在進行產品可靠度的分析及改善時，通常需要做產品的抽樣壽命實驗，希望能利用已觀測到的產品壽命來估計參數與評估產品品質是否達到所需的水準。本文首先考慮以逐步第一失敗設限計畫，探討浴缸型壽命分配中參數的最大概似估計量與其相關特性、信賴區間及信賴區域。此外，在製造業中，製程能力指標常被用來評估產品品質是否達到所需的水準，而壽命績效指標C_L就是評估工具之一。再以服從浴缸型壽命分配的逐步第一失敗設限樣本來建構壽命績效指標C_L之最大概似估計量(MLE)，進而利用C_L之估計量求得壽命績效指標C_L之信賴區間，同時發展一檢定程序，以評估產品之壽命是否達到所需的水準，進而針對壽命績效指標的檢定力及信賴區間進行蒙地卡羅模擬(Monte Carlo simulation)。最後，將以數值範例說明如何應用所提出之方法，分析評估產品的壽命績效指標是否達到所要求的水準。

During the analysis and improvement of product’s reliability, we usually need to do sampling test. We can take advantage of the product’s lifetime which has been observed to estimate parameters and to assess whether the product quality to meet the required level. In this article, we consider the progressively first-failure-censored scheme to investigate the maximum likelihood estimator (MLE), exact and approximate confidence intervals and an exact confidence region for the parameters of the bathtub-shaped distribution. In addition, process capability indices are often used to assess whether the product quality to meet the required level, and the lifetime performance index is one of the useful tools for evaluating the performance. Also, we construct a MLE of the lifetime performance index C_L based on the progressive first-failure-censored sample under the bathtub-shaped distribution. The MLE of C_L is then utilized to develop the confidence interval of  C_L and the hypothesis testing procedure in the condition of known  L to determine whether the lifetime performance of products adhere to the required level. And some the Monte Carlo simulations are made for the computation of the power of the test and the confidence level. Finally, two numerical examples are to illustrate how to apply the proposed method to analyze the performance index to assess the product's lifetime to meet the required level.

目錄
目錄......................................................I
表目錄..................................................III
圖目錄.................................................XIII
第一章 緒論...............................................1
  1.1 前言................................................1
  1.2 研究目的............................................1
  1.3 文獻探討............................................2
   1.3.1 製程能力指標的文獻探討...........................2
   1.3.2 設限型態................ ........................4
  1.4 本文架構............................................5
第二章 在逐步第一失敗設限下對具有浴缸型分配的參數進行估計.............................................6
  2.1 參數的最大概似估計.................................6
  2.2 參數的近似區間估計.................................9
  2.3 參數之精確的區間估計...............................10
  2.4 數值範例...........................................12

第三章 利用逐步第一失敗設限樣本評估具有浴缸型分配之產品的壽命績效指標........................................14
  3.1 產品的壽命績效指標與製程良率.......................14
   3.1.1 產品的壽命績效指標..............................14
   3.1.2 產品的製程良率..................................15
  3.2 壽命績效指標之估計量...............................16
  3.3 壽命績效指標C_L的檢定程序..........................17
  3.4 壽命績效指標C_L的檢定力...........................19
   3.4.1 壽命績效指標之檢定力函數.......................19
   3.4.2 檢定力函數的蒙地卡羅模擬與其模擬值得比較........19
  3.5 壽命績效指標的信賴區間............................21
  3.6 壽命績效指標之信賴水準的蒙地卡羅模擬..............22
  3.7 數值範例..........................................24
第四章 結論與未來研究方向................................62
  4.1 結論...............................................62
  4.2 未來研究方向.......................................62
參考文獻................................................63
附錄.................................................66

表目錄
表3.1 壽命績效指標C_L與製程良率P_r.......................15
表3.2 在顯著水準α=0.01、c*=0.1(0.1)0.9及m=2(1)65下，具有浴缸型分配之產品壽命績效指標的臨界值C_0...................26
表3.3 在顯著水準α=0.05、c*=0.1(0.1)0.9及m=2(1)65下，具有浴缸型分配之產品壽命績效指標的臨界值C_0...................28
表3.4 在給定α=0.01、c*=0.1、L=0.02、β=0.8、n=30、m=12及逐步移除R=(18,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................30
表3.5 在給定α=0.01、c*=0.1、L=0.02、β=0.8、n=30、m=12及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,18)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................30
表3.6 在給定α=0.01、c*=0.1、L=0.02、β=0.8、n=30、m=12及逐步移除R=(3,0,3,0,3,0, 3,0,3,0,3,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................31
表3.7 在給定α=0.01、c*=0.1、L=0.02、β=0.8、n=30、m=18及逐步移除R=(12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE................................31
表3.8 在給定α=0.01、c*=0.1、L=0.02、β=0.8、n=30、m=18及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE................................32
表3.9 在給定α=0.01、c*=0.1、L=0.02、β=0.8、n=30、m=18及逐步移除R=(2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE..........................32
表3.10在給定α=0.05、c*=0.1、L=0.02、β=0.8、n=30、m=12及逐步移除R=(18,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................33
表3.11 在給定α=0.05、c*=0.1、L=0.02、β=0.8、n=30、m=12及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,18)下，和在不同 和 下壽命績效指標的檢定力真實值 與模擬平均(P ̂(c_1)) ¯及MSE......33
表3.12在給定α=0.05、c*=0.1、L=0.02、β=0.8、n=30、m=12及逐步移除R=(3,0,3,0,3,0, 3,0,3,0,3,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................34
表3.13 在給定α=0.05、c*=0.1、L=0.02、β=0.8、n=30、m=18及逐步移除R=(12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................34
表3.14 在給定α=0.05、c*=0.1、L=0.02、β=0.8、n=30、m=18及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................35
表3.15 在給定α=0.05、c*=0.1、L=0.02、β=0.8、n=30、m=18及逐步移除R=(2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE..............................35
表3.16 在給定α=0.01、c*=0.1、L=0.02、β=2、n=30、m=12及逐步移除R=(18,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.....36
表3.17 在給定α=0.01、c*=0.1、L=0.02、β=2、n=30、m=12及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,18)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.....36
表3.18 在給定α=0.01、c*=0.1、L=0.02、β=2、n=30、m=12及逐步移除R=(3,0,3,0,3,0, 3,0,3,0,3,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.......37
表3.19 在給定α=0.01、c*=0.1、L=0.02、β=2、n=30、m=18及逐步移除R=(12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE............................37
表3.20 在給定α=0.01、c*=0.1、L=0.02、β=2、n=30、m=18及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE..........................38
表3.21 在給定α=0.01、c*=0.1、L=0.02、β=2、n=30、m=18及逐步移除R=(2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.............................38
表3.22 在給定α=0.05、c*=0.1、L=0.02、β=2、n=30、m=12及逐步移除R=(18,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE........39
表3.23 在給定α=0.05、c*=0.1、L=0.02、β=2、n=30、m=12及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,18)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.....39
表3.24 在給定α=0.05、c*=0.1、L=0.02、β=2、n=30、m=12及逐步移除R=(3,0,3,0,3,0, 3,0,3,0,3,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.......40
表3.25 在給定α=0.05、c*=0.1、L=0.02、β=2、n=30、m=18及逐步移除R=(12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................40
表3.26 在給定α=0.05、c*=0.1、L=0.02、β=2、n=30、m=18及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE............................41
表3.27 在給定α=0.05、c*=0.1、L=0.02、β=2、n=30、m=18及逐步移除R=(2,0,0,2,0,0,2,0,0,2,0,0,2,0,0,2,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.............................41
表3.28 在給定α=0.01、c*=0.1、L=0.02、β=2、n=50、m=30及逐步移除R=(20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE..........42
表3.29 在給定α=0.01、c*=0.1、L=0.02、β=2、n=50、m=30及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........42
表3.30 在給定α=0.01、c*=0.1、L=0.02、β=2、n=50、m=30及逐步移除R=(2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................43
表3.31 在給定α=0.01、c*=0.1、L=0.02、β=2、n=50、m=20及逐步移除R=(30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................................43
表3.32 在給定α=0.01、c*=0.1、L=0.02、β=2、n=50、m=20及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................................44
表3.33 在給定α=0.01、c*=0.1、L=0.02、β=2、n=50、m=20及逐步移除R=(3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE..............................................44
表3.34 在給定α=0.05、c*=0.1、L=0.02、β=2、n=50、m=30及逐步移除R=(20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................45
表3.35 在給定α=0.05、c*=0.1、L=0.02、β=2、n=50、m=30及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................45
表3.36 在給定α=0.05、c*=0.1、L=0.02、β=2、n=50、m=30及逐步移除R=(2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0,2,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...................................46
表3.37 在給定α=0.05、c*=0.1、L=0.02、β=2、n=50、m=20及逐步移除R=(30,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................................46
表3.38 在給定α=0.05、c*=0.1、L=0.02、β=2、n=50、m=20及逐步移除R=(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,30)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE...........................................47
表3.39 在給定α=0.05、c*=0.1、L=0.02、β=2、n=50、m=20及逐步移除R=(3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0,3,0)下，和在不同c_1和k下壽命績效指標的檢定力真實值P(c_1)與模擬平均(P ̂(c_1)) ¯及MSE.............................................47
表3.40 在形狀參數β=0.8、每組個數k=1、顯著水準α=0.01、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.......................................50
表3.41 在形狀參數β=0.8、每組個數k=1、顯著水準α=0.05、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.....................................51
表3.42 在形狀參數 、每組個數k=5、顯著水準α=0.01、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.......................................52
表3.43 在形狀參數 、每組個數k=5、顯著水準α=0.05、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.......................................53
表3.44 在形狀參數 、每組個數k=7、顯著水準α=0.01、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.....................................54
表3.45 在形狀參數 、每組個數k=7、顯著水準α=0.05、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.......................................55
表3.46 在形狀參數β=2、每組個數k=1、顯著水準α=0.01、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯........................................56
表3.47 在形狀參數β=2、每組個數k=1、顯著水準α=0.05、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.........................................57
表3.48 在形狀參數β=2、每組個數k=5、顯著水準α=0.01、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.........................................58
表3.49 在形狀參數β=2、每組個數k=5、顯著水準α=0.05、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.........................................59
表3.50 在形狀參數β=2、每組個數k=7、顯著水準α=0.01、目標值C_L=0.8和下規格界限L=0.02下，C_L之信賴區間的平均信賴水準(1-α ̂ ) ¯.........................................60
表3.51 在形狀參數β=2、每組個數k=7、顯著水準α=0.05、目標值C_L=0.8和下規格界限L=0.02下， 之信賴區間的平均信賴水準(1-α ̂ ) ¯.........................................61
圖目錄
圖2.1 當λ=1時之機率密度函數曲線.........................7
圖2.2 當λ=2時之機率密度函數曲線.........................7
圖2.3 當λ=1時之故障率函數曲線...........................7
圖2.4 當λ=2時之故障率函數曲線...........................7
圖2.5 (β,λ)之95%的信賴區域..............................13
圖3.1 在顯著水準α=0.01、c*=0.1、β=0.8、n=30、m=12和k=5下，在不同移除序列R下，壽命績效指標的檢定力真實值P(c_1)與模擬值(P ̂(c_1)) ¯之圖形.....................48
圖3.2 在顯著水準α=0.01、c*=0.1、β=2、n=30、m=12和k=5下，在不同移除序列R下，壽命績效指標的檢定力真實值P(c_1)與模擬值(P ̂(c_1)) ¯之圖形...........................49

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