# 臺灣博碩士論文加值系統

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 本論文主旨在研究雙尾設限資料的尺度參數估計式的統計性質與應用。這個估計式適用一般的位置-尺度分配(location-scale distributions)，並且它的變異數有一個容易使用的解析式。應用時，可以先固定樣本數與左尾設限數，然後控制變異數小於預設值，就可以決定右尾設限數。有關不對稱的壽命資料，作者將處理韋伯分配(Weibull distributions)形狀參數的推估問題；這等同於處理極值分配(extreme-value distributions)尺度參數的推估問題。有關對稱的隨機資料，作者將分別導出trimmed mean與Winsorized mean的標準誤的解析式，並以常態分配為例，研究統計量的性質。另外，論文中經常用到順序統計量的期望值、變異數與共變異數，也將提出一般的近似公式。
 In this article, we will study a robust scale estimator for location-scale distributions with Type II doubly censored data. Its standard error will be derived analytically. Determining censoring numbers under controlling the standard error is studied. For asymmetric distributions, an estimator of the shape parameter of the Weibull distribution will be discussed. Equivalently, we will study the estimator of the scale parameter of Extreme Value distribution. For symmetric distributions, the standard errors of trimmed mean and Winsorized mean will be studied. Some analytical expressions for the means, variances, and covariances of order statistics are derived for our estimators. In the cases of the standard Extreme Value distribution and the standard normal distribution are also discussed.
 COVERABSTRACTCONTENTSTABLESFIGURES1.Introduction2.Generalized Tukey λ Distributions and Moments of Order Statistics3.Inference for Shape Parameter on Weibull Distributions with Doubly Censored Data4.Inference for Trimmed Mean and Winsorized Mean on Symmetric Location-Scale Distributions with Doubly Censored Data5.ConclusionsReferences
 REFERENCES[1] Arnold, B. C., Balakrishnan, N. and Ngaraja, H. N.(1992), A First Course in Order Statistics, John Wiley &Sons, Inc.[2] Bain, L. J. (1972), "Inferences Based on CensoredSampling From the Weibull or Extreme-ValueDistribution," Technometrics, 14, 693-702.[3] Bain, L. J. and Engelhadrt, M. (1991), Statistical Analysisof Reliability and Life- Testing Models, second edition,Marcel Dekker, Inc.[4] Balakrishnan, N. and Cohen, A. C. (1990), OrderStatistics and Inference -- Estimation Methods, AcademicPress, Inc.[5] Balakrishnan, N. and Chan, P. S. (1992), "Order Statisticsfrom Extreme Value Distribution, I: Tables of Means,Variances and Covariances," Commun. Statist.-- Simula.,21(4), 1199-12117.[6] Bryson, M. C. and Siddiqui, M. M. (1969), "SomeCriteria for Aging", Journal of the American StatisticalAssociation, 64, 1472-1483.[7] David, H. A. (1981), Order Statistics, second edition,John Wiley & Sons, Inc.[8] Engelhardt, M. (1975), "On Simple Estimation of theParameters of the Weibull or Extreme-Value Distribution,"Technometrics, 17, 369-374.[9] Engelhardt, M. and Bain, L. J. (1973), "Some Completeand Censored Sampling Results for the Weibull orExtreme-Value Distribution," Technometrics, 15, 541-549.[10] Engelhardt, M. and Bain, L. J. (1974), "Some Results onPoint Estimation for the Two-Parameter Weibull orExtreme-Value Distribution," Technometrics, 16, 49-56.[11] Harter, H. L. and Balakrishnan, N. (1996), CRCHandbook of Tables for the Use of Order Statistics inEstimation, CRC Press, Inc.[12] Hoaglin, D. C., Mosteller, F. and Tukey, J. W. (editors)(1983), Understanding robust and exploratory dataanalysis, John Wiley & Sons, Inc.[13] Huang, D.Y. and Lin, C. C. (1997), "Multiple DecisionProcedures for Testing Homogeneity of Normal MeansWith Unequal Unknown Variances," Advances inStatistical Decision Theory and Applications (eds., S.Panchapakesan and N.Balakrishnan) Birkhauser, Boston.[14] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995),Continuous Univariate Distributions, Vol. I & II, secondedition, John Wiley & Sons, Inc.[15] Lawless, J. F. (1981), Statistical Models and Methodsfor Lifetime Data, John Wiley & Sons, Inc.[16] Lehmann, E. L. (1983), Theory of Point Estimation,John Wiley & Sons, Inc.[17] Mann, N.R. , Scheuer, E.M. , and Singpurwalla, N. D.(1974), Methods for Statistical Analysis of Reliability andlife data, John Wiley & Sons, Inc.[18] Nelson, W. (1982), Applied life Data Analysis, JohnWiley & Sons, Inc.[19] White, J. S. (1969), "The Moments of log-Weibull OrderStatistics," Technometrics, 11, 373-386.[20] Wolfram, S. (1988), MathematicaTM -- A System forDoing Mathematics by Computer, Addison-WesleyPublishing Company.TABLES1. Values of A(r,s,n) 212. Values of for n = 43, r = 0(1)5 and s = 0(1)6 253. Values of C.V. of for n = 43, r = 0(1)5 and s = 0(1)6 264. Values of C.V. of for n = 19, r = 0(1)5 and s = 0(1)5 275. Relative efficiency of variance and asymptotic varianceof Trimmed mean under normal distribution 366. Simulated data from N(60,100) 377. Estimates of normal distribution parameters 38FIGURES1. Probability density function of Weibull distributions 122. Hazard rate function of Weibull distributions 123. Variances of order statistics of standard extreme value distribution 134. Lifetime interval of Y(n-s,n)-Y(i:n) 155. Standara extreme value distribution Z 176. Generalized Tukey lambda distribution T 177. Comparison of Z and T 188. Histogram of example 1 (5 lasses) 249. Histogram of example 1 (7 lasses) 24
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