當品質特徵來自一個偏態分配時，本論文針對調整的加權標準差R 管制圖、加權變異數R 管制圖和偏態修正的R 管制圖進行數值研究及比較。在考慮製程分配為Weibull、gamma和lognormal分配下。本論文比較上述三種不同的R 管制圖的型一誤差與名目值0.27%的接近程度。當製程資料來自平均數已知的指數分配時，研究的結果顯示調整的加權標準R 管制圖不論在型一誤差及型二誤差都比其他的兩個R 管制圖接近真值。

The thesis conducts a numerical study in comparing the performance of the adjusted weighted standard deviation R chart, weighted variance R chart, and the skewness correlation R chart with the underlying distribution of quality characteristic is skewed. Assume that the underlying distribution is Weibull, gamma, and lognormal, we compare the Type I Risk of those R chart how to be close to the nominal value 0.27%. If the process distribution is exponential with known mean, we show that not only the Type I Risk, but also the Type II Risk of the adjusted weighted standard deviation R chart are closer to those of the exact R chart than both weighted variance R chart and skewess correlation R chart.

Contents

1. Introduction	1
2. The Proposed R Chart	5
2.1 The Weighted Standard Deviation Method	5
2.2 Adjusted Weighted Standard Deviation R Chart	13
3. Numerical Studies and Example	25
3.1 The Parameters are Known	26
3.2 The Parameters are Unknown	28
3.3 Comparisons with the Exact Method for the Exponential Distribution	33
3.4 Example	36
4. Conclusions	41
Bibliography	42

 


List of Figures

1. Original asymmetric and derived distributions: (a) p.d.f. of   (b) p.d.f. of the upper side and (c) p.d.f. of the lower side .…………………………...…………..6
2. Approximations of  : (a) upper part, (b) lower part and (c) original and approximated distributions ……………………………………………….....……..9
3. Ranges of   observations, (a) range   of   observations, (b) range   
of copied   observations from the upper part, and (c) range   of 
copied   observations from the lower part ………………………………..…12
4. The plots of  ,  , and  : (a) n=3, (b) n=5, (c) n=7, and (d) n=10 …………………………………………………………………………..…20
5. The scatter plot of  v.s.   ………………………………..………21
6. Type I risks of exponential R charts ……………….……………….………...…...34
7. Type II risks of exponential R chart with (a)  and (b) …………..35
8. Distribution of Concentration of Residue Resulting from a Chemical Process ……………………………………………………………….……………39
9. R Charts …………………………………………………………………………...40
 



List of Tables

1. The OLS estimates of b1, b2, and b3 ……………………….……………..………19
2. The performance of   and   for evaluating the   under Weibull, lognormal, and gamma distribution………..…………………………………….22
3. False alarm rates for parameters known …………..………………………..…….30
4. False alarm rates for parameters unknown……………………………………..…31
5. Chemical process data from Cowden (1957) ……………………………………..38

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