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研究生:郭孟憲
研究生(外文):Meng-Hsien Kuo
論文名稱:加速壽命實驗在逆高斯分配下的最佳化設計與正確推論
論文名稱(外文):The Optimum Plan and Accurate Inference for Accelerated Life Test under Inverse Gaussian Distribution
指導教授:鄭順林鄭順林引用關係
指導教授(外文):Shuen-Lin Jeng
學位類別:碩士
校院名稱:東海大學
系所名稱:統計學系
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2001
畢業學年度:89
語文別:英文
論文頁數:59
中文關鍵詞:加速壽命實驗逆高斯分配
外文關鍵詞:Accelerated Life TestInverse Gaussian Distribution
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Industry develops very quickly in recent years. High quality products are requested with high reliability. So how to use methods in short time to estimate the life of object is interested. Accelerated life test is a practical method, which place product into a test in some special (accelerated) state by characters, for example, with high temperature and voltage. We obtain information in accelerated conditions and estimate exact life time in the normal operating conditions. This method does not only save time but also save cash.
Here we consider the optimum accelerated plan that employs the minimization of standard deviation for estimation to find the proper low and high stress for testing. The best compromise plan is a design that minimize the standard deviation of the target estimator with three stress in equal space and with same allocation of observations on the stress.
We derive the Fisher information and employ the large sample theory to obtain the standard deviation of $\widehat{t_{p}}$ (p-th quantile of the distribution). The method based on the normal approximation theory to obtain the stress and allocation of the plan may not result in the real optimum state when sample size is less than 1000. So we propose to find the optimum stress and allocation by simulation.
In this thesis, we overcome some predicament in generating
Inverse Gaussian random variables. For the most common
procedure, one uses uniform random variable and solve
the inverse of CDF to obtain the required random variable.
This is quite time consuming, so we use the method
provided by Michael (1976) based on Chi-square
distribution with degree of freedom 1.
It is known that the bias problem could be serious when
censoring is presented. We use the bias correction by
applying bootstrap procedure to correct the estimate of
$\widehat{t_{p}}$.

Industry develops very quickly in recent years. High quality products are requested with high reliability. So how to use methods in short time to estimate the life of object is interested. Accelerated life test is a practical method, which place product into a test in some special (accelerated) state by characters, for example, with high temperature and voltage. We obtain information in accelerated conditions and estimate exact life time in the normal operating conditions. This method does not only save time but also save cash.
Here we consider the optimum accelerated plan that employs the minimization of standard deviation for estimation to find the proper low and high stress for testing. The best compromise plan is a design that minimize the standard deviation of the target estimator with three stress in equal space and with same allocation of observations on the stress.
We derive the Fisher information and employ the large sample theory to obtain the standard deviation of $\widehat{t_{p}}$ (p-th quantile of the distribution). The method based on the normal approximation theory to obtain the stress and allocation of the plan may not result in the real optimum state when sample size is less than 1000. So we propose to find the optimum stress and allocation by simulation.
In this thesis, we overcome some predicament in generating
Inverse Gaussian random variables. For the most common
procedure, one uses uniform random variable and solve
the inverse of CDF to obtain the required random variable.
This is quite time consuming, so we use the method
provided by Michael (1976) based on Chi-square
distribution with degree of freedom 1.
It is known that the bias problem could be serious when
censoring is presented. We use the bias correction by
applying bootstrap procedure to correct the estimate of
$\widehat{t_{p}}$.

1:Introduction
2:Model
3:Asymptotic Normality
4:Accurate Standard Deviation
5:Simulation Procedure
6:Bias Correction by Bootstrap
7:Conclusion
8:Future Research

1: C. J. Adcock. (1997) "Sample size determination: a review"}, The Statistician 45, No. 2, pp. 261-283.
2: R. S. Chhikara and J. L. Folks. (1977) "The Inverse Gaussian Distribution as a Lifetime Model", Technometrics}, Vol. 19, No. 4, pp. 461-468.
3: R. S. Chhikara and J. L. Folks. (1989) "The Inverse Gaussian Distribution", New York: Marcel Dekker, INC.
4: Kjell A. Doksum and Arnljot H$\acute{o}$yland.(1992) "Models for Variable-Stress Accelerated Life Testing Experiments Based on Wiener Processes and the Inverse Gaussian Distribution", Technometrics, Vol. 34, No. 1, pp. 74-82.
5: Bradley Efron and Robert J. Tibshirani. (1993) "An Introduction to the Bootstrap", New York: Chapman and Hall.
6: Luis A. Escobar and William Q. Meeker. (1998) "Fisher Information Matrices with Censoring, Truncation,and Explanatory Variables", Statisitca Sinica 8, pp. 221-237.
7: Shuen-Lin Jeng and William Q. Meeker. (2000) "Comparisons of Approximate Confidence Interval Procedures for Type I Censored Data", Technometrics}, Vol. 42, No. 2, pp. 135-159.
8: Norman L. Johnson and Samuel Kotz. (1970) "Continuous Univariate Distributions-1", New York: John Wiley and Sons, Inc.
9: William Q. Meeker, Jr. (1984) "A Comparison of Accelerated Life Test Plans for Weibull and Lognormal Distributions and Type I Censoring", Technometrics, Vol. 26, No. 2, pp. 157-171.
10: William Q. Meeker and Luis A. Escobar. (1998) "Statistical Methods for Reliability Data", New York: John Wiley and Sons, Inc.
11: Ulrich Menzefricke. (1992) "Designing Accelerated Life Tests when There is Type II Censoring", {\em Commum. Statist.-Theory Meth}, 21(9), 2569-2590.
12: John R. Michael, William R. Schucany and Roy W. Haas*. (1976) "Generating Random Variates Using Transformations with Multiple Roots", The American Statistician, Vol. 30, No. 2, pp. 88-90.
13: Wayne Nelson and Thomas J. Kielpinski. (1976) "Theory for Optimum Censored Accelerated Life Tests for Normal and Lognormal Life Distributions", Technometrics, Vol. 18, No. 1, pp. 105-114.
14: Wayne Nelson and William Q Meeker. (1978) "Theory for Optimum Accelerated Censored Life Tests for Weibull and Extreme Value Distributions", Technometrics, Vol. 20, No. 2, pp.171-177.
15: Wayne Nelson. (1990) "Accelerated Testing", New York:
John Wiley and Sons, Inc.
16: A. Onar and W.J. Padgett. (2000) "Inverse Gausian Accelerated Test Models Based on Cumulative Damage" , J. Statist. Comput. Simul, Vol. 64, pp. 233-247.
17: G. A. Whitmore and M. Yalovsky. (1978) "A Normalizing Logarithmic Transformation for Inverse Gaussian Random Variables", Technometrics}, Vol. 20, No. 2, pp. 207-208.

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