加速不變原則之偏斜-t過程

以作者查詢圖書館館藏

、以作者查詢臺灣博碩士

、以作者查詢全國書目

、勘誤回報

、線上人數：133

、訪客IP：3.16.207.215

姓名

黃鴻緒(Hung-Hsu Huang) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

加速不變原則之偏斜-t過程
(Skew-t Processes Based on the Acceleration Invariance Principle)

相關論文

★ 穩定過程之衰變分析

★ Hougaard 過程之衰變分析

檔案

[Endnote RIS 格式]

[Bibtex 格式]

[相關文章]

[文章引用]

[完整記錄]

[館藏目錄]

至系統瀏覽論文 ( 永不開放)

摘要(中)

衰變試驗常用於推論高可靠度產品之壽命資訊，其中隨機過程因具有物理/化學的機制及工程上的解釋，從而廣泛地使用在衰變模型的建構上。本研究提出一非線性偏斜-t過程，可描述產品間變異、量測誤差、偏斜及厚尾特徵之衰變路徑，將文獻上常見的維納、偏斜維納及學生-t等過程涵蓋為特例。本文詳細探討此過程首次通過門檻值之產品壽命分布，並提供計算其機率密度函數之演算法的收斂性。在線性偏斜-t過程假設下，可推導出產品壽命之機率密度函數、累積分布函數與平均失效時間的解析式。除此之外，藉由加速不變原則理論分析所提過程之加速衰變模型中，加速變數與模型參數之間的關聯性。進而論敘此加速衰變模型之可辨別性，並提供參數估計之演算法。最後輔以兩組實際例子驗證所提模型之可行性。

摘要(英)

Degradation tests are often used to assess the lifetime of highly reliable products. Stochastic processes are widely used in the construction of degradation models due to the physical/chemical mechanisms and engineering interpretation. This research proposes a non-linear skew-t process, including Wiener, skew-Wiener and Student-t processes as special cases, under consideration for modelling the between-unit variability, measurement errors, skewed and heavy-tailed degradation paths. The first-passage-time distribution of the skew-t degradation-based process is discussed in detail, along with the convergence of an algorithm for calculating its density function. Under the assumption of linear skew-t process, the product′s lifetime probability density function, cumulative distribution function and mean-time-to-failure are derived in closed forms. In addition, the relationship between the accelerated variables and model parameters of the accelerated degradation model is obtained by using the acceleration invariance principle. The identifiability of the accelerated degradation model is investigated and two EM-type algorithms are proposed for the parameter estimation. Finally, the proposed accelerated degradation model is verified by two real datasets.

關鍵字(中)

★ 拔靴法
★ 共軛分布
★ 逆高斯分布
★ 積分方程式
★ 隨機效應

關鍵字(英)

★ bootstrap
★ conjugate distribution
★ inverse Gaussian distribution
★ integral equation
★ random effects

論文目次

摘要 i

Abstract ii

誌謝 iii

目錄 iv

圖目錄 vi

表目錄 vii

第一章緒論 1
1.1 背景介紹與研究動機 .............................. 1
1.2 文獻回顧 ....................................... 2
1.3 研究方法 ....................................... 3
1.4 本文架構 ....................................... 3

第二章偏斜-t 過程之衰變模型 4
2.1 偏斜-t 過程 .................................... 4
2.2 壽命分布 ....................................... 7
2.3 線性偏斜-t 過程性質 ............................ 10

第三章加速不變原則之偏斜-t 加速衰變模型 16
3.1 加速不變原則 ................................... 16
3.2 參數估計 ....................................... 19
3.2.1 EM 演算法 ................................ 20
3.2.2 ECM 演算法 ............................... 23
3.3 壽命資訊之估計 .................................. 25
3.4 適合度檢定 ..................................... 25

第四章實例分析 26
4.1 分析流程 ........................................ 26
4.2 紅外發光二極管資料分析 ........................... 27
4.3 發光二極管資料分析 ............................... 33

第五章結論與未來研究方向 42

參考文獻 43

附錄證明 46
A.1 定理2.2 ........................................ 46
A.1.1 線性偏斜-t 過程壽命分布之PDF ............... 46
A.1.2 線性偏斜-t 過程壽命分布之CDF ............... 47
A.1.3 線性偏斜-t 過程之MTTF ..................... 53
A.2 系理2.1 ........................................ 54
A.3 定理3.1 ........................................ 55
A.4 系理3.1 ........................................ 61
A.5 定理3.2 ........................................ 61
A.6 性質3.1 ........................................ 62

參考文獻

[1] 張孟筑(2017). 應用累積暴露模式至單調過程之加速衰變模型，國立中央大學統計研究所，碩士論文。

[2] Abramowitz, M. and Stegun, I. A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables, Washington, D. C.: US Government
Printing Office.

[3] Azzalini, A. (2005). The skew-normal distribution and related multivariate families, Scandinavian Journal of Statistics, 32, 159–188.

[4] Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution, Journal of the Royal Statistical Society: Series B
(Statistical Methodology), 61, 579–602.

[5] Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution, Biometrika, 83, 715–726.

[6] Chhikara, R. S., Folks, J. L. (1989). The inverse Gaussian distribution: Theory, methodology, and applications, New York: Marcel Dekker.

[7] 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:
Series B (Methodological), 39, 1–22.

[8] 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.

[9] Geller, M. and Ng, E. W. (1969). A table of integrals of the exponential integral, Journal of Research of the National Bureau of Standards, 71, 1–20.

[10] Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of integrals, series, and products, San Diego: Elsevier.

[11] Hamada, M. S., Wilson, A. G., Reese, C. S. and Martz, H. F. (2008). Bayesian reliability, New York: Springer-Verlag.

[12] Henze, N. (1986). A probabilistic representation of the skew-normal distribution Scandinavian Journal of Statistics, 13, 271–275.

[13] Liao, H. and Elsayed, E. A. (2006). Reliability inference for field conditions from accelerated degradation testing, Naval Research Logistics, 53, 576–587.

[14] Lin, T. I., Lee, J. C. and Hsieh, W. J. (2007). Robust mixture modeling using the skew t distribution, Statistics and Computing, 17, 81–92.

[15] Meeker, W. Q. and Escobar, L. A. (1998). Statistical methods for reliability data, New York: John Wiley & Sons.

[16] Meng, X. L. and Rubin, D. B. (1993). Maximum likelihood estimation via the ECM algorithm: A general framework, Biometrika, 80, 267–278.

[17] Nelson, W. (1980). Accelerated life testing-step-stress models and data analyses, IEEE Transactions on Reliability, 29, 103–108.

[18] Owen, D. B. (1956). Tables for computing bivariate normal probabilities, The Annals of Mathematical Statistics, 27, 1075–1090.

[19] Owen, D. B. (1980). A table of normal integrals, Communications in Statistics-Simulation and Computation, 9, 389–419.

[20] Peng, C. Y. (2015). Inverse Gaussian processes with random effects and explanatory variables for degradation data, Technometrics, 57, 100–111.

[21] Peng, C. Y. and Cheng, Y. S. (2016). Threshold degradation in R using iDEMO, Computational Network Analysis with R: Applications in Biology, Medicine and
Chemistry, Germany: John Wiley & Sons, 83–124.

[22] Peng, C. Y. and Cheng, Y. S. (2020). Student-t processes for degradation analysis, Technometrics, 62, 223–235.

[23] Peng, C. Y. and Tseng, S. T. (2009). Mis-specification analysis of linear degradation models, IEEE Transactions on Reliability, 58, 444–455.

[24] Peng, C. Y. and Tseng, S. T. (2013). Statistical lifetime inference with skew-Wiener linear degradation models, IEEE Transactions on Reliability, 62, 338–350.

[25] Pieruschka, E. (1961). Relation between lifetime distribution and the stress level causing the failures, Sunnyvale: Lockheed Missiles and Space.

[26] Singpurwalla, N. D. (1995). Survival in dynamic environments, Statistical Science, 10, 86–103.

[27] Wang, H. W. and Kang, R. (2020). Modeling of degradation data via Wiener stochastic process based on acceleration factor constant principle, Applied Mathematical Modelling, 84, 19–35.

[28] Wang, H. W. and Xi, W. J. (2016). Acceleration factor constant principle and the application under ADT, Quality and Reliability Engineering International, 32, 2591–2600.

[29] Whitmore, G. A. (1986). Normal-gamma mixtures of inverse Gaussian distributions, Scandinavian Journal of Statistics, 13, 211–220.

[30] Yang, G. (2007). Life cycle reliability engineering, New York: John Wiley & Sons.

指導教授

樊采虹彭健育(Tsai-Hung Fan Chien-Yu Peng)

審核日期

2021-8-10

推文