利用廣義p值、廣義信賴區間及特徵函數的統計推論(I)

行政院國家科學委員會專題研究計畫 期中進度報告

利用廣義 p 值, 廣義信賴區間及特徵函數的統計推論(1/3)

中 華 民 國 93 年 5 月 7 日

行政院國家科學委員會專題研究計畫期中報告

利用廣義 p 值, 廣義信賴區間及特徵函數的統計推論(1/3)

計畫編號：NSC 92-2118-M-009-008

執行期限：2003 年 8 月 1 日至 2004 年 7 月 31 日

主持人：李昭勝博士 國立交通大學財務金融研究所

一、On Modeling Data from Degradation Sample Paths over Time

此乃與博士生林宗儀教授（目前任教於東海大學統計系）合作的文章。本文已發 表於 Australian & New Zealand Journal of Statistics (an SCI journal)。其中、英文之 摘要如下。 （一）、中文摘要 本文主要對衰退資料的建模應用成長曲線模型加上 Box-Cox 轉換、隨機效應、及 ARMA (p,q)相關來分析一組衰退的資料。導出最佳概似估計及利用最佳線性不偏 預測法求出未來值的預測。本文從預測的觀點比較所提之方法與一個非線性衰退 模型。我們也比較不同樣本長度下之斷點(failure times)的預測。 （二）、英文摘要

This paper is mainly concerned with modelling data from degradation sample paths over time. It uses a general growth curve model with Box–Cox transformation, random effects and ARMA (p, q) dependence to analyse a set of such data. A maximum likelihood estimation procedure for the proposed model is derived and future values are predicted, based on the best linear unbiased prediction. The paper compares the proposed models with a nonlinear degradation model from a prediction point of view. Forecasts of failure times with various data lengths in the sample are also compared.

（三）、報告內容

In this paper, we are concerned with modelling degradation data such as measurements of the growth of fatigue cracks. In general, engineers need to produce units of material with acceptable reliability and at an acceptable life-cycle cost. Most material accumulates irreversible damage during its life, which leads to failure. The

cumulative damage reduces the reliability of the material as time increases. To maintain an acceptable reliability in the unit, inspections and repairs must be made, which increase life-cycle costs. Thus, cumulative damage plays a very important role in the design of the unit.

For degradation data such as the fatigue crack growth data described above, it is very important to develop a model capable of predicting the fatigue crack growth and, most important of all, predicting the time to failure. Then engineers can order repair or replacement before the failure actually occurs. Once the unit or material has failed, it is too late to repair, and failure could cause heavy physical and/or economic losses.

This type of data is quite typical in studies such as accelerated life testing, because the product usually takes a long time to wear out. One important characteristic of the observations obtained in degradation studies is that they are measurements of several units, and each unit is measured over time. The measurements on a single unit are not independent because they are time-series in nature. If there are only a few measurements on the unit, say fewer than 20, then the dependence may be too hard to estimate. Fortunately, such data are usually obtained for several similar and independent units. Also, the linearity of growth function can be enhanced by the well known Box–Cox transformation (Box & Cox, 1964), as seen in Figure 1(b). These phenomena occur in many studies including technology substitutions as reported by Keramidas & Lee (1990). This paper predicts that a general growth curve model having ARMA(p, q) dependence coupled with the Box–Cox transformation can be applied to degradation data. A model is proposed and compared, in terms of its prediction accuracy and failure time prediction, with the degradation model of Lu & Meeker (1993) using the fatigue crack data of Bogdanoff & Kozin (1985). The failure time is the time to grow a crack from 0.90 inches to the critical crack length of 1.60 inches. A credible prediction of failure time is important, particularly to engineers.

From the results presented in Section 4 we know that if there are only a few measurements on each unit, it may be too hard to estimate the autocorrelation. Also, the data in each unit are time-series in nature and hence are not independent. Therefore, we can use the general growth curve model with ARMA(p, q) covariance structures to analyse this kind of data, using measurements from similar units to get better prediction results. The advantage of our modelling for this type of data is evident in the comparisons of forecast accuracy in future values and in failure times.

As remarked in Rochon (1992), ARMA(p, q) covariance structures are worth considering and may have better performance than AR(1) dependence in many applications. For the modelling of degradation data, with appropriate ARMA(p, q) covariance structure and coupled with random effects and the Box–Cox

transformation, our modelling approach makes the prediction results quite appealing.

（四）、參考文獻

Barndorff-Nielsen, O.E. & Schou, G. (1973). On the reparameterization of autoregressive models by partial autocorrelations. J. Multivariate Anal. 3, 408–419.

Bogdanoff, J.L. & Kozin, F. (1985). Probabilistic Models of Cumulative Damage. NewYork:Wiley.

Box, G.E.P. & Cox, D.R. (1964). An analysis of transformations (with discussion). J.

Roy. Statist. Soc. Ser.B 26, 211–252.

Box, G.E.P., Jenkins, G.M. & Reinsel, G.C. (1994). Time Series Analysis Forecasting

and Control. 3rd

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Dempster, A.P., Laird, N.M. & Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39, 1–38.

Diggle, P.J. (1988). An approach to the analysis of repeated measurement. Biometrics

44, 959–971.

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Statist. Assoc. 70, 320–328.

Jennrich, R.I. & Schluchter, M.D. (1986). Unbalanced repeated–measures models with structured covariance matrices. Biometrika 42, 805–820.

Keramidas, E.M. & Lee, J.C. (1990). Forecasting technological substitutions with concurrent short time series. J. Amer. Statist. Assoc. 85, 625–632.

Laird, N.M. & Ware, J.H. (1982). Random effects models for longitudinal data.

Biometrics 38, 963–974.

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Liu, C. & Rubin, D.B. (1994). The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81, 633–648.

Lu, C.J. & Meeker, W.Q. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics 35, 161–174.

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Monahan, J.F. (1984).A note on enforcing stationarity in autoregressive moving average models. Biometrika 71, 403–404.

Morrell, C.H., Pearson, J.D., Carter, H.B. & Brant, L.J. (1995). Estimating unknown transition times using a piecewise non-linear mixed-effects model in men with prostate cancer. J. Amer. Statist. Assoc. 90, 45–53.

Paris, P.C. & Erdogan, F. (1963). A critical analysis of crack propagation laws. J.

Basic Engineering 85, 528–534.

Potthoff, R.F. & Roy, S.N. (1964). A generalized multivariate analysis of variance model useful especially for growth curve problems. Biometrika 51, 313–326.

Rao, C.R. (1987). Prediction of future observations in growth curve model. Statis. Sci.

2, 434–471.

Rochon, J. (1992). ARMA covariance structures with time heteroscedasticity for repeated measures experiments. J. Amer. Statist. Assoc. 87, 777–784.

Verbeke, G., Lesaffre, E. & Brant, L.J. (1998). The detection of residual serial correlation in linear mixed models. Statist. Med. 17, 1391–1402.

（五）、計畫成果自評

本 研 究 成 果 發 表 於 SCI 的 期 刊 Australian & New Zealand Journal of Statistics，這是個頗被肯定的國際期刊。

二、Bayesian Estimation for Time Series Regressions with Applications

此乃與博士生林淑惠教授(目前任教於台中技術學院)合作的文章。本文已發表於 Journal of Statistical Planning and Inference (an SCI journal)。其中、英文摘要如下。

（一）、中文摘要 根據廣義 p 值及廣義信賴區間的概念，我們研發有關兩個常態分配母體平均數比 例(ratio)的假設檢定與信賴區間。我們利用兩個不同的方法來取得關鍵值 (pivotal qualities)。當中之一是直接利用平均數的比例來找關鍵值。另一方 法則是將此問題當作類似 Behrens-Fisher 之問題來處理，然後利用假設檢定之 法來建構信賴區間。我們的模擬結果發現本文所提之信賴區間優於其他兩種方 法。這個優點更顯示於兩個母體之變異數相當不同之情況。 （二）、英文摘要

Based on the generalized p-values and generalized confidence interval developed by Tsui and Weerahandi (1989), Weerahandi (1993), respectively, hypothesis testing and confidence intervals for the ratio of means of two normal populations are developed to solve Fieller’s problems. We use two different procedures to find two potential generalized pivotal quantities. One procedure is to find the generalized pivotal quantity based directly on the ratio of means. The other is to treat the problem as a pseudo Behrens-Fisher problem through testing the two-sided hypothesis on θ, and then to construct the 1−α generalized confidence interval as a counterpart of generalized p-values. Illustrative examples show that the two proposed methods are numerically equivalent for large sample sizes. Furthermore, our simulation study shows that confidence intervals based on generalized p-values without the assumption of identical variance are more efficient than two other methods, especially in the situation in which the heteroscedasticity of the two populations is serious.

implement, but in most applications involving nuisance parameters it is not easy or impossible to fnd A(x) and B(x) so as to satisfy the above equation for all possible values of the nuisance parameters. The idea in generalized confidence intervals is to make this possible by making probability statements relative to the observed sample, as done in Bayesian and nonparametric methods. In other words, we allow the functions A(·) and B(·) to depend not only on the observable random vector X but also on the observed data xobs.

In this article, we propose two different exact generalized approaches based on generalized p-values and generalized confidence intervals to solve the well-known Fieller-Creasy problem, which is widely used in many important research areas such

as bioassay and bioequivalence. Under homogeneous case, Fieller’s solution gives exact coverage probability for all parameters. Unfortunately, in the presence of serious heteroscedasticity, the methods under the restriction of identical variance can not yield decent confidence intervals. Through the proposed methods in this article, an exact 1 − α generalized confidence intervals for the ratio of two means can be obtained under unequal variances and unequal sample sizes. According to our findings, the existing procedures ignoring the mild heteroscedasticity will perform well. However, they will perform very poorly in the situation in which serious heteroscedasticity is present. Thus our proposed methods are very valuable in practice, especially when the two variances are quite different.

（四）、參考文獻

Berger R. L. and Hsu J. C. (1996), “Bioequivalence Trials, Intersection Union Tests and Equivalence Confidence Sets,” Statistical Science, 11(4), 283-319.

Chow, S. C. and Liu, J. P. (1992), “Design and Analysis of Bioavailability and Bioequivalence Studies,” New York: Marcel Dekker, Inc.

Cox, C. P. (1985), “Interval Estimates for the Ratio of the Means of Two Normal Populatins with Variances Related to Means,” Biometrics, 41, 261-265.

Fieller, E. C. (1944), “A Fundamental Formula in the Statistics of Biological Assay, and Some Applications,” Quarterly Journal of Pharmacy and Pharmacology, 17, 117-123.

Fieller, E. C. (1954), “Some Problems in Interval Estimation,” J. Roy. Statist. Soc, Series B 16, 175-185.

Finney, D. J. (1978), “Statistical Method in Biological Assay,” London: Griffin, 3rd ed.

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Koschat, M. A. (1987), “A Characterization of the Fieller Solution,” The Annals of

Statistics, Vol. 15, No. 1, 462-468.

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Tsui, K. W., and Weerahandi, S. (1989), “Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters,” Journal of American

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Weerahandi, S. (1993), “Generalized Confidence Intervals,” Journal of American

Statistical Association, 88, 899-905.

（五）、計畫成果自評

本研究成果乃計畫所提的一部分，刊登的期刊是SCI統計期刊當中不錯的雜誌， 相當值得。

三、On the Inverse of the Autocorrelation Matrix for an AR (p) Process

此乃與博士生王仁聖及林宗儀教授（目前任教於東海大學統計系）合作的文章。 本文已發表於Journal of Chinese Statistical Association。這是個CIS的期刊。其中英 文之摘要如下。 （一）、中文摘要 假設C_N =(ρ_t−s ), t,s=1,...,N,是穩定p次自我迴歸向量X_N =

(

)

（四）、參考文獻

（五）、計畫成果自評

本研究發表於Journal of Chinese Statistical Association。這是個CIS的期 刊，所得的方法非常有助於time series、multivariate linear model、及growth curve model等參數估計。