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題名 論混合常態分配之估計 作者 李瑞蘭 貢獻者 李隆安
李瑞蘭日期 1991
1990上傳時間 2-May-2016 17:02:53 (UTC+8) 摘要 混合模式(mixture m。del) 參數的估計問題，研究迄今已近一世紀，自Pearson (1894) 提出動差法來估計兩個混合常態模式的參數，往後有許多學者繼續這個範疇的研究，且提出他們不同的估計法。而近半世紀以來，混合常態分配被廣泛應用在許多學科的進一步研究分析上，因此，它的參數估計的意義就更顯重要。 我們欲估計兩個混合常態分配的參數，但原參數空間有識別性的問題，估計參數時會產生Chiang 等人所提的不好的現象，經過適當轉換後的新參數空間，則無識別性的問題。本文第一章簡單地介紹吾師李隆安博士所提的四個空間轉換函數和轉換後所對應的新參數空間。在第二章中，先簡單地介紹一些估計兩個混合常態參數的方法，然後再介紹現今最常用來估計兩個混合常態參數的方法- MLE ，其中包括估計MLE 的EM 演算法、牛頓法和最佳化循序演算法，並將Burnham (1988) 提出求解有限混合分配的最大概似估計值的縮減公式，應用在EM 演算法上，可知道五維參數EM 反覆求解法即三維參數EM 反覆求解法，可以縮減計算問題的大小。同時也介紹拋磚引玉法，其為一精深再抽樣的方法，它有系統的改進原始估計值及估計改進後估計式的變異數。在第三章中，對摸擬所需之資料組、反覆求解的式子和數值問題的解決方法做介紹。第四章中，針對EM法配合求解MLE 的縮減公式和最佳化循序演算法估計所得之兩個混合常態分配參數估計值做分析，並做原參數空間和新參數空間上EM 法收斂快慢的比較；此外並將應用拋磚引玉法，取一些特殊形式的統計量來估計參數的結果做記錄。 第五章，為結論與進一步研究。 參考文獻 Beran,R. (1977), Minimum Hellinger Distance Estimates for Parametric Modles.The Annals of Statistics,5 ,445-463 Brenton R. Clarke (1989,Feb), An unbiased minimum distance estimator of the proportion parameter in a mixture of two normal distribution. Statistics & Probability Letters. Vol 7, No 4,275-281 Bryant,J.L. and Paulson,A.S. (1983), Estimating of Mixing Proportions via Distance between Characteristic Functions.Communications in Statistics,A12,1009-1029 Burnham,K.P.(1988), A Comment on Maximum Likelihood Estimation for Finite Mixture of Distributions.Biom,J. Vol 30, No 3,379-384 Cassie,R.M. (1954), Some Uses of Probability Paper in the Analysis of Size Frequency Distributions. Austral,J. of Marine and Freshwater Res,5,513-22 Chiang,C.L. (1951),On the Design of Mass Medical Surveys. Human Biol. 23,242-271 Choi,K. and Bulgren,W.G. (1968), An Estimation Procedure for Mixtures of Distributions.Journal of the Royal Statistical Society,Ser.B, 30,444-460. Cohen,A.C. (1967), Estimation in Mixtures of two Normal Distribution.Technometrics,9,l5-28 Day, N. E. (1969), Estimating the Components of a Mixture Normal Distributions.Biometrika,56,463-474 Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977), Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society Sere B 39, 1-38 D.M. Titterington; A.F.M. Smith; U.E. Markov, Statistical Analysis of Finite Mixture Distributions 1985 Wiley NY:UK 243P Everitt, B.S. and Hand, D.J. (1981), Finite Mixture Distributions. Chapman and Hall Everitt, B.S. (1984), Maximum Likekihood Estimation of the Parameters in a Mixture of Two Univariate Normal Distributions: A Comparison of Different Algorithms.The Statistican,33,205-215 Fabir. , Rossi, (1983), Decompsition of Mixture Via a Generalized E-M Method.Metron,133-146 Fisher, R.A. (1922), On the Mathematical Foundations of Theoretical Statistics.reprinted in contributions to Mathematical Statistics (by R.A.Fisher)(1950), J. Wiley & Sons, New York Fowlkes, E.B. (1979), Some Methods for Studying the Mixture of Two Normal(Lgnormal) Distributions. Journal of the American StatisticalAssociation, 74,561-575 Harding, J.P. (1949), The Use of Probability Paper for the Graphical Analysis of Polymodal Frequency Distributions. J. of the Marine Biol. Ass. of the UK, 28, 141-53 Hathaway, R.J. (1985), A Constrained Formulation of Maximum Likelihood Estimation for Normal Mixture Distributions. The Annals of Statistics,vol 13, No 2,795-800 Hathaway, R.J. (1986,a), A Constrained EM Algorithm for Univariate Normal Mixtures. Journal of Statistical Computation and Simulation, 23,211-230 Hathaway, R.J. (1986,b), Another Interpretation of the EM Algorithm for Mixture Distributions. Statistics & Probability Letters,4,53-56 James C. Fu (1989), Method of Kim-Zam : An Algorithm for Computing the Maximum Likelihood Estimator. Statistics & Probability Letters,8,289-296 James C. FU, Lung-An Li (1989), Method of Pao-Zhuan-Yin-Yu: A method of Stochastic Point Estimate. Kiefer, N.M. (1978), Comment Journal of the American Statistical Associations,December,P 744 Kiefer, N.M. (1978), Discrete Parameter Variation: Efficient Estimation of a Switching Regression Model. Econometrica 46, 427-434 Kumar,K.D., Nicklin,E.H.,and Paulson,A.S. (1979), Comment on `Estimating Mixtures of Normal Distributions and switching Distributions` by Quandt and Ramsey. Journal of the American Statistical Association, 74,52-55 Liang,W. (1989), Strong Consistency of Constrained Maximum Likelihood Estimator of Finite Normal Mixture. Bulletin of Institute of Mathematics Academia Sinica, Vol 17, No 4,299-304 Louis,T.A. (1982), Finding the Observed Information Matrix When Using the EM Algorithm. Journal of the Royal Statistical Society, Ser.B 44,226-233 Macdonald,P.D.M. (1971), Comment on `An Estimation Procedure for Mixture of Distributions` by Choi and Bulgren. Journal of the Royal Statistical Society, Ser.B,33, 326-329 Pearson,K. (1894), Contribution to the Mathematical Theory of Evolution.Philosophical Transactions of the Royal Society of London,Ser.A, 71-110 Quandt,R.E. and Ramsey,J.B. (1978), Estimating Mixtures of Normal Distributions and Switching Regressions. Journal of the American Statics tical Association 73, 730-738 Robertson,C.A. and Fryer,J.G. (1972), A Comparison of Some Methods for Estimating Mixed Normal Distributions. Biometrika,59,639-648 Tan,W.Y. and Chang,W.C. (1972), Some Comparisons of Method of Moments and the Method of Maximum Likelihood in Estimating Parameters of a Mixture of Two Normal densities. J.Amer Statist Assoc. 67,702-708 Woodward,W.A., Parr,W.C., Schucany, W.R., and Lindsey,H (1984), A Comparison of Minimum Distance and Maximum Likelihood Estimation of a Mixture Proportion. Wu,C.F.J. (1983), On the Convergence Properities of the EM Algorithm. The Annals of Stastistics,Vol 11, No 1, 95-103 李友錚(1985)，利用最小Hellinger距離估計混合模式的參數，清大工業工程碩士論文 劉培熙(1987)，最小距離估計法應用於離散機率分配混合模式之研究，清大工業工程所碩士論文 黃瓊玉(1990)，論混合常態之可識別性，政大統計所碩士論文 描述 碩士
國立政治大學
統計學系資料來源 http://thesis.lib.nccu.edu.tw/record/#B2002005031 資料類型 thesis dc.contributor.advisor 李隆安 zh_TW dc.contributor.author (Authors) 李瑞蘭 zh_TW dc.creator (作者) 李瑞蘭 zh_TW dc.date (日期) 1991 en_US dc.date (日期) 1990 en_US dc.date.accessioned 2-May-2016 17:02:53 (UTC+8) - dc.date.available 2-May-2016 17:02:53 (UTC+8) - dc.date.issued (上傳時間) 2-May-2016 17:02:53 (UTC+8) - dc.identifier (Other Identifiers) B2002005031 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/89644 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 統計學系 zh_TW dc.description.abstract (摘要) 混合模式(mixture m。del) 參數的估計問題，研究迄今已近一世紀，自Pearson (1894) 提出動差法來估計兩個混合常態模式的參數，往後有許多學者繼續這個範疇的研究，且提出他們不同的估計法。而近半世紀以來，混合常態分配被廣泛應用在許多學科的進一步研究分析上，因此，它的參數估計的意義就更顯重要。 我們欲估計兩個混合常態分配的參數，但原參數空間有識別性的問題，估計參數時會產生Chiang 等人所提的不好的現象，經過適當轉換後的新參數空間，則無識別性的問題。本文第一章簡單地介紹吾師李隆安博士所提的四個空間轉換函數和轉換後所對應的新參數空間。在第二章中，先簡單地介紹一些估計兩個混合常態參數的方法，然後再介紹現今最常用來估計兩個混合常態參數的方法- MLE ，其中包括估計MLE 的EM 演算法、牛頓法和最佳化循序演算法，並將Burnham (1988) 提出求解有限混合分配的最大概似估計值的縮減公式，應用在EM 演算法上，可知道五維參數EM 反覆求解法即三維參數EM 反覆求解法，可以縮減計算問題的大小。同時也介紹拋磚引玉法，其為一精深再抽樣的方法，它有系統的改進原始估計值及估計改進後估計式的變異數。在第三章中，對摸擬所需之資料組、反覆求解的式子和數值問題的解決方法做介紹。第四章中，針對EM法配合求解MLE 的縮減公式和最佳化循序演算法估計所得之兩個混合常態分配參數估計值做分析，並做原參數空間和新參數空間上EM 法收斂快慢的比較；此外並將應用拋磚引玉法，取一些特殊形式的統計量來估計參數的結果做記錄。 第五章，為結論與進一步研究。 zh_TW dc.description.tableofcontents 摘要....................1 第一章 緒論....................2 1-1 混合常態的可識別性....................2 1-2 空間轉換函數....................3 第二章 混合常態分配的參數估計....................22 2-1 動差法、圖形估計法、動差母函數法、最小距離估計法....................22 2-2 最大概似法....................24 2-2-1 MLE 的一致性....................26 2-2-2 MLE 求解的縮減公式....................26 2-3 牛頓法....................27 2-4 EM 演算法.................... 28 2-5 最佳化循序演算法....................31 2-6 目前所提出之方法已有的比較結果....................33 2-7 拋磚引玉法....................34 第三章 模擬程序....................37 3-1 原始資料的產生....................37 3-2 EM 演算法....................42 3-3 最佳化循序演算法....................43 3-4 拋磚引玉法....................44 3-5 解決數值之問題....................44 第四章 模擬結果分析 4-1 EM演算法....................47 4-2 最佳化循序演算法....................47 4-3 拋磚引玉法....................53 第五章 結論與進一步研究....................55 5-1 結論....................55 5-2 進一步研究....................55 參考文獻....................57 zh_TW dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#B2002005031 en_US dc.title (題名) 論混合常態分配之估計 zh_TW dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) Beran,R. (1977), Minimum Hellinger Distance Estimates for Parametric Modles.The Annals of Statistics,5 ,445-463 Brenton R. Clarke (1989,Feb), An unbiased minimum distance estimator of the proportion parameter in a mixture of two normal distribution. Statistics & Probability Letters. Vol 7, No 4,275-281 Bryant,J.L. and Paulson,A.S. (1983), Estimating of Mixing Proportions via Distance between Characteristic Functions.Communications in Statistics,A12,1009-1029 Burnham,K.P.(1988), A Comment on Maximum Likelihood Estimation for Finite Mixture of Distributions.Biom,J. Vol 30, No 3,379-384 Cassie,R.M. (1954), Some Uses of Probability Paper in the Analysis of Size Frequency Distributions. Austral,J. of Marine and Freshwater Res,5,513-22 Chiang,C.L. (1951),On the Design of Mass Medical Surveys. Human Biol. 23,242-271 Choi,K. and Bulgren,W.G. (1968), An Estimation Procedure for Mixtures of Distributions.Journal of the Royal Statistical Society,Ser.B, 30,444-460. Cohen,A.C. (1967), Estimation in Mixtures of two Normal Distribution.Technometrics,9,l5-28 Day, N. E. (1969), Estimating the Components of a Mixture Normal Distributions.Biometrika,56,463-474 Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977), Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society Sere B 39, 1-38 D.M. Titterington; A.F.M. Smith; U.E. Markov, Statistical Analysis of Finite Mixture Distributions 1985 Wiley NY:UK 243P Everitt, B.S. and Hand, D.J. (1981), Finite Mixture Distributions. Chapman and Hall Everitt, B.S. (1984), Maximum Likekihood Estimation of the Parameters in a Mixture of Two Univariate Normal Distributions: A Comparison of Different Algorithms.The Statistican,33,205-215 Fabir. , Rossi, (1983), Decompsition of Mixture Via a Generalized E-M Method.Metron,133-146 Fisher, R.A. (1922), On the Mathematical Foundations of Theoretical Statistics.reprinted in contributions to Mathematical Statistics (by R.A.Fisher)(1950), J. Wiley & Sons, New York Fowlkes, E.B. (1979), Some Methods for Studying the Mixture of Two Normal(Lgnormal) Distributions. Journal of the American StatisticalAssociation, 74,561-575 Harding, J.P. (1949), The Use of Probability Paper for the Graphical Analysis of Polymodal Frequency Distributions. J. of the Marine Biol. Ass. of the UK, 28, 141-53 Hathaway, R.J. (1985), A Constrained Formulation of Maximum Likelihood Estimation for Normal Mixture Distributions. The Annals of Statistics,vol 13, No 2,795-800 Hathaway, R.J. (1986,a), A Constrained EM Algorithm for Univariate Normal Mixtures. Journal of Statistical Computation and Simulation, 23,211-230 Hathaway, R.J. (1986,b), Another Interpretation of the EM Algorithm for Mixture Distributions. Statistics & Probability Letters,4,53-56 James C. Fu (1989), Method of Kim-Zam : An Algorithm for Computing the Maximum Likelihood Estimator. Statistics & Probability Letters,8,289-296 James C. FU, Lung-An Li (1989), Method of Pao-Zhuan-Yin-Yu: A method of Stochastic Point Estimate. Kiefer, N.M. (1978), Comment Journal of the American Statistical Associations,December,P 744 Kiefer, N.M. (1978), Discrete Parameter Variation: Efficient Estimation of a Switching Regression Model. Econometrica 46, 427-434 Kumar,K.D., Nicklin,E.H.,and Paulson,A.S. (1979), Comment on `Estimating Mixtures of Normal Distributions and switching Distributions` by Quandt and Ramsey. Journal of the American Statistical Association, 74,52-55 Liang,W. (1989), Strong Consistency of Constrained Maximum Likelihood Estimator of Finite Normal Mixture. Bulletin of Institute of Mathematics Academia Sinica, Vol 17, No 4,299-304 Louis,T.A. (1982), Finding the Observed Information Matrix When Using the EM Algorithm. Journal of the Royal Statistical Society, Ser.B 44,226-233 Macdonald,P.D.M. (1971), Comment on `An Estimation Procedure for Mixture of Distributions` by Choi and Bulgren. Journal of the Royal Statistical Society, Ser.B,33, 326-329 Pearson,K. (1894), Contribution to the Mathematical Theory of Evolution.Philosophical Transactions of the Royal Society of London,Ser.A, 71-110 Quandt,R.E. and Ramsey,J.B. (1978), Estimating Mixtures of Normal Distributions and Switching Regressions. Journal of the American Statics tical Association 73, 730-738 Robertson,C.A. and Fryer,J.G. (1972), A Comparison of Some Methods for Estimating Mixed Normal Distributions. Biometrika,59,639-648 Tan,W.Y. and Chang,W.C. (1972), Some Comparisons of Method of Moments and the Method of Maximum Likelihood in Estimating Parameters of a Mixture of Two Normal densities. J.Amer Statist Assoc. 67,702-708 Woodward,W.A., Parr,W.C., Schucany, W.R., and Lindsey,H (1984), A Comparison of Minimum Distance and Maximum Likelihood Estimation of a Mixture Proportion. Wu,C.F.J. (1983), On the Convergence Properities of the EM Algorithm. The Annals of Stastistics,Vol 11, No 1, 95-103 李友錚(1985)，利用最小Hellinger距離估計混合模式的參數，清大工業工程碩士論文 劉培熙(1987)，最小距離估計法應用於離散機率分配混合模式之研究，清大工業工程所碩士論文 黃瓊玉(1990)，論混合常態之可識別性，政大統計所碩士論文 zh_TW
