韋伯分配分佈已廣泛應用於許多領域，尤其是可靠度分析及產品的壽命分析等，是所有分配中較符合浴缸型分配，並且估計其參數是一個重要的問題，因此考慮到估計問題。
　　但是，當資料的數據是受污染的情況，估計其參數不具穩健性。在此論文中，吾人針對此問題運用minimum Hellinger distance estimate (MHDE)估計參數。眾所周知，MHDE不僅具有效性，且具有穩健性，還提供了數值模擬的基礎上最大概似估計法和MHDE及其比較。 
　　最後，提出一些延伸的Weibull模型，推廣成四個參數型態之分配。

Weibull distribution has been widely applied in many areas and estimations of its parameters are an important issue. Thus are let of literature. Consider the estimation problem; however, when the data is contaminated most of the proposed estimations do not possess the property of robustness. In this thesis, we apply minimum Hellinger distance estimate (MHDE) for this problem. As is well-known, MHDE provides not only the first order efficiency, but also robustness, we have also provided numerical simulations based on MLE and MHDE and its comparisons have been made.
Finally, some extension of the Weibull model has also been proposed.

目錄
致謝	I
中文摘要	II
英文摘要	III
目錄	IV
表目錄	V
圖目錄	VI
第一章	緒論	1
1.1	研究動機	1
第二章	韋伯分配	3
2.1	韋伯分配之介紹	3
2.2	韋伯分配之應用	6
2.3	韋伯分配之一些統計性質	8
2.4	韋伯分配之推廣	11
2.5	韋伯分配參數之圖形	12
第三章	估計方法及函數介紹	15
3.1	核函數(Kernel function)	15
3.2	帶寬值的決定	16
3.3	Hellinger最短距離法(Minimum Hellinger distance)	18
第四章	韋伯分配之參數估計	22
4.1	最大概似估計法	22
4.2	高斯核函數及帶寬值的選擇(Gauss Kernel function)	23
4.3	Hellinger最短距離法(Minimum Hellinger distance)	25
第五章	模擬	27
5.1	模擬方法之設計	27
5.2	模擬之結果比較	28
第六章	結論	30
參考文獻	31

表目錄
表 1：MLE與MHDE之參數比較表(N=30,Λ=2,Β=0.5)	28
表 2：MLE與MHDE之參數比較表(N=70,Λ=2,Β=0.5)	28
表 3：MLE與MHDE之參數比較表(N=30,Λ=2,Β=2)	28
表 4：MLE與MHDE之參數比較表(N=70,Λ=2,Β=2)	29
表 5：MLE與MHDE之參數比較表(N=30,Λ=2,Β=4)	29
表 6：MLE與MHDE之參數比較表(N=70,Λ=2,Β=4)	29

圖目錄
圖 1: Β<1、Β=1、Β>1	13
圖 2: Λ=2、Β=0.5、Β=2、Β=4	13
圖 3: Λ=2、Β=0.5、Β=2、Β=4、Δ=1.5	13
圖 4: Λ=2、Β=0.5、Β=2、Β=4、Γ=3	14

參考文獻
中文文獻
1.	龔平、曾心傳、嚴尊國，2001，發震時間的指數分佈、Gamma分佈和Weibull分佈之間關係的研究，西北地震學報，數理统計與管理。21(2) :  P315。
2.	範琦，2003，Weibull分佈的資訊熵在地震預報中的應用研究 ，西北地震學報。25(2) : P315.75。
3.	黃勁松、王高，2008，Weibull分佈在新產品市場滲透研究中的應用拓展，數理統計與管理。27(2) : 320-328。
4.	鄭傳奇、楊勁，2001，應用WEIBULL分佈函數建立非房室線性藥動學模型，廣東藥學院學報。17(3)：P213
5.	陳春香，2007，壽險業房貸提前清償之風險-韋伯分配之應用，碩士論文，國立高雄第一科技大學。
6.	瓦拉第．韋伯，2007，「新韋伯分析手冊第四版」，鼎茂出版社。

英文文獻
[1]	Ahmad, I. A. and Fan, Y. (2001) , “Optimal bandwidth for kernel density estimators of functions of Observations, ”Statistics and Probability Letter, 513, 245-251.
[2]	B. U. Park and W. C. Kim (1994) , “Asymptotically best bandwidth selectors in kernel density estimation ,”Statistics and Probability Letters, 19, 119-127.
[3]	Balakrishnan, N. and Kocherlakota, S. (1985) , “On the double Weibull distribution: Order statistics and estimation ,”Sankhya, 47, 161-176.
[4]	Basu, A., and Lindsay, B. G. (1994) , “Minimum Disparity Estimation for Continuous Models :Efficiency, Distributions and Robustness ,”Annals of the Institute of Statistical Mathematics,46, 638-705.
[5]	Beran, R. J. (1977) , “Minimum Hellinger Distance Estimates for Parametric Models ,”The Annals of Statistics, 5, 445-463.
[6]	Bowman, K.O. and Shendon, L.R. (2000) , “Maximum likelihood and Weibull distribution,” Far East J. Theory Statist.
[7]	Cohen, A. C. (1965) , “The maximum likelihood estimation in the Weibull distribution,”Technometrics, 8, 579-588
[8]	Cohen, A. C. (1973) , “The reflected Weibull distribution,” Technomertrics, 15, 867-873
[9]	Cohen, A. C., Whitten, B.J. and Ding Y. (1984) , “ Modified moment estimation for the three-parameter Weibull distribution,” J. Quality Technology, 17, (2), 92-99.
[10]	Cutler, A., Corderon, B. and O. I. (1996) , “Minimum Hellinger Distance Estimation for Finite Mixture Models,” Journal of the American Statistical Association, 91, 1716-1723.
[11]	Devroye, L. (1983) , “ The equivalence of weak, strong and complete convergence in L1 for kernel density estimates [J] , ” Annals of Statistics, 11. 896-904.
[12]	Flygare, M. E., Austin, J. A. and Buckwalter, R. M. (1985) , “ Mmaximum likelihood estimation for the 2-parameter Weibull distribution based on interval data, ”IEEE Transactions on Reliability,Vol R-34,1,57-59
[13]	Gumbel, E.J. (1958) . Statistics of Extremes, Columbia University Press, New York.
[14]	Huang, J. S. and Wang, G. (2008) , “ Application Extension of Weibull Distribution for New Product Market Penetration Research, ”Application of Statistics and Management .
[15]	Johnson, N. L., Kotz, S. and Balakrishnan, N. “Cntinuous univariate distributions, ”1,Second Edition, John Wiley&Sons, New York.
[16]	Kanazawa, Y. (1993) , “Hellinger distance and Kullback-leibler loss for the kernel density estimator, ”Statistics and Probability Letters, 18, 315-321.
[17]	Lindsay, B. G. (1994) , “Efficiency versus Robustness: The Case for Minimum Hellinger Distance and Related Methods,”The Annals of Statistics, 22, 1081-1114.
[18]	M. M. Brooks (1991) , “Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions, ”Stochastic Processes and Their Applications, 38, 157-165
[19]	Mudholkar, G.S. and Kollia.CD.(1994) , “Generalized Weibull family:A structural analysis, ”Communicaltions in Satistics-Series A:Theory and Methods,25,1149-1171
[20]	Murthy, D. N. P., Xie, M.M. and Jiang, R.(2004).Weibull Models. John Wiley and Sons, Hoboken, New Jersey.
[21]	P. Hall and J. S. Marron(1988) ,“Choice of kernel order in density estimation, ” Annals of Statistics, 16, 161-173.
[22]	Parzen, E. (1962) ,“ On estimation of a probability density function and mode, ” Ann.Math.Staist., 33, 1065-1076
[23]	Powel, J. L. and Stoker, T. M. (1996) ,“ Optimal bandwidth choice for density-weighted　averages, ” Journal of Econometrics, 75, 291-316
[24]	Ranneby B. (1984) , “The maximum spacing method: an estimated method related to the maximum likelihood method, ” Scand. J. Statists. 11, 93-112.
[25]	Rosenblatt, M. (1956) , “ Remarks on some nonparametric estimates of a density function , ”Ann.Math.Statist.27, 832-837.
[26]	Royland, W. D. (1972) , “ Maximum Likelihood Estimation of the Parameters of the Weibull distribution by Modified , ”IEEE transaction on reliability , Vol R-21,2,89-93
[27]	Rudemo, M. (1982) , “ Empirical choice of histograms and kernel density estimators, ”Scand. J. Statist. 9, 65-78
[28]	S. J. Sheather, (1990) , “ Kernel quantile estimation, ”Journal of the American Statistical Association, 85, 410-419
[29]	Sarhan and Zaindin (2008) , “Parameters Estimation of the Modified Weibull Distribution, ”Applied Mathematical Sciences, Vol. 3, 11, 541 - 550
[30]	Silverman, B, W. (1986) , Density Estimation for Statistics and Data Analysis, Chapman and Hall.
[31]	Simpson, D. G. (1987) ,“Minimum Hellinger Distance Estimation for the Analysis of Count Data,”Journal of the American Statistical Association, 82, 807-807.
[32]	Stefan Stefanescu: “Estimating the three parameters of Weibull model with restrictive location values, ” Economic Computation and Economic Cybernetics Studies and Research, XXXV, 1-4, 91-99.
[33]	Terrell, G. R. (1990) , “The maximum smoothing principle in density estimation, ” Journal of American Statistics Association, 85, 470-477
[34]	Turlach, B.A. (1993) , “Bandwidth Selection in Kernel Density Estimation: A Review,CORE and Institute de Statistique
[35]	Wolfowitz, J., (1952) , “ Consistent Estimation of the parameter of a Linear Structural Relation, ” Skandinavisk Aktuarietidskrift, 35, 132-157.
[36]	Wolfowitz, J., (1954) , “Estimation by the Minimum Distance Method in Nonparametric Difference Equations, ” Annals of Mathematical Statistics, 25,
[37]	W. J. Padgett, (1987) , “Asymptotically optimal bandwidth selection for kernel density estimators from randomly right censored samples, ” Annals of Statistics, 15, 1520-1535
[38]	Wand, M. P. and Jones, M. C. (1995), Kernel smoothing. Chapman and Hall, London.
[39]	Weibull, W. (1939) , “A statistical theory of the strength of material, ” Ingeniors Vetenskaps Akakemien Handlingar ,15:293-297.
[40]	Yang, S. (1993) , “ A central limit theorem for the integrated square error of the kernel density estimators with randomly censored data, ”J. Statist. Plan. Inference, 37: 127-143.