Bi-optimal Quantile-based Test Planning for Accelerated Degradation Test Based on a Wiener Process

Ya-Shan Cheng and Chien-Yu Peng

doi:10.5705/ss.202025.0285

Abstract

Under limited resources, the widely used Vtq-optimal test

plan determines the sample size, termination time, and number of

measurements by minimizing the approximate variances of the estimated q-quantile, tq, for highly reliable products.

This approach is

economically efficient when the Vtq-optimal test plan simultaneously

satisfies another optimality criterion through an appropriate choice of

q. Therefore, we theoretically study a bi-optimal quantile-based test

plan based on a Wiener process, which achieves 100% efficiency for

two optimality criteria.

The necessary and sufficient conditions for

its existence and uniqueness are derived, which can then be used to

determine the optimal test configuration for accelerated degradation

tests. Two numerical examples are presented to illustrate the practical applicability of the proposed bi-optimal quantile-based test plan.

Key words and phrases: Cost-constrained optimization, D-optimal frequency, inverse Gaussian distribution, monotonicity, signal-to-noise ratio

Information

Preprint No.	SS-2025-0285
Manuscript ID	SS-2025-0285
Complete Authors	Ya-Shan Cheng, Chien-Yu Peng
Corresponding Authors	Chien-Yu Peng
Emails	chienyu@stat.sinica.edu.tw

References

Atkinson, A. C., Donev, A. N., and Tobias, R. D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press, New York.
Bagdonavičius, V. and Nikulin, M. S. (2001). Accelerated Life Models: Modeling and Statistical Analysis. Chapman & Hall/CRC, Boca Raton.
Boulanger, M. and Escobar, L. A. (1994). Experimental design for a class of accelerated degradation tests. Technometrics 36, 260–272.
Cheng, Y. S., Chen, Y., and Lee, M. L. T. (2025). Longitudinal survival analysis using first hitting time threshold regression: with applications to Wiener processes. Stats 8, Article 32.
Cheng, Y. S. and Peng, C. Y. (2012). Integrated degradation models in R using iDEMO. Journal of Statistical Software 49, 1–22.
Cheng, Y. S. and Peng, C. Y. (2024). Optimal test planning for heterogeneous Wiener processes. Naval Research Logistics 71, 509–520.
Chernoff, H. (1953). Locally optimal designs for estimating parameters. Annals of Mathematical Statistics 24, 586–602.
Chhikara, R. S. and Folks, L. (1989). The Inverse Gaussian Distribution: Theory, Methodology, and Applications. Marcel Dekker, New York. D’Hondt, M., Fedorova, M., Peng, C. Y., Gevaert, B., Taevernier, L., Hoffmann, R., and De
Spiegeleer, B. (2014). Dry heat forced degradation of buserelin peptide: Kinetics and degradant profiling. International Journal of Pharmaceutics 467, 48–59.
Dong, Y. S. and Peng, C. Y. (2025). Importance inference of optimal test planning for degradation analysis. IEEE Transactions on Reliability 74, 4426–4440.
Hu, C. H., Lee, M. Y., and Tang, J. (2015). Optimum step-stress accelerated degradation test for Wiener degradation process under constraints. European Journal of Operational Research 241, 412–421.
Lee, I. C., Tseng, S. T., and Hong, Y. (2020). Global planning of accelerated degradation tests based on exponential dispersion degradation models. Naval Research Logistics 67, 469–483.
Liao, C. M. and Tseng, S. T. (2006). Optimal design for step-stress accelerated degradation tests. IEEE Transactions on Reliability 55, 59–66.
Lim, H. (2015). Optimum accelerated degradation tests for the gamma degradation process case under the constraint of total cost. Entropy 17, 2556–2572.
Lim, H. and Yum, B. J. (2011). Optimal design of accelerated degradation tests based on Wiener process models. Journal of Applied Statistics 38, 309–325.
Meeker, W. Q., Escobar, L. A., and Pascual, F. G. (2022). Statistical Methods for Reliability Data. 2nd Edition. John Wiley & Sons, New York.
Nelson, W. B. (2004). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. John Wiley & Sons, New York.
Peng, C. Y. and Cheng, Y. S. (2021). Profile optimum planning for degradation analysis. Naval Research Logistics 68, 951–962.
Prasad, M., Gopika, V., and Shridharan, A. (2020). Application of Hougaard stochastic model for flow-accelerated corrosion wall thinning in an orifice. Life Cycle Reliability and Safety Engineering 9, 349–352.
Prasad, M., Gopika, V., Sridharan, A., Parida, S., and Gaikwad, A. J. (2018). Hougaard process stochastic model to predict wall thickness in flow accelerated corrosion. Annals of Nuclear Energy 117, 247–258.
Pukelsheim, F. (1993). Optimal Design of Experiments. John Wiley & Sons, New York.
R Core Team (2025). R: a language and environment for statistical computing, Vienna: R foundation for statistical computing. Available at http://www.R-project.org/
Taha, H. A. (2017). Operations Research: An Introduction. 10th Edition. Pearson Education, New York.
Tsai, C. C., Tseng, S. T., and Balakrishnan, N. (2012). Optimal design for gamma degradation processes with random effects. IEEE Transactions on Reliability 61, 604–613.
Tseng, S. T. and Lee, I. C. (2016). Optimum allocation rule for accelerated degradation tests with a class of exponential-dispersion degradation models. Technometrics 58, 244–254.
Tseng, S. T., Tsai, C. C., and Balakrishnan, N. (2011). Optimal sample size allocation for accelerated degradation test based on Wiener process. In Methods and Applications of Statistics in Engineering, Quality Control, and the Physical Sciences (Ed. N. Balakrishnan), pp. 330–343. John Wiley & Sons, New York.
Wang, Y. H. and Wong, S. H. D. (2015). Modelling accelerated degradation test and shelf-life prediction of dye-sensitized solar cells with different types of solvents. Solar Energy 118, 600–610.
Whitmore, G. A. and Yalovsky, M. (1978). A normalizing logarithmic transformation for inverse Gaussian random variables. Technometrics 20, 207–208.
Wu, S. J. and Chang, C. T. (2002). Optimal design of degradation tests in presence of cost constraint. Reliability Engineering and System Safety 76, 109–115.
Wu, C. F. J. and Hamada, M. (2021). Experiments: Planning, Analysis, and Optimization. 3rd Edition. John Wiley & Sons, New York.
Yang, C. H., Hsu, Y. H., and Hu, C. H. (2024). Mis-specification analyses and optimum degradation test plan for Wiener and inverse Gaussian processes. Communications in Statistics— Theory and Methods 53, 700–717.
Yang, G. (2007). Life Cycle Reliability Engineering. John Wiley & Sons, New York.
Yu, H. F. and Tseng, S. T. (1999). Designing a degradation experiment. Naval Research Logistics 46, 689–706. Ya-Shan Cheng Graduate Institute of Statistics, National Central University, Taoyuan 320317, Taiwan.

Acknowledgments

The authors are grateful to the Editor, the Associate Editor, and anonymous referees for their insightful and constructive comments. This work of

Ya-Shan Cheng and Chien-Yu Peng was supported in part by the National

Science and Technology Council of Republic of China (Taiwan) under Grant

Number NSTC-114-2118-M-008-003-MY2 and NSTC-112-2118-M-001-005-

MY2, respectively.

Supplementary Materials

The online Supplementary Material contains the Fisher information matrix

of the accelerated degradation model based on a Wiener process, the proofs

of Theorems 1, 3, 4, Corollaries 1, 2, Proposition 1 and the derivation of

(4.5).

Supplementary materials are available for download.

[1] Atkinson, A. C., Donev, A. N., and Tobias, R. D. (2007). Optimum Experimental Designs, with SAS. Oxford University Press, New York.

[2] Bagdonavičius, V. and Nikulin, M. S. (2001). Accelerated Life Models: Modeling and Statistical Analysis. Chapman & Hall/CRC, Boca Raton.

[3] Boulanger, M. and Escobar, L. A. (1994). Experimental design for a class of accelerated degradation tests. Technometrics 36, 260–272.

[4] Cheng, Y. S., Chen, Y., and Lee, M. L. T. (2025). Longitudinal survival analysis using first hitting time threshold regression: with applications to Wiener processes. Stats 8, Article 32.

[5] Cheng, Y. S. and Peng, C. Y. (2012). Integrated degradation models in R using iDEMO. Journal of Statistical Software 49, 1–22.

[6] Cheng, Y. S. and Peng, C. Y. (2024). Optimal test planning for heterogeneous Wiener processes. Naval Research Logistics 71, 509–520.

[7] Chernoff, H. (1953). Locally optimal designs for estimating parameters. Annals of Mathematical Statistics 24, 586–602.

[8] Chhikara, R. S. and Folks, L. (1989). The Inverse Gaussian Distribution: Theory, Methodology, and Applications. Marcel Dekker, New York. D’Hondt, M., Fedorova, M., Peng, C. Y., Gevaert, B., Taevernier, L., Hoffmann, R., and De

[9] Spiegeleer, B. (2014). Dry heat forced degradation of buserelin peptide: Kinetics and degradant profiling. International Journal of Pharmaceutics 467, 48–59.

[10] Dong, Y. S. and Peng, C. Y. (2025). Importance inference of optimal test planning for degradation analysis. IEEE Transactions on Reliability 74, 4426–4440.

[11] Hu, C. H., Lee, M. Y., and Tang, J. (2015). Optimum step-stress accelerated degradation test for Wiener degradation process under constraints. European Journal of Operational Research 241, 412–421.

[12] Lee, I. C., Tseng, S. T., and Hong, Y. (2020). Global planning of accelerated degradation tests based on exponential dispersion degradation models. Naval Research Logistics 67, 469–483.

[13] Liao, C. M. and Tseng, S. T. (2006). Optimal design for step-stress accelerated degradation tests. IEEE Transactions on Reliability 55, 59–66.

[14] Lim, H. (2015). Optimum accelerated degradation tests for the gamma degradation process case under the constraint of total cost. Entropy 17, 2556–2572.

[15] Lim, H. and Yum, B. J. (2011). Optimal design of accelerated degradation tests based on Wiener process models. Journal of Applied Statistics 38, 309–325.

[16] Meeker, W. Q., Escobar, L. A., and Pascual, F. G. (2022). Statistical Methods for Reliability Data. 2nd Edition. John Wiley & Sons, New York.

[17] Nelson, W. B. (2004). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. John Wiley & Sons, New York.

[18] Peng, C. Y. and Cheng, Y. S. (2021). Profile optimum planning for degradation analysis. Naval Research Logistics 68, 951–962.

[19] Prasad, M., Gopika, V., and Shridharan, A. (2020). Application of Hougaard stochastic model for flow-accelerated corrosion wall thinning in an orifice. Life Cycle Reliability and Safety Engineering 9, 349–352.

[20] Prasad, M., Gopika, V., Sridharan, A., Parida, S., and Gaikwad, A. J. (2018). Hougaard process stochastic model to predict wall thickness in flow accelerated corrosion. Annals of Nuclear Energy 117, 247–258.

[21] Pukelsheim, F. (1993). Optimal Design of Experiments. John Wiley & Sons, New York.

[22] R Core Team (2025). R: a language and environment for statistical computing, Vienna: R foundation for statistical computing. Available at http://www.R-project.org/

[23] Taha, H. A. (2017). Operations Research: An Introduction. 10th Edition. Pearson Education, New York.

[24] Tsai, C. C., Tseng, S. T., and Balakrishnan, N. (2012). Optimal design for gamma degradation processes with random effects. IEEE Transactions on Reliability 61, 604–613.

[25] Tseng, S. T. and Lee, I. C. (2016). Optimum allocation rule for accelerated degradation tests with a class of exponential-dispersion degradation models. Technometrics 58, 244–254.

[26] Tseng, S. T., Tsai, C. C., and Balakrishnan, N. (2011). Optimal sample size allocation for accelerated degradation test based on Wiener process. In Methods and Applications of Statistics in Engineering, Quality Control, and the Physical Sciences (Ed. N. Balakrishnan), pp. 330–343. John Wiley & Sons, New York.

[27] Wang, Y. H. and Wong, S. H. D. (2015). Modelling accelerated degradation test and shelf-life prediction of dye-sensitized solar cells with different types of solvents. Solar Energy 118, 600–610.

[28] Whitmore, G. A. and Yalovsky, M. (1978). A normalizing logarithmic transformation for inverse Gaussian random variables. Technometrics 20, 207–208.

[29] Wu, S. J. and Chang, C. T. (2002). Optimal design of degradation tests in presence of cost constraint. Reliability Engineering and System Safety 76, 109–115.

[30] Wu, C. F. J. and Hamada, M. (2021). Experiments: Planning, Analysis, and Optimization. 3rd Edition. John Wiley & Sons, New York.

[31] Yang, C. H., Hsu, Y. H., and Hu, C. H. (2024). Mis-specification analyses and optimum degradation test plan for Wiener and inverse Gaussian processes. Communications in Statistics— Theory and Methods 53, 700–717.

[32] Yang, G. (2007). Life Cycle Reliability Engineering. John Wiley & Sons, New York.

[33] Yu, H. F. and Tseng, S. T. (1999). Designing a degradation experiment. Naval Research Logistics 46, 689–706. Ya-Shan Cheng Graduate Institute of Statistics, National Central University, Taoyuan 320317, Taiwan.