臺灣博碩士論文加值系統

衰變模型目前已廣泛地被用來評估高可靠度產品的壽命資訊。當產品存在一與壽命具高度相關之品質特徵值(Quality Characteristics, QC) 時, 則可透過此品質特徵值的衰變量來估計產品壽命分配, 因此, 評估產品壽命的準確度與產品衰變路徑的建模工作有著極密切的關連性。本文將針對衰變試驗所面臨的建模及相關資料分析進行研究。
(i) 文中首先提出一般化線性衰變模型, 其優點是可以同時將測試產品之間的差異性、衰變量與時間的關聯性及量測誤差等因素納入考慮。在此衰變模型下, 可推導出產品壽命之分配, 並進一步探討當衰變模型(隨機效應模型及Wiener 過程模型) 之間發生誤判時, 對產品平均壽命(mean-time-to-failure, MTTF) 之影響。一般而言, 在大樣本的假設下, 衰變模型之間的誤判對MTTF 之影響並不嚴重, 然而, 在小樣本或測試時間較短時, 其誤判的影響程度將不容忽略。
(ii) 當試驗總成本受限時, 本文採用平均壽命估計量之變異數極小化為準則, 提出一簡便演算法來決定執行前述衰變試驗所需之最適樣本數、測試頻率以及量測次數, 進而可精確地推估產品可靠度資訊。最後, 藉由敏感度(sensitivity) 分析,可探討衰變模型中之參數及試驗成本發生變動時, 對最佳實驗配置之影響。
(iii) 從實證資料可發現, 衰變模型中單位時間的衰變率(mean degradation rate)大多為正值且其分配並非左右對稱, 故採用偏斜常態(skew-normal) 分配將更適合用來描述產品的衰變路徑。在隨機效應為偏斜常態分配之假設下, 本文利用EM (Expectation-Maximization) 演算法來估計模型中之參數, 並推導出產品
之壽命分配。此外, 文中亦探討模型選擇(model selection) 問題, 由laser 的實證資料, 可發現此衰變模型較傳統常態假設下的隨機效應模型更具穩健性(robustness)。
(iv) 在新產品研發階段, 欲進行可靠度壽命測試時, 通常僅有少數的測試樣本可供使用。此時如何建構在經濟且有效的逐步應力(step-stress) 或連續型應力(progressivestress)之加速衰變試驗, 是生產者所面臨的重要課題。針對非線性衰變路徑且在連續型應力加速下, 本文採用累積暴露(cumulative exposure) 模型來建構出連
續應力與正常應力之間的時間轉換函數, 進而可推估產品在正常應力下之壽命分配, 此成果對縮短衰變試驗的時間將有顯著的貢獻。

Degradation models have been widely used to assess the lifetime information of highly reliable products. The performance of a degradation analysis strongly depends on the modeling of product’s degradation path. For designing and analyzing the degradation tests of highly reliable products, we study the following four topics in this thesis.
(i) Motivated by a real data set, we propose a general linear degradation model in which the unit-to-unit variation and time-dependent structure are simultaneously
considered. For this model, the product’s mean-time-to-failure (MTTF) can be obtained under some regular conditions. Furthermore, we also address the effects of model mis-specification on the prediction of product’s MTTF. It shows that the effect of model mis-specification on
product’s MTTF predictions is not critical when the sample size is large enough. However, when the sample size and termination time are not large enough, a simulation study shows that these effects are not negligible.
(ii) Under the proposed linear degradation model, we study the problem of optimal test plans. Under the constraint that the total experimental cost does not exceed a pre-determined budget, the optimal decision variables
such as sample size, sample frequency and terminational time are solved by minimizing the variance of the estimated MTTF of the lifetime distribution of the product. Moreover, we also assess the robustness of this degradation model through sensitivity analysis and address the effects of variety of parameters and cost conditions on the optimal test plans.
(iii) Motivated by a laser data, we relax the normal assumption of random-effect to fit realistic data sets. In this topic, we construct a skew-Wiener linear degradation model and derive the closed-form expression of the lifetime
distribution. Because the likelihood functions for such a degradation model are analytically intractable, we develop an EM type algorithm to efficiently obtain the maximum likelihood estimators for this model.
(iv) For highly reliable products with very few test units on hand, we use the concept of cumulative exposure model to formulate a typical progressive stress accelerated degradation test (PSADT) problem. An analytical expression
of the product’s lifetime distribution can then be obtained by using the first passage time of its degradation path. Next, an analytical performance comparison between the PSADT and the constant stress degradation test
under same special cases is present. The comparison includes the product’s MTTF, median lifetime, and variations of lifetime during different stresses.

1 緒論1
1.1 前言. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 衰變模型之簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 混合效應模型. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 隨機過程模型. . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 研究主題與動機. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 研究架構. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 文獻回顧8
2.1 衰變模型為一般化Wiener 過程之壽命分配. . . . . . . . . . . . . . 8
2.2 Cauchy 主值觀念下之負一階動差. . . . . . . . . . . . . . . . . . . 12
2.3 衰變模型誤判之理論基礎. . . . . . . . . . . . . . . . . . . . . . . 15
3 線性衰變模型之誤判分析20
3.1 動機例子與問題描述. . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 產品壽命之密度及分配函數. . . . . . . . . . . . . . . . . . . . . . 24
3.3 線性衰變模型之統計推論. . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 參數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 MLE 及MdTTF 之信賴區間. . . . . . . . . . . . . . . . . 27
3.3.3 MdTTF 之漸近分配. . . . . . . . . . . . . . . . . . . . . . 28
3.4 回顧動機例子. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 模型誤判效應之分析. . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5.1 有限樣本及終止時間之模擬分析. . . . . . . . . . . . . . . . 32
3.6 小結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 附錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7.1 定理3.1 之證明. . . . . . . . . . . . . . . . . . . . . . . . 40
3.7.2 定理3.2 之證明. . . . . . . . . . . . . . . . . . . . . . . . 41
3.7.3 線性衰變模型無截距項時之統計推論. . . . . . . . . . . . . 42
3.7.4 (3.23) 之導證. . . . . . . . . . . . . . . . . . . . . . . . . 44
3.7.5 (3.24) 之導證. . . . . . . . . . . . . . . . . . . . . . . . . 45
3.7.6 定理3.5 之證明. . . . . . . . . . . . . . . . . . . . . . . . 46
4 線性衰變模型之最佳實驗配置47
4.1 前言. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 衰變試驗之最佳化問題. . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.1 成本函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.2 最佳化模式. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.3 最佳實驗配置之演算法. . . . . . . . . . . . . . . . . . . . . 49
4.3 舉例說明. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.1 最佳實驗配置. . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.2 敏感度分析. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 小結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.5 附錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5.1 性質4.1 之證明. . . . . . . . . . . . . . . . . . . . . . . . 55
5 偏斜之Wiener 衰變模型與分析56
5.1 資料驗證. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 產品壽命之密度及分配函數. . . . . . . . . . . . . . . . . . . . . . 58
5.3 參數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.1 EM 演算法. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3.2 ECM 演算法. . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 回顧動機例子. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.5 小結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.6 附錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.1 定理5.1 之證明. . . . . . . . . . . . . . . . . . . . . . . . 66
5.6.2 定理5.7 之證明. . . . . . . . . . . . . . . . . . . . . . . . 68
5.6.3 (5.11) 之導證. . . . . . . . . . . . . . . . . . . . . . . . . 69
6 連續應力之非線性衰變模型建構分析70
6.1 動機例子. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 PSADT 之統計模型. . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 產品壽命分配. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.1 壽命分配T0 . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3.2 參數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4 PSADT 與NSDT 壽命分配之關聯. . . . . . . . . . . . . . . . . . 77
6.4.1 變異項為τ (η(S)t) . . . . . . . . . . . . . . . . . . . . . . . 77
6.4.2 變異項為τ (t) = 1 −M(t) . . . . . . . . . . . . . . . . . . 79
6.5 小結. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.6 附錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.6.1 定理6.2 之證明. . . . . . . . . . . . . . . . . . . . . . . . 83
6.6.2 定理6.7 之證明. . . . . . . . . . . . . . . . . . . . . . . . 84
7 廣義非線性模型及後續研究工作85
7.1 衰變模型之最佳實驗配置. . . . . . . . . . . . . . . . . . . . . . . 85
7.1.1 ADT 衰變模型之建構與統計推論. . . . . . . . . . . . . . . 85
7.2 廣義非線性衰變模型. . . . . . . . . . . . . . . . . . . . . . . . . . 89
參考文獻93
個人簡歷101

[1] Arellano-Valle R. B., Bolfarine, H. and Lachos, V. H. (2005), “Skew-normal
linear mixed models,” Journal of Data Science, 3, 415-438.
[2] Azzalini, A. (1985), “A class of distributions which includes the normal
ones,” Scandinavian Journal of Statistics, 12, 171-178.
[3] Azzalini, A. (1986), “Further results on a class of distributions which includes
the normal ones,” Statistica, 46, 199-208.
[4] Azzalini, A. (2005), “The skew-normal distribution and related multivariate
families,” Scandinavian Journal of Statistics, 32, 159-188.
[5] Azzalini A. and Capitanio A. (1999), “Statistical applications of the multivariate
skew-normal distributions,” Journal of the Royal Statistical Society.
B, 61, 579-602.
[6] Azzalini, A. and Dalla Valle, A. (1996), “The multivariate skew-normal distribution,”
Biometrika, 83, 715-726.
[7] Bae, S. J. and Kvam, P. H. (2004), “A nonlinear random coefficients model
for degradation testing,” Technometrics, 46, 460-469.
[8] Bagdonavicius, V. and Nikulin, M. (2000), “Estimation in degradation models
with explanatory variables,” Lifetime Data Analysis, 7, 85-103.
[9] Bagdonavicius, V. and Nikulin, M. (2002), Accelerated Life Models: Modeling
and Statistical Analysis, Chapman & Hall/CRC, New York.
[10] Boulanger, M. and Escobar, L. A. (1994), “Experimental design for a class
of accelerated degradation tests,” Technometrics, 36, 260-272.
[11] Chao, M. T. (1999), “Degradation analysis and related topics: Some
thoughts and a review,” The Proceedings of the National Science Council.
A, 23, 555-566.
[12] Chhikara, R. S. and Folks, L. (1989), The Inverse Gaussain Distribution.
Theory, Methodology, and Applications, Marcel Dekker, New York.
[13] Chow, Y. S. and Teicher, H. (1997), Probability Theory: independence, in-
terchangeability, martingales, 3rd ed, Springer-Verlag, New York.
[14] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977), “Maximum likelihood
from incomplete data via the EM algorithm,” Journal of the Royal
Statistical Society. B, 39, 1-22.
[15] Di Nardo, E., Nobile, A. G., Pirozzi, E. and Ricciardi, L. M. (2001), “A
computational approach to first-passage-time problems for Gauss-Markov
processes,” Advances in Applied Probability, 33, 453-482.
[16] Di Nardo, E., Nobile, A. G., Pirozzi, E. and Ricciardi, L. M. (2003), “On the
asymptotic behavior of first passage time densities for stationary Gaussian
processes and varying boundaries,” Methodology and Computing in Applied
Probability, 5, 211-233.
[17] Doksum, K. A. and H´oyland, A. (1992), “Model for variable-stress accelerated
life testing experiments based on Wiener processes and the inverse
Gaussian distribution,” Technometrics, 34, 74-82.
[18] Durham, S. D. and Padgett, W. J. (1997), “A cumulative damage model for
system failure with application to carbon fibers and composites,” Techno-
metrics, 39, 34-44.
[19] Ellison, B. E. (1964), “Two theorems for inferences about the normal distribution
with applications in acceptance sampling,” Journal of the American
Statistical Association, 59, 89-95.
[20] Feller, W. (1971), An Introduction to Probability Theory and Its Application,
Vol. 2, John Wiley & Sons, New York.
[21] Gertsbakh, I. B. and Kordonskiy, Kh. B. (1969), Models of Failure, Springer-
Verlag, New York.
[22] Gill, P. E., Murray, W. and Wright, M. H. (1981), Practical Optimization,
Academic Press, London.
[23] Gourieroux, C. and Monfort, A. (1995), Statistics and Econometric Models:
Volumn 1-2, Cambridge University Press, New York.
[24] Harville, D. A. (1977), “Maximum likelihood approaches to variance component
estimation and to related problems,” Journal of the American Statistical
Association, 72, 320-338.
[25] Hendry, D. F. (1995), Dynamic Econometrics, Oxford University Press, New
York.
[26] Henze, N. (1986), “A probabilistic representation of the skew-normal distribution,”
Scandinavian Journal of Statistics, 13, 271-275.
[27] Hoel, P. G., Port, S. C. and Stone, C. J. (1972), Introduction to Stochastic
Processes, Waveland Press, Illinois.
[28] Lawless, J. F. (2002), Statistical Models and Methods for Lifetime Data, John
Wiley & Sons, New York.
[29] Lawless, J. F. and Crowder, M. J. (2004), “Covariates and random effects
in a gamma process model with application to degradation and failure,”
Lifetime Data Analysis, 10, 213-227.
[30] LeCam, L. (1953), “On some asymptotic properties of maximum likelihood
estimates and related Bayes’ estimates,” University of California Publica-
tions in Statistics, 1, 277-330.
[31] Li, Q. and Kececioglu, D. B. (2004), “Optimal design of accelerated degradation
tests,” International Journal of Materials and Product Technology,
20, 73-90.
[32] Liao, C. M. and Tseng, S. T. (2006), “Optimal design for step-stress accelerated
degradation tests,” IEEE Transactions on Reliability, 55, 59-66.
[33] Lu, J. (1995), “Degradation processes and related reliability models,” Un-
published Ph.D. Thesis, McGill University, Canada.
[34] Lu, C. J. and Meeker, W. Q. (1993), “Using degradation measures to estimate
a time-to-failure distribution,” Technometrics, 35, 161-174.
[35] Lu, J. C., Park, J. and Yang, Q. (1997), “Statistical inference of a time-tofailure
distribution derived from linear degradation data,” Technometrics,
39, 391-400.
[36] Meeker, W. Q. and Escobar, L. A. (1998), Statistical Methods for Reliability
Data, John Wiley & Sons, New York.
[37] Meeker, W. Q., Escobar, L. A. and Lu, C. J. (1998), “Accelerated degradation
tests: modeling and analysis,” Technometrics, 40, 89-99.
[38] Meng, X. L. and Rubin, D.B. (1993), “Maximum likelihood estimation via
the ECM algorithm: a general framework,” Biometrika, 80, 267-278.
[39] Nelson, W. (1990), Accelerated Testing: Statistical Models, Test Plans, and
Data Analyses, John Wiley & Sons, New York.
[40] Ng, T. S. (2008), “An application of the EM algorithm to degradation modeling,”
IEEE Transactions on Reliability, 57, 2-13.
[41] O’Hagan, A. and Leonard, T. (1976), “Bayes estimation subject to uncertainty
about parameter constraints,” Biometrika, 63, 201-203.
[42] Onar, A. and Padgett, W. J. (2000), “Inverse Gaussian accelerated test
models based on cumulative damage,” Journal of Statistical Computation
and Simulation, 66, 233-247.
[43] Owen, D. B. (1980), “A table of normal integrals,” Communications in
Statistics: Part B - Simulation and Computation, 9, 389-419.
[44] Park, S. J., Yum, B. J. and Balamurali, S. (2004), “Optimal design of stepstress
degradation tests in the case of destructive measurement,” Quality
Technology & Quantitative Management, 1, 105-124.
[45] Padgett, W. J. (1998), “A multiplicative damage model for strength of fibrous
composite materials,” IEEE Transactions on Reliability, 47, 46-52.
[46] Padgett, W. J. and Tomlinson, M. A. (2002), “A cumulative damage model
for strength of materials when initial damage is a gamma process,” Journal
of Statistical Theory and Applications, 1, 1-14.
[47] Park, C. and Padgett, W. J. (2005a), “Accelerated degradation models for
failure based on geometric Brownian motion and gamma processes,” Lifetime
Data Analysis, 11, 511-527.
[48] Park, C. and Padgett, W. J. (2005b), “New cumulative damage models for
failure using stochastic processes as initial damage,” IEEE Transactions on
Reliability, 54, 530-540.
[49] Park, C. and Padgett, W. J. (2006), “Stochastic degradation models with
several accelerating variables,” IEEE Transactions on Reliability, 55, 379-
390.
[50] Pascual, F. G. (2006), “Theory for accelerated life test plans robust to misspecification
of stress-life relationship,” Technometrics, 48, 11-25.
[51] Pascual, F. G. and Montepiedra, G. (2005), “Accelerated life testing under
distribution misspecification: biased estimation and test planning,” IEEE
Transactions on Reliability, 54, 43-52.
[52] Patel, R. C. (1965), “Estimates of parameters of truncated inverse Gaussian
distribution,” Annals of the Institute of Statistical Mathematics, 17, 29-33.
[53] Peng, C. Y. (2008), “The first negative moment in the sense of the Cauchy
principal value,” Statistics & Probability Letters, in press.
[54] Peng, C. Y., and Tseng, S. T. (2008), “Mis-specification analysis of linear
degradation models,” Submitted for publication.
[55] Quenouille, M. H. (1956), “Notes on bias in estimation,” Biometrika, 43,
353-360.
[56] Schott, J. R. (2005), Matrix Analysis for Statistics, 2nd ed, John Wiley &
Sons, New York.
[57] Seshadri, V. (1999), The Inverse Gaussian Distribution: Statistical Theory
and Applications, Springer-Verlag, New York.
[58] Tseng, S. T. and Liao, C. M. (1998), “Optimal design for a degradation
test,” International Journal of Operations and Quantitative Management, 4,
293-301.
[59] Tseng, S. T. and Peng, C. Y. (2004), “Optimal burn-in policy by using
integrated Wiener process,” IIE Transactions, 36, 1161-1170.
[60] Tseng, S. T. and Peng, C. Y. (2007), “Stochastic diffusion modeling of degradation
data,” Journal of Data Science, 5, 315-333.
[61] Tseng, S. T. and Wen, Z. C. (2000), “Step-stress accelerated degradation
analysis for highly reliable products,” Journal of Quality Technology, 32,
209-216.
[62] Tseng, S. T. and Yu, H. F. (1997), “A termination rule for degradation
experiment,” IEEE Transactions on Reliability, 46, 130-133.
[63] Voinov, V. G. (1985), “Unbiased estimation of powers of the inverse of mean
and related problems,” Sankhy‾a, B, 47, 354-364.
[64] Wald, A. (1949), “Note on the consistency of the maximum likelihood estimate,”
Annals of Mathematical Statistics, 60, 595-603.
[65] White, H. (1982), “Maximum likelihood estimation of misspecified models,”
Econometrica, 50, 1-25.
[66] Whitmore, G. A. (1986), “Normal-gamma mixtures of inverse Gaussian distributions,”
Scandinavian Journal of Statistics, 13, 211-220.
[67] Whitmore, G. A. (1995), “Estimating degradation by a Wiener diffusion
process subject to measurement error,” Lifetime Data Analysis, 1, 307-319.
[68] Whitmore, G. A., Crowder, M. I. and Lawless, J. F. (1998), “Failure inference
from a marker process based on a bivariate model,” Lifetime Data Analysis,
4, 229-251.
[69] Whitmore, G. A. and Schenkelberg, F. (1997), “Modeling accelerated degradation
data using Wiener diffusion with a scale transformation,” Lifetime
Data Analysis, 3, 27-45.
[70] Wu, C. F. J. (1983), “On the convergence properties of the EM algrithm,”
Annals of Statistics, 11, 95-103.
[71] 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.
[72] Yu, H. F. and Tseng, S. T. (1998), “On-line procedure for terminating an
accelerated degradation test,” Statistica Sinica, 8, 207-220.
[73] Yu, H. F. and Tseng, S. T. (1999), “Designing a degradation experiment,”
Naval Research Logistics, 46, 689-706.
[74] Yu, H. F. and Tseng, S. T. (2004), “Designing a degradation experiment with
a reciprocal Weibull degradation rate,” Quality Technology & Quantitative
Management, 1, 47-63.
[75] Zacks, S. (1971), The Theory of Statistical Inference, John Wiley & Sons,
New York.