. (1996). Optimum 3-Step Step-Stress Tests. IEEE Transactions on Reliability, 45(2), 341-345.
[7]	Khamis, I. H. (1997). Optimum M-Step, Step-Stress Design with K Stress Variables. Communicationsin Statistics, Simulation and Computation, 26(4), 1301-1313.
[8]	Khamis I.H., and Higgins J.J. (1998)., A new model for step-stress testing, IEEE Transactions on Reliability, 47,131
[9]	Lu, M.W. & Rudy, R.J. (2002.) Step-stress accelerated test, International Journal of Materials & Product Technology, 17 (5-6), 425–434.
[10]	McSorley, E.O., Lu, J.C. & Li, C.S. (2002). Performance of parameter-estimates in step stress accelerated life-tests with various sample-sizes, IEEE Transactions on Reliability, 51, (3), 271–277.
[11]	Miller, R., and Nelson, W. B. (1983). Optimum Simple Step-Stress Plans for Accelerated Life Testing. IEEE Transactions on Reliability, R-32(1), 59-65
[12]	Nelson, W. (1990), Accelerated Testing, Statistical Models, Test Plans and Data Analysis. John Wiley & Sons.
[13]	Xiong, C. (1999) Step stress model with threshold parameter, Journal of Statistical Computation and Simulation, 63, 349–360.

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APA Style

Sadia Anwar, Sana Shahab, Arif Ul Islam. (2015). Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution. American Journal of Theoretical and Applied Statistics, 4(3), 163-169. https://doi.org/10.11648/j.ajtas.20150403.22

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ACS Style

Sadia Anwar; Sana Shahab; Arif Ul Islam. Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution. Am. J. Theor. Appl. Stat. 2015, 4(3), 163-169. doi: 10.11648/j.ajtas.20150403.22

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AMA Style

Sadia Anwar, Sana Shahab, Arif Ul Islam. Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution. Am J Theor Appl Stat. 2015;4(3):163-169. doi: 10.11648/j.ajtas.20150403.22

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@article{10.11648/j.ajtas.20150403.22,
  author = {Sadia Anwar and Sana Shahab and Arif Ul Islam},
  title = {Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {4},
  number = {3},
  pages = {163-169},
  doi = {10.11648/j.ajtas.20150403.22},
  url = {https://doi.org/10.11648/j.ajtas.20150403.22},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150403.22},
  abstract = {This article contains the optimum 3 step stress accelerated life test under cumulative exposure model. The lifetimes of test units are assumed to follow a generalized Pareto distribution. The scale parameter of the used failure time distribution at the constant stress level is assumed to have a log-linear and quadratic relationship with the stress. A comparison between linear plan and quadratic plan by maximum likelihood estimators for the different sample sizes is shown in the table. The optimum test plans is obtained by minimizing the asymptotic variance of the maximum likelihood estimator of the   percentile of the lifetime distribution at normal stress condition for the model parameters. Tables of optimum times of changing stress level for both plans are also obtained.},
 year = {2015}
}

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TY  - JOUR
T1  - Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
AU  - Sadia Anwar
AU  - Sana Shahab
AU  - Arif Ul Islam
Y1  - 2015/05/11
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150403.22
DO  - 10.11648/j.ajtas.20150403.22
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 163
EP  - 169
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20150403.22
AB  - This article contains the optimum 3 step stress accelerated life test under cumulative exposure model. The lifetimes of test units are assumed to follow a generalized Pareto distribution. The scale parameter of the used failure time distribution at the constant stress level is assumed to have a log-linear and quadratic relationship with the stress. A comparison between linear plan and quadratic plan by maximum likelihood estimators for the different sample sizes is shown in the table. The optimum test plans is obtained by minimizing the asymptotic variance of the maximum likelihood estimator of the   percentile of the lifetime distribution at normal stress condition for the model parameters. Tables of optimum times of changing stress level for both plans are also obtained.
VL  - 4
IS  - 3
ER  -

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Author Information

Sadia Anwar

Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, U.P, India
Sana Shahab

Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, U.P, India
Arif Ul Islam

Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, U.P, India

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APA Style

Sadia Anwar, Sana Shahab, Arif Ul Islam. (2015). Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution. American Journal of Theoretical and Applied Statistics, 4(3), 163-169. https://doi.org/10.11648/j.ajtas.20150403.22

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ACS Style

Sadia Anwar; Sana Shahab; Arif Ul Islam. Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution. Am. J. Theor. Appl. Stat. 2015, 4(3), 163-169. doi: 10.11648/j.ajtas.20150403.22

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AMA Style

Sadia Anwar, Sana Shahab, Arif Ul Islam. Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution. Am J Theor Appl Stat. 2015;4(3):163-169. doi: 10.11648/j.ajtas.20150403.22

Copy | Download

@article{10.11648/j.ajtas.20150403.22,
  author = {Sadia Anwar and Sana Shahab and Arif Ul Islam},
  title = {Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {4},
  number = {3},
  pages = {163-169},
  doi = {10.11648/j.ajtas.20150403.22},
  url = {https://doi.org/10.11648/j.ajtas.20150403.22},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150403.22},
  abstract = {This article contains the optimum 3 step stress accelerated life test under cumulative exposure model. The lifetimes of test units are assumed to follow a generalized Pareto distribution. The scale parameter of the used failure time distribution at the constant stress level is assumed to have a log-linear and quadratic relationship with the stress. A comparison between linear plan and quadratic plan by maximum likelihood estimators for the different sample sizes is shown in the table. The optimum test plans is obtained by minimizing the asymptotic variance of the maximum likelihood estimator of the   percentile of the lifetime distribution at normal stress condition for the model parameters. Tables of optimum times of changing stress level for both plans are also obtained.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modeling of Optimum 3 Step Stress Accelerated Life Testing for Generalized Pareto Distribution
AU  - Sadia Anwar
AU  - Sana Shahab
AU  - Arif Ul Islam
Y1  - 2015/05/11
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150403.22
DO  - 10.11648/j.ajtas.20150403.22
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 163
EP  - 169
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20150403.22
AB  - This article contains the optimum 3 step stress accelerated life test under cumulative exposure model. The lifetimes of test units are assumed to follow a generalized Pareto distribution. The scale parameter of the used failure time distribution at the constant stress level is assumed to have a log-linear and quadratic relationship with the stress. A comparison between linear plan and quadratic plan by maximum likelihood estimators for the different sample sizes is shown in the table. The optimum test plans is obtained by minimizing the asymptotic variance of the maximum likelihood estimator of the   percentile of the lifetime distribution at normal stress condition for the model parameters. Tables of optimum times of changing stress level for both plans are also obtained.
VL  - 4
IS  - 3
ER  -

Copy | Download