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System No. U0002-1207200911325500
Title (in Chinese) 陳氏分佈族之統計推論
Title (in English) Inferences of Chen’s family
Other Title
Institution 淡江大學
Department (in Chinese) 管理科學研究所碩士班
Department (in English) Graduate Institute of Management Science
Other Division
Other Division Name
Other Department/Institution
Academic Year 97
Semester 2
PublicationYear 98
Author's name (in Chinese) 葉純志
Author's name(in English) Chun-Chih Yeh
Student ID 696620433
Degree 碩士
Language Traditional Chinese
Other Language
Date of Oral Defense 2008-06-23
Pagination 39page
Committee Member advisor - Wen-Tao Huang
co-chair - 陳基國
co-chair - 鄧文舜
Keyword (inChinese) 核函數
最短距離
浴缸型
Keyword (in English) kernel
minimum Hellinger distance
bathtub
Other Keywords
Subject
Abstract (in Chinese)
陳氏於2000年時提出了一個大的分佈族,其故障率函數呈現浴缸形分佈,不幸的是這個參數估計並非真正的最大概似估計量。因此,本文中提出了最短Hellinger距離來估計參數。最短Hellinger 距離擁有一些好的特性,它不僅擁有有效性還具有穩健性的特性。當數據資料受到汙染,穩健性的估計是一個適當的選擇。在數值模擬時將分別針對最大概似估計量與最短Hellinger 距離進行參數估計,並建議參數估計的方法。
最後推廣陳氏模型,並針對推廣模型進行檢定。
Abstract (in English)
Chen (2000) proposed a big family of distributions which is suitable for life model since its hazard rate function has a bathtub shape. Unfortunately, the proposed estimate for the parameter is not an exact MLE. In this thesis, we propose the minimum Hellinger distance estimate (MHDE) for the parameters involved in the family. This MHDE has good properties; it possesses not only the first order efficiency, but also robustness. When the data is contaminated, robust estimate is an appropriate choice. Some numerical simulations have been carried out both for the case of maximum likelihood like estimate and MHDE. Some improved estimate has been proposed.
Finally, some extension of the Chen’s model has also been made.
Other Abstract
Table of Content (with Page Number)
目錄
致謝     	I
中文摘要 	II
英文摘要 	III
目錄     	IV
表目錄   	VI
圖目錄   	VII
第一章	緒論	1
1.1	研究背景與動機	1
1.2	問題架構	3
第二章	核函數估計量與Hellinger 距離	7
2.1	核函數估計的相關簡介	7
2.2	Hellinger最短距離相關簡介	12
2.3	基於Hellinger最短距離參數估計之理論基礎及其評論	13
第三章	參數推論	17
3.1.	Hellinger 最短距離之參數估計	17
3.2.	Chen’s模型之約略MLE	19
3.3.	Chen模型的推廣及參數 之檢定問題	20
第四章	數值分析	23
4.1.	參數模擬設計方法	23
4.2.	數值分析結果	24
4.3.	參數模型檢定之檢定表	34
4.4.	數值實例	34
第五章	結論	36
5.1.	主要研究成果	36
5.2.	未來研究方向	36
參考文獻 	37

表目錄
表4. 1參數估計之MEAN、MSE、SD、BIAS、CORRELATION(N=30,K=27)	25
表4. 2  參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=30,K=30)	25
表4. 3 參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=50,K=45)	25
表4. 4 參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=50,K=50)	26
表4. 5 參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=100,K=90)	26
表4. 6參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=100,K=100)	26
表4. 7參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=300,K=270)	27
表4. 8參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=300,K=300)	27
表4. 9參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=500,K=450)	27
表4. 10參數估計之MEAN、MSE、SD、BIAS、CORRELATION (N=500,K=500)	28
表4. 11 VN樣本觀測值之臨界點檢定表	34

圖目錄
圖 1.   之故障率函數曲線	4
圖 2.  之故障率函數曲線	4
圖 3.  之故障率函數曲線	5
圖 4. MLE法之平均數走勢圖(實際值(LANDA=0.5,BETA=0.4))	28
圖 5. MHDE法之平均數趨勢圖(實際值(LANDA=0.5,BETA=0.4))	29
圖 6. MLE法與MHDE法之MEAN綜合比較(實際值(LANDA=0.5,BETA=0.4))	29
圖 8. MLE法之MSE趨勢圖(實際值(LANDA=0.5,BETA=0.4))	30
圖 9. MHDE法之MSE趨勢圖(實際值(LANDA=0.5,BETA=0.4))	30
圖 10. MLE法與MHDE法之MSE綜合比較(實際值(LANDA=0.5,BETA=0.4))	31
圖 11. MLE法與MHDE法之SD趨勢圖	31
圖 12. MLE法與MHDE法之BIAS趨勢圖	32
圖 13. MLE法與MHDE法之CORRELATION趨勢圖	32
References
參考文獻
[1]	Beran, R., 1997, Minimum Hellinger distance estimates for parameteric models, The Annals of Statistics, 5(3), 445-463.
[2]	Chen Z., 2000, A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function, Statistics & Probability Letters, 49,155-161.
[3]	Cressie, N. and Read, T. R. C., 1984, Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. B 46, 440-464.
[4]	Deheuvels, P., 1977, Estimation non-paramétrique de la densité par histogrammes generalizes, Rev. Stat. Appl., 25, 5-42.
[5]	Devroye, L. & Gyorfi, L., 1985, Nonparameteric density estimation:the L1 view, Wiley, New York.
[6]	Devroye, L., 1989, The double kernel method in density estimation, Ann. Inst. Henri Poincare, 25, 533-580.
[7]	Hall, P. and Wand, M.P., 1988a, Minimizing L1 distance in nonparameteric density estimation, Journal of Multivariate Analysis, 26, 59-88.
[8]	Hall, P. and Wand, M.P., 1988b, On the minimization of absolute distance in kernel density estimation, Statistics & Probability Letters, 6 311-314.
[9]	Leemis L.M., 1986, Lifetime distribution identities,” IEEE Trans. Reliability 35,170-174.
[10]	Lindsay, B. G., 1994, Efficiency versus robustness:The case for minimum Hellinger distance and related methods, Ann. Statist. 22, 1081-1114.
[11]	Mudholkar G.S. and Srivastava D.K., 1993, Exponentiated Weibull family for analyzing bathtub failure-rate data, IEEE Trans. Reliability 42,299-302.
[12]	Smith, K., 1916, On the ‘best’ values of the constants in the frequency distributions, Biometrika 11, 262–276.
[13]	Sinha, S. K., 1986, Reliability and Life Testing, John Wiley & Sons, Inc., New York.
[14]	Silverman, B. W., 1986, Density Estimation for Statistics and Data Analysis, Chapman and Hall.
[15]	Turlach, B.A., 1993, Bandwidth Selection in Kernel Density Estimation:A Review, CORE and Institut de Statistique.
[16]	Wolfowitz, J., 1952, Consistent Estimation of the parameter of a Linear Structural Relation, Skandinavisk Aktuarietidskrift, 35, 132-157.
[17]	Wolfowitz, J., 1954, Estimation by the Minimum Distance Method in Nonparametric Difference Equations, Annals of Mathematical Statistics, 25, 203-217.
[18]	Wolfowitz, J., 1957, The Minimum Distance Method, Annals of Mathematical Statistics, 28, 75-88.
[19]	Wu, J. W., 2004, Statistical Inference About the shape Parameter of the New Two-parameter Bathtub-shaped Lifetime Distribution, Qual. Reliab. Engng. Int. 2004; 20:607-616.
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