工業統計或生物統計上，壽命的分佈在許多情況下以指數分佈加以描述。為此，估計或檢定其壽命相關參數，是一重要課題。壽命的實驗常花費很長時間。尤其在工業成品上，因為工業技術的長足發展，壽命資料的收集常是非常耗時。因此在統計上設法加以改良，其中之一即為設限資料的設計。本文主要分別探討設限資料下壽命均值的檢定與估計問題。有關設限資料的檢定問題，以往學者以檢測實驗停止時所觀測到元件壽命的總個數或以壽命均值的最大概度估計量 (Maximum Likelihood Estimator, MLE）作為其檢定量。本文在型I設限資料下，提出一個新的檢定量，而該檢定量是基於兩個統計量，分別為壽命均值的最大概度估計量及觀測到的元件個數。在數值分析上並與其他的檢定方法做比較。另一方面，有關設限資料的估計問題，以往學者所提出的設限設計，主要在設法縮短實驗時間下估計產品的壽命均值，但因為沒有提出損失函數，所以在兼顧縮短時間及更精確估計壽命之目標難以達成。為此，本文提出合理的損失函數以決策理論方式加以探討如何達成精確估計壽命的同時又兼具縮減實驗時間及元件個數的成本。此外，本文也提出了一個新的設限設計。

對於設限資料下之壽命檢定問題，本文所提出的檢定量 φ*=f(overset{wedge}{theta},M_T)$ 為壽命均值的最大概度估計量 $overset{wedge}{theta}$ 及觀測到壽命元件的個數 $M_T$ 之函數。除理論部份外，以數值分析驗證在樣本個數分別為10,30及50的情況下，壽命均值分別為θ≧0.6, θ≧0.75及θ≧0.8時，本文所提出的檢定力都優於Spurrier and Wei（1980）的檢定（α=0.05）。

在設限設計方面，本文提出了損失函數為壽命估計量的平方誤差、實驗時間以及實驗個數之加權線性組合，並計算其風險函數。對於新近所廣為討論的一般化型I、一般化型II以及統合混合設限設計作了深入的探討。本文在這些設計上之參數提出了最適解。此外，本文亦提出另一混合型設限設計。在各設計之參數為最適解時，本文在一些情況下作了數值上的風險比較。本文所提之混合型設限設計在許多情況下都優於已知設限設計。

In many situations, life time is often described by exponential family. Accordingly, inference for the exponential mean is one of important issues. Due to great breakthrough in technology in last twenty years, life times of industrial products often become much longer than they were. Therefore, in life testing experiment, it often spends lots of time to collect life time data and this often exhausts lots of budget. To overcome this dilemma, various censoring schemes have been proposed. This leads to our main goal to study inferences concerning exponential mean θ based on censored data in this dissertation.

Our main work consists of two parts which are respectively, based on censored data, the hypotheses testing on θ, and the maximum likelihood estimator for θ. For the first part, we consider H_{0}:θ≧θ0 versus H_{1}:θ<θ0 for given θ0. We propose a new test φ* which is based on respectively two statistical quantities $overset{wedge}{theta}$ and $M_T$, the MLE of θ and the number of observed items based on type-I censoring. Some numerical comparisons with that of Spurrier and Wei (1980) have been carried out and it is found that when respectively θ≧0.6, θ≧0.75 and θ≧0.8, φ* is superior to that of Spurrier and Wei (1980) in the sense of power when sample sizes are respectively 10, 30 and 50 under type I error α=0.05.

In the second part, we consider a decision-theoretic approach with a loss function which is a weighted linear combination of squared error, duration of experiment and total number of items for experiment. We have derived the risks for various censoring schemes such as generalized type-I hybrid, generalized type-II hybrid and the unified hybrid censoring schemes. Some numerical optimal solutions for parameters associated with various schemes have been tabulated. Furthermore, a so-called combined hybrid censoring scheme has been proposed. Numerical comparisons in the sense of superiority (in terms of its risk) among those schemes have also been studied. It is found that the proposed combined hybrid censoring scheme is most superior in several situations.

目錄...I
圖目錄...III
表目錄...IV
使用符號一覽表...V

目錄
第一章 緒論...1
1.1 研究動機與目的...1
1.2 本文架構...4
第二章 文獻探討...5
第三章 型I設限下標準指數母體均值之局部最適性檢定...9
3.1 前言...9
3.2 基於型I設限資料下之局部最適性檢定...10
3.2.1 模式之建立...10
3.2.2 局部最適性檢定...11
3.3 φ* 的一些統計性質...14
3.3.1 概度比檢定...14
3.3.2 檢定之不偏性...14
3.3.3 與貝氏檢定的類似性...15
3.4 臨界值之計算...17
3.5 檢定力之比較...20
3.6 數值範例...23
第四章 另一混合設限設計之新抽樣模式...30
4.1 前言...30
4.2 組合混合型設限設計及其均值估計...31
4.3 不同目標下之損失函數及其最適解...37
4.3.1 考慮估計誤差和檢測時間之損失函數...37
4.3.2 考慮估計誤差、檢測時間和檢測數量成本之損失函數...39
4.3.3 檢測樣本在考慮剩餘價值下之損失函數...41
4.4 最適解之演算法...44
4.5 模擬的最適解和數值比較...45
第五章 結論...65
5.1 主要研究成果...65
5.2 未來研究方向...66
參考文獻...68
附錄...74
A. 定理4.1之證明...74
B. 定理4.2之證明...76
C. 定理4.3之證明...77
D. 定理4.4之證明...81
E. 定理4.5之證明...85

圖目錄
4.1 效率比較圖 ($n=10,c_1=c_2=0.5$)...60
4.2 效率比較圖 ($n=20,c_1=c_2=0.5$)...60
4.3 效率比較圖 ($c_1=c_2=c_3=1/3$)...61
4.4 效率比較圖 ($c_1=c_3=0.5,c_2=0$)...61
4.5 效率比較圖 ($c_1=0.5,c_2=0.3,c_3=0.2$)...62
4.6 效率比較圖 ($c_1=0.6,c_2=c_3=0.2$)...62
4.7 效率比較圖 ($c_1=c_2=c_3=1/3,c_4=1/6$)...63
4.8 效率比較圖 ($c_1=c_3=0.5,c_2=0,c_4=0.25$)...63
4.9 效率比較圖 ($c_1=0.5,c_2=0.3,c_3=0.2,c_4=0.1$)...64
4.10 效率比較圖 ($c_1=0.6,c_2=c_3=0.2,c_4=0.1$)...64

表目錄
3.1 $c(n,alpha,R)$ 值($alpha=0.01$, $0.025, 0.05$)...24
3.2 φ* 與 φSW 之檢定力比較 ($n=10(10)50$, $alpha=0.05, θ0=1$)...25
4.1 最適解 $(T_{10},T_{20},k_{0},r_{0})$ 及其風險值 ($n=10,c_1=c_2=0.5$)...48
4.2 最適解 $(T_{10},T_{20},k_{0},r_{0})$ 及其風險值 ($n=20,c_1=c_2=0.5$)...50
4.3 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=c_2=c_3=1/3$)...52
4.4 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=c_3=0.5,c_2=0$)...53
4.5 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=0.5,c_2=0.3,c_3=0.2$)...54
4.6 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=0.6,c_2=c_3=0.2$)...55
4.7 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=c_2=c_3=1/3,c_4=1/6$)...56
4.8 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=c_3=0.5,c_2=0,c_4=0.25$)...57
4.9 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值
($c_1=0.5,c_2=0.3,c_3=0.2,c_4=0.1$)...58
4.10 最適解 $(T_{10},T_{20},k_{0},r_{0},n_{0})$ 及其風險值 ($c_1=0.6,c_2=c_3=0.2,c_4=0.1$)...59

[1] Balakrishnan, N., Abbas Rasouli and Sanjari Farsipour, N. (2008). Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution. Journal of Statistical Computation and Simulation, 78 (5), 475-488.

[2] Balakrishnan, N. and Han, D. (2008). Exact inference for a simple step-stress model with competing risks for failure from exponential distribution under type-II censoring. Journal of Statistical Planning and Inference, 138 (12), 4172-4186.

[3] Balakrishnan, N. and Lin, C. T. (2005). Exact inference and prediction for k-sample exponential case under type-II censoring. Journal of Statistical Computation and Simulation, 75 (5), 315-331.

[4] Balakrishnan, N. and Xie, Q. (2007a). Exact inference for a simple step-stress model with type-II hybrid censored data from the exponential distribution. Journal of Statistical Planning and Inference, 137 (8), 2543-2563.

[5] Balakrishnan, N. and Xie, Q. (2007b). Exact inference for a simple step-stress model with type-I hybrid censored data from the exponential distribution. Journal of Statistical Planning and Inference, 137 (11), 3268-3290.

[6] Banerjee, A. and Kundu, D. (2008). Inference based on type-II hybrid censored data from a weibull distribution. IEEE Transactions on Reliability, 57 (2), 369-378.

[7] Bartholomew, D. J. (1963). The sampling distribution of an estimate arising in life testing. Technometrics, 5, 361-374.

[8] Chandrasekar, B., Childs, A. and Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring. Naval Research Logistics, 51, 994-1004.

[9] Chen, J. W., Chow, W. L., Wu, H. L. and Zhou, H. B. (2004). Designing acceptance sampling schemes for life testing with mixed censoring. Naval Research Logistics, 51, 597-612.

[10] Chen, S. and Bhattacharyya, G. K. (1988). Exact confidence bounds for an exponential parameter under hybrid censoring. Communications in Statistics: Theory and Methods, 17, 1857-1870.

[11] Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. (2003). Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, 319-330.

[12] Choy, S. T. B. (1992). An algorithm for optimal. Naval Research Logistics, 33, 513-522.

[13] Cohen, A. C. (1963). Progressively censored samples in the life testing. Technometrics, 5, 327-339.

[14] Engelhardt, M. and Bain, L. J. (1978). Tolerance limits and confidence limits on reliability for the two-parameter exponential distribution. Technometrics, 20, 37-39.

[15] Epstein, B. (1954). Truncated life tests in the exponential case. Ann. Math. Statist., 25, 555-564.

[16] Grubbs, F. E. (1971). Approximate fiducial bounds on reliability for the two parameter negative exponential distribution. Technometrics, 13, 873-876.

[17] Guenther, W. C., Patil, S. A. and Uppuluri, V. R. R. (1976). One-sided content tolerance factors for the two parameter exponential distribution, Technometrics, 18, 333-340.

[18] Gupta, S. S. and Liang, T. (1993). Selecting the best exponential population based on type-I censored data: a Bayesian approach. In Advances in Reliability (Ed. Asit P. Basu), Elsevier Science Publishers B. V., 171-180.

[19] Huang, W. T. and Chen, H. S. (1992). Estimation of the exponential mean under type-I censored sampling. Journal of Statistical Planning and Inference, 33, 187-196.

[20] Huang, W. T. and Lin, Y. P. (2004). Bayesian sampling plans for exponential distribution based on uniform random censored data. Computational Statistics and Data Analysis, 44 (4), 669-691.

[21] Kocherlakota, S. and Balakrishnan, N. (1986). One- and two-sided sampling plans based on the exponential distribution. Naval Research Logistics, 33, 513-522.

[22] Lam, Y. (1994). Bayesian variable sampling plans for the exponential distribution with type-I censoring. Ann. Statist., 22 (2), 696-711.

[23] Lam, Y. and Choy, S. T. B. (1995). Bayesian variable sampling plans for the exponential distribution with uniformly distributed random censoring. Journal of Statistical Planning and Inference, 47, 277-293.

[24] Lin, Y. P., Liang, T. and Huang, W. T. (2002). Bayesian sampling plans for exponential distribution based on type-I censoring data. Ann. Inst. Statist. Math., 54 (1), 100-113.

[25] Pierce, D. A. (1973). Fiducial frequency, and Bayesian inference on reliability for the two-parameter negative exponential distribution. Technometrics, 15, 249-253.

[26] Sinha, S. K. (1986). Reliability and life testing. Wiley Eastern Ltd., New Delhi.

[27] Spurrier, J. D. and Wei, L. J. (1980). A test of the parameter of the exponential distribution in the type-I censoring case. J. Amer. Statist. Assoc., 75, 405-409.

[28] Yang, G. and Sirvanci, M. (1977). Estimation of a time-truncated exponential parameter used in life testing. J. Amer. Statist. Assoc., 72, 444-447.

[29] Yuen, H. K. and Tse, S. K. (1996). Parameters estimation for Weibull distribution lifetime under progressive censoring with random removals. Journal of Statistical Computation and Simulation, 55, 57-71.