最近幾年來，由於高科技產品，例如:智慧型手機和平板電腦等的盛行，消費者對於產品的品質要求越加嚴格，在產業高度競爭的時代，廠商該如何提升製程能力，是品管上很重要的工作。在實務上，製程能力指標(process capability indices, PCIs)被廣泛應用在評估製程的績效，進而不斷地提升產品品質及製程能力。
本研究假設產品的壽命服從Weibull分配時，在逐步型I區間設限下，計算出壽命績效指標 之最大概似估計量，並探討其漸近分配與檢定力函數，在規格下限L已知的情形下，利用此估計量及兩種拔靴法，發展出三個新的假設檢定程序，以判定壽命績效是否達到預期的能力水準。最後，我們用兩個數值實例說明如何使用本研究所提出的檢定程序。

In recent years, due to the prevalence of smart phones and tablet PCs, the consumers require more stringent product quality in the highly competitive commercial market. In practice, process capability indices (PCIs) has been widely used to assess the performance of the process, and then continues to be employed to improve the product quality and process capability.
This research is focusing on the lifetime of products following the Weibull distribution. The maximum likelihood estimator is used to estimate the lifetime performance index (C_L)  based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also investigated. We use this estimator and two kinds of bootstrap methods to develop three kinds of new hypothesis testing algorithmic procedure in the condition of known lower specification limit L. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.

目錄	I
表目錄	III
圖目錄	IV
第一章 緒論	1
1.1 研究動機與目的	1
1.2 文獻探討	3
1.2.1 製程能力指標之發展	3
1.2.2 設限樣本	5
1.3 本文架構	7
第二章 Weibull分配壽命績效指標與其估計	8
2.1 產品的壽命績效指標C_L	10
2.2壽命績效指標的估計量	13
第三章 壽命績效指標的檢定演算程序	18
3.1 壽命績效指標的檢定演算程序	18
3.2 壽命績效指標檢定的檢定力	22
第四章  模擬與數值實例分析	29
4.1 數值實例	29
4.2 模擬範例	34
第五章  結論與未來研究	38
5.1 結論	38
5.2 未來研究	39
參考文獻	40
附錄	42
表目錄
表 2.1 壽命績效指標C_L值對應之製程良率Pr 	12
附表1 當規格下限L=0.025，總觀測時間T=0.5，觀測樣本數n=30、40、50、60，設限樣本數m=5、6及逐步移除率p=0.05時，在目標值C_0=0.8和顯著水準alpha=0.1下，檢定力函數h(c_1)在c_1=0.75,0.8(0.0125),0.9,0.95的數值	42
附表2 當規格下限L=0.025，總觀測時間T=0.5，觀測樣本數n=30、40、50、60，設限樣本數m=5、6及逐步移除率p=0.05時，在目標值C_0=0.8和顯著水準alpha=0.05下，檢定力函數h(c_1)在c_1=0.75,0.8(0.0125),0.9,0.95的數值	44
圖目錄 
圖1.1 逐步型I區間設限圖	6
圖2.1 雙參數在lambda=1,2時Weibull分配之機率密度函數圖	9
圖2.2 雙參數在lambda=1,2時Weibull分配之故障率函數圖	9
圖3.1 當alpha=0.1、m=5、n=30及p=0.05下，對不同檢定方法的檢定力函數。	25
圖3.2 當alpha=0.1、n=30及p=0.05下，對不同的設限樣本m=(5,6)時的檢定力函數。	25
圖3.3 當alpha=0.1、m=5及p=0.05下，對不同總樣本n=(30,40,50,60)時的檢定力函數。	26
圖3.4 當alpha=0.05、m=5、n=30及p=0.05下，對不同檢定方法的檢定力函數	26
圖3.5 當alpha=0.05、n=30及p=0.05下，對不同的設限樣本m=(5,6)時的檢定力函數。	27
圖3.6 當alpha=0.05、m=5及p=0.05下，對不同總樣本n=(30,40,50,60)時的檢定力函數。	27
圖3.7 當m=5、n=30及p=0.05下，對不同alpha=0.05,0.1的檢定力函數。	28
圖4.1 不同beta下之p-value	30

[1]  Boyles, R. A. (1991), The Taguchi capability index, Journal of Quality Technology, 23(1), pp. 17–26. 
[2]  Cohen, A. C. (1963), Progressively Censored Samples in Life Testing,Technometrics, 5(3), pp. 327–339.
[3]  Caroni, C. (2002), The correct “ball bearings” data, Lifetime Data Analysis, 8, pp.395-399.
[4]  Chan, L. K., Cheng, S. W. and Spiring, F. A. (1988), A new measure of process capability Cpm, Journal of Quality Technology, 20(3), pp.162-175.
[5] Efron, B. (1982), The Jackknife, the Bootstrap and Other Re-samplingPlans, CBMS-NSF Regional Conference Series in Applied Mathematics, 38, SIAM, Philadelphia, PA.
[6]  Frechet, M . (1927): Sur la loi de probabilit de l'ecart maximum, Ann. Soc. Polon. Math. (Cracovie), 6, pp. 93-116
[7]  Gill, M. H. and Gastwirth, J. L. (1978), A sacle-free goodness-of-fit Test for the Exponential Distribution Based on the Gini Statistic, Journal of the Royal Statistical Society, Series B(Methodological), 40, pp. 350–357.
[8] Hall, P. (1988), Theoretical comparison of bootstrap confidence intervals, Annals of Statistics, 16, pp. 927–953.
[9] Hong, C. W., Lee, W. C. and Wu, J. W. (2012),Computational Procedure of Performance Assessment of Lifetime Index of Products for the Weibull Distribution with the Progressive First-Failure Censored Sampling Plan, Journal of Applied Mathematics, Article ID 717184.  
[10] Hong, C. W., Wu, J. W., and Cheng, C. H. (2008), Computational procedure of performance assessment of lifetime index of Pareto lifetime businesses based on confidence interval, Applied Soft Computing , 8, no. 1, pp. 698–705.
[11] Juran, J. M. (1974), Journal Quality Control Handbook, 3nded Edition, McGraw-Hill, New York.
[12] Kane, V. E. (1986), Process capability indices, Journal of Quality Technology, 18, pp. 41–52.
[13] Lawless, J. F. (2003), Statistical Models and Methods for Lifetime Data, (2nded), New York, John Wiley.
[14] Montgomery, D. C. (1985), Introduction to statistical quality control, John Wiley and Sons, New York.
[15] Pearn, W. L., Kotz, S. and Johnson, N. L. (1992), Distributional and inferential properties of process capability indices, Journal of Quality Technology, 24(4), pp. 216–231.
[16] Tong, L. I., Chen, K. S. and Chen, H. T. (2002), Statistical testing for assessing the performance of lifetime index of electronic components with exponential distribution, International Journal of Quality＆Reliability Management, 19(7), pp. 812–824.
[17] Tonyng, H. K. and Wang, Z. (2011), Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval censored sample, Journal of Statistical Computation and Simulation,79, No. 2, 145–159.
[18] Weibull, W. A. (1951), A statistical distribution function of wide applicability,  Journal of Applied Mechanics ,18,293-297.
[19] Wu, J. W., Lee, H. M. , Lee, W. C. and  Lei, C. L. (2011), Computational procedure of assessing lifetime performance index of Weibull lifetime products with the upper record values, Mathematics  and Computers in Simulation , 81, 1177–1189
[20] Wu, J. W., Lee, H. M. and Lei, C. L. (2007), Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution, Applied Mathematics and Computation, 190, pp. 116–125.