由於科技日益漸進,全球化的競爭和市場壓力下,廠商必須在高可靠度產品之研發有所推動,以期在激烈的市場中佔有一席之地,尤其是在追求顧客導向的今日,確保產品品質更為塑造企業形象的關鍵一環,因此,一個主要的問題便是如何取得產品可靠度的資訊;傳統的統計方法對於現今高可靠的產品已不適用,一個可行的方法是,使用產品的衰變資料;技術上,當產品某項品質特徵值衰退到一定程度時,我們便宣稱產品已失效,利用此種方法則可估計產品的壽命,這一篇論文使用Wiener隨機過程來配適產品的衰變路徑,並且使用初始的衰變資料作為估計參數的資料;對於產品壽命分配期望值與變異數的MLE和UMVUE兩個估計量,我們比較其效率上差異,並做了一些推測;在產品壽命分配上,由於沒有真實衰變資料(我們使用初始衰變資料),UMVUE似乎不可能得到;我們proposed一個可用的估計量,並且用模擬結果說明這個估計量比MLE還要精準且穩定,最後則是LED例子的模擬結果,以及ALT的應用.

Rapid advances in technology, development of highly sophisticated products, intense global competition, and increasing customer expectations have put new pressures on manufacturers to produce high-quality products. Customers expect purchased products to be reliable and safe. In other words, customer expect products be able to perform their intended function under usual operating conditions, for some specified period of time. The problems that come along are how to obtain the information of product reliability. But, today, many products are designed and manufactured to function for a long period of time before they fail. The problem of obtaining failure data to apply traditional statistical tools to predict lifetime distribution becomes more difficult. An alternative is to use data on a quality characteristic (QC) whose degradation over time is highly correlated with product failure. If degradation paths can be modeled properly, and the product fails when the QC’s degradation path first passes a critical value, then predicting the product’s failure time or the lifetime can be made without actually observing the product’s failure. Instead one will obtain what will be termed as the initial data from the early stage of reliability testing. A Wiener process is typical for describing a degradation process since it allows non-constant variance and non-zero correlation among data collected over time. In this paper we first review how a Wiener process was used to describe the continuous degradation path of a quality characteristic of the product. We compare the MLES and UMVUES of the mean and variance of lifetime distribution of the product, all based on initial data. Next, we propose an estimator of the lifetime distribution and compare it with the corresponding MLE. Then using an example in the literature, we demonstrate that our proposed estimator of the lifetime distribution is better than the MLE. The example is about the light intensity of light emitting diodes (LEDs). The data we use to obtain various estimates are all collected only from the product’s initial observed degradation path.

Bhattacharyya, K. G. and Fries, A. (1982), “Fatigue Failure Models- Birnbaum- Sanuders vs. Inverse Gaussian,” IEEE Transactions on Reliability, 31, 439-440.
Birnbaum, Z. W. and Sanuders, S. C. (1969), “A New Family of Life Distributions,” Journal of Applied Probability, 6, 319-327.
Blischke, W. R. and Murthy, D. N. P. (2000), Reliability: Modeling, Prediction, and Optimization, John Wiley and Sons: New York.
Chang, D. S., (2000), “Optimal Burn-in Decision For Products With an Unimodel Failure Rate Function,” European Journal of Operational Research, 126, 534-540.
Chhikara, R. S. and Folks, L. (1974), “Estimation of the Inverse Gaussian Distribution Function,” Journal of the American Statistical Association Journal, 69, 250-254.
Chhikara, R. S. and Folks, L. (1989), The Inverse Gaussian Distribution: Theory and Methodology, and Applications, Marcel Dekker, inc.
Chiu, C. Y. (2002), “Estimating Lifetime Distribution Based on Degradation Data
for Highly Reliable Products,” MS Thesis, Institute of Statistics, National Tsing-Hua University, Taiwan.
Cressie, N., Davis, A. S., Folks, J. L., and Policello II, G. E. (1981), “The Moment-Generating Function and Negative Integer Moments,” American Statistician, 35, 148-150.
Doksum, K. A., and Hoyland, A., (1992), “Models for Variable-Stress Accelerated Life Testing experiments Based on Wiener Processes and the Inverse Gaussian Distribution,” Technometrics, 34(1), 74-82.
Folks, J. L. and Chhikara, R. S. (1978), “The Inverse Gaussian Distribution and Its Application--A Review,” Journal of the Royal Statistical Society, Ser. B, 40, 263-289.
Iwase, K., and Seto, N. (1983), “Uniformly Minimum Variance Unbiased Estimation for the Inverse Gaussain Distribution,” Journal of the American Statistical association Journal, 78, 660-663.
Johnson, N. L., and Kotz, S. (1990), Distributions in Statistics: Continuous Univariate Distributions 1, Boston : Houghton Mifflin.
Kuo, W. and Kuo, Y., (1983), "Facing the Headaches of Early Failures: A State-of-the-Art Review of Burn-in Decisions," Proceedings of the IEEE, 71, 11, 1257-1266.
Lawrence, M. J., (1966), “An Investigation of the Burn-in Problem,” Technometrics, 8(1), 61-71.
Lu, C. J. and Meeker, W.Q., (1993), “Using Degradation Measures to Estimate a Time-to-Failure Distribution,” Technometrics, 35, 161-174.
Luke, Y. L., (1969), “ The Special Functions and Their Approximations, Vol. 1, Academic process, New York.
Meeker, W. Q., and Escobar, L. A., (1993), “A Review of Recent Research and Current Issues in Accelerated Testing,” International Statistical Review, 61(1), 147-168.
Meeker, W.Q., Escobar, L. A., and Lu, C. J., (1998), “Accelerated Degradation Tests: Modeling and Analysis,” Technometrics, 40, 89-99.
Nelson, W., (1990), Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, John Wiley & Sons: New York.
Nguyen, D. G., and Nurthy, D. N. P., (1982), “Optimal Burn-in Time to Minimize Cost for Products Sold under Warranty,” IIE Transactions, 14(3), 167-174.
Oksendal, B., (1999), Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer-Verlag: Berlin.
Shiau, J.-J. H. and Lin, H.-H. (1999). "Analyzing Accelerated Degradation Data by Nonparametric Regression." IEEE Transactions on Reliability, Vol. 48, No.2, 149-158.
Tseng, S.T., Tang, J., and Ku, I. H., (2003), “Determination of Optimal Burn-in Parameters and Residual Life For Highly Reliable Products,” Naval Research Logistics, 50, 1-14.
Tseng, S. T., and Yu, H. F., (1997), “A Termination Rule for Degradation Experiment,” IEEE Transactions on Reliability, R-46(1), 130-133.
Tweedie, M. C. K., (1957), “Statistical Properties of Inverse Gaussian Distributution I, II,” Annals of Mathematical Statistics, (June) 362-377, (September) 696-705.
Watson, G. S., and Wells, W. T., (1961), “On the Probability of Improving the Mean Useful Life of Items by Eliminating Those with Short Lives,” Technometrics, 3(2), 281-298.
Wu, A. J. and Shao, J., (1999), “Reliability Analysis using the Least Squares Method in Nonlinear Mixed-Effect Degradation Models,” Statistica Sinica, 9, 855-877.