臺灣博碩士論文加值系統

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

Bibliography
Bai, D. S. and Choi, I. S. (1995), and R control charts for skewed populations. Journal of Quality Technology, 27, pp.120-131.
Bittanti, S., Lovera, M., Moriraghi, L. (1998), Application of non-normal process capacility indices to semiconductor quality control. IEEE Transactions On Semiconductor Manufacturing, 11, pp. 296-303.
Chan, L. K. and Cui, H. J. (2003), Skewness correction and R charts for skewed distributions. Vol. 50, Naval Research Logistics, pp. 1-19.
Chang, Y. S. and Bai, D. S. (2001), Control charts for positively-skewed populations with weighted standard deviations. Quality and Reliability Engineering International, Vol.17, pp. 397-406.
Chang, Y. S., Choi, I. S. and Bai, D. S. (2002), Process capability indices for skewed populations. Quality and Reliability Engineering International, Vol.20, pp. 31-46.
Chang, Y. S. and Bai, D. S. (2004), A Multivariate T2 control chart for skewed populations using weighted standard deviations. Quality and Reliability Engineering International, Vol.20, pp. 31-46.
Choobineh, F. and Branting D. (1986), A simple approximation for semivariance. European Journal of Operational Research, 27, pp.364-370.
Choobineh, F. and Ballard, J. L. (1987), Control-limits of QC charts for skewed distribution using weighted variance. IEEE Transactions on Reliability, 36, p.473-477.
Cowden DJ. (1957), Statistical Method in Quality Control. Prentice-Hall: Englewood Cliffs, NJ, 1957.
Ferrell, E. B. (1958), Control chart for log-normal universe. Industrial Quality Control, 15, pp.4-6.
Gunter, W. H. (1989), The use and abuse of Cpk, 2/3. Quality Progress, 22, pp. 108-109.
Lucas, J. M. (1985), Counted data CUSUM''s. Technometrics, 27, pp.129-144.
Nelson, P. R. (1979), Control chart for Weibull process with standard given. IEEE Transactions on Reliability, 28, pp.383-387.
Pyzdek, T. (1995), Why normal distribution aren’t-all that normal. Quality Engineering, 7, pp. 769-777.
Tsai, T.-R. (2004), Adjusted weighted standard deviation R chart for skewness distributions, Technical Report, No. 1, Department of Statistics, Tamkang University.
Vardeman, S. and Ray, D. (1985), Average run lengths for CUSUM schemes when observations are exponentially distributed. Technometrics, 27, pp.145-150.