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凸函數最佳化在統計問題上的應用 / Convex optimization: A statistical application劉世凰 Unknown Date (has links)
近年來,凸函數最佳化相關的理論與實務已漸趨完善並廣泛應用在各種不同的領域上。已知針對限制條件下之最大概似估計量(Maximum Likelihood Estimator,簡寫MLE)求解的統計問題,一般都是先求解在無限制條件下之全域最大概似估計量(global MLE),若所求得之解能滿足給定的限制條件時,則代表我們的確得到所要的結果;但若所求得之解不能滿足限制條件時,我們就必須考量於此限制條件下之求解區域的最大概似估計量(local MLE),而其計算通常趨於複雜。在本研究中,我們嘗試藉由凸函數最佳化的理論與方法在受限最大概似估計量的求解上。首先針對一組2X2列聯表(contingency table)資料,給定限制條件為勝算比(odds ratio,簡寫OR)不小於1情況下,欲求各聯合機率之受限最大概似估計量。接下來則討論針對3X2列聯表資料,給定兩個區域勝算比(local OR)皆不小於1之限制條件,求取各聯合機率的受限最大概似估計量。我們最終整理歸納出一套分析方法,並將此歸納結果拓展到對於任意J不小於2之JX2列聯表中之受限最大概似估計量計算問題上。本研究中所提出的求解方法包括將決策變數重新參數化,忽略原始的線性限制等式,並另外在原始目標問題中加入某個懲罰項,使其新的最佳化問題滿足凸函數最佳化問題的條件。接下來利用凸函數最佳化之理論,列出其Karush-Kuhn-Tucker 條件,再藉其中的互補差餘條件(complementary slackness)來分析求得理論最佳解。最後我們得出當懲罰項之相對應的係數為n時,則其所求得之最佳解即為此統計問題中之受限最大概似估計量。
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2x2列聯表模型下MLE與MPLE之比較 / The comparison between MLE and MPLE under two-by two contingency table models郭名斬 Unknown Date (has links)
Arnold and Strauss (1991) 探討2x2列聯表中的3個方格 (cell) 有相同機率θ的問題,他們比較了參數θ的最大概似估計值與最大擬概似估計值,發現參數θ的最大概似估計值與最大擬概似估計值是不相同的。在本論文中,我們將2x2列聯表中的3個方格的參數值 (機率值),從限制為相同θ,放寬為成某種比例,並證明了在一般情況下參數θ的最大概似估計值與最大擬概似估計值也不相同。我們也提出一些使參數θ的最大概似估計值及最大擬概似估計值相同的特殊條件,諸如三個方格內的觀察值跟機率值成比例或格子內的觀察值有某些特定值。本論文也透過電腦模擬的結果,發現最大概似估計式較最大擬概似估計式來得精確,而且當參數θ在參數空間之中點附近時,最大概似估計值與最大擬概似估計值的差異為最大。 / Arnold and Strauss (1991) study the cases that three of the four cells in the 2x2 contingency table have the same cell probability θ. In particular, Arnold and Strauss (1991) compare the maximum likelihood estimate (MLE) and maximum pseudolikelihood estimate (MPLE) of the parameter θ. They find that MLE and MPLE of the parameter are not the same. In this thesis, we relax the assumptions so that those three cell probabilities may not be the same and each is proportional to a parameter θ. We find that, in general, MLE’s of θ are still not the same as MPLE’s of θ. Some special cases that make MLE the same as MPLE are also given. We also find, through computer simulations, that MLE’s are accurate than MPLE’s and that the difference between MLE and MPLE is getting larger when the parameter θ is closer to the midpoint of its space.
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多元自迴歸條件異質變異數之模型設定研究欉清全, Genius Tung Unknown Date (has links)
經濟理論明白揭示,在不確定下,金融性資產的選擇不僅要考慮其未來報
酬率的平均值,更需將風險程度納入決策過程中。而最佳風險測度為預測
誤差的變異數(Variance of Forec ast Error)。傳統實証方法均視變異
數為固定常數,實無法掌握變異數具有條件異質性的特點。為了到達此目
的,Engle(1982) 提出向量自迴歸條件異質變異數(ARCH)模型,此模型假
定條件變異數不再是固定常數而是過去干擾項平方的線型函數,為實証方
法上一項偉大的突破。在考慮多個變數的聯立動態體系中,由於跨方程式
間可以互相提供額外的訊息,往往可以增加估計的效率性,直覺上比單變
數的設定更能掌握資料的實際情形。故往後的學者便提出了多元自迴歸條
件異質變異數(Multivariate ARCH) 模型,此一模型亦有其缺點存在,因
其待估計參數過多,形成自由度嚴重減少,將導致估計值缺乏效率性。所
以如何利用可獲得的有限資料對模型進行更有效率的估計方式,此為研究
Multivaria te ARCH的重要課題。本文將對Multivariate ARCH做一系列
的介紹,並利用VAR 的貝氏方法對參數進行估計。而多元因素AR CH模型
也是探討的重點。
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遺漏值存在時羅吉斯迴歸模式分析之研究 / Logistic Regression Analysis with Missing Value劉昌明, Liu, Chang Ming Unknown Date (has links)
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具有額外或不足變異的群集類別資料之研究 / A Study of Modelling Categorical Data with Overdispersion or Underdispersion蘇聖珠, Su, Sheng-Chu Unknown Date (has links)
進行調查時,最後的抽樣單位常是從不同的群集取得的,而同一群集內的樣本對象,因背景類似而對於某些問題常會傾向相同或類似的反應,研究者若忽略這種群內相關性,仍以獨立性樣本進行分析時,因其共變異數矩陣通常會與多項模式的共變異數矩陣相差懸殊,而造成所謂的額外變異或不足變異的現象。本文在不同的情況下,提出了Dirichlet-Multinomial模式(簡稱DM模式)、擴展的DM模式、以及兩種平均數-共變異數矩陣模式,以適當的彙整所有的群集資料。並討論DM與EDM模式中相關之參數及格機率之最大概似估計法,且分別對此兩種平均數-共變異數矩陣模式,提出求導廣義最小平方估計的程序。此外,也針對幾種特殊的二維表及三維表結構,探討對應的參數及格機率之估計方法。並提出計算簡易的Score統計檢定量以判斷群內相關(intra-cluster correlation)之存在性,及判斷資料集具有額外或不足變異,而對於不同母體的群內相關同質性檢定亦提出討論。 / This paper presents a modelling method of analyzing categorical data with overdispersion or underdispersion. In many studies, data are collected from differ clusters, and members within the same cluster behave similary. Thus, the responses of members within the same cluster are not independent and the multinomial distribution is not the correct distribution for the observed counts. Therefore, the covariance matrix of the sample proportion vector tends to be much different from that of the multinomial model. We discuss four different models to fit counts data with overdispersion or underdispersion feature, witch include Dirichlet-Multinomial model (DM model), extended DM model (EDM model), and two mean-covariance models. Method of maximum-likelihood estimation is discussed for DM and EDM models. Procedures to derive generalized least squares estimates are proposed for the two mean-covariance models respectively. As to the cell probabilities, we also discuss how to estimate them under several special structures of two-way and three-way tables. More easily evaluated Score test statistics are derived for the DM and EDM models to test the existence of the intra-cluster correlation. And the test of homogeneity of intra-cluster correlation among several populations is also derived.
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利用混合模型估計風險值的探討阮建豐 Unknown Date (has links)
風險值大多是在假設資產報酬為常態分配下計算而得的,但是這個假設與實際的資產報酬分配不一致,因為很多研究者都發現實際的資產報酬分配都有厚尾的現象,也就是極端事件的發生機率遠比常態假設要來的高,因此利用常態假設來計算風險值對於真實損失的衡量不是很恰當。
針對這個問題,本論文以歷史模擬法、變異數-共變異數法、混合常態模型來模擬報酬率的分配,並依給定的信賴水準估算出風險值,其中混合常態模型的參數是利用準貝式最大概似估計法及EM演算法來估計;然後利用三種風險值的評量方法:回溯測試、前向測試與二項檢定,來評判三種估算風險值方法的優劣。
經由實證結果發現:
1.報酬率分配在左尾臨界機率1%有較明顯厚尾的現象。
2.利用混合常態分配來模擬報酬率分配會比另外兩種方法更能準確的捕捉到左尾臨界機率1%的厚尾。
3.混合常態模型的峰態係數值接近於真實報酬率分配的峰態係數值,因此我們可以確認混合常態模型可以捕捉高峰的現象。
關鍵字:風險值、厚尾、歷史模擬法、變異數-共變異教法、混合常態模型、準貝式最大概似估計法、EM演算法、回溯測試、前向測試、高峰 / Initially, Value at Risk (VaR) is calculated by assuming that the underline asset return is normal distribution, but this assumption sometimes does not consist with the actual distribution of asset return.
Many researchers have found that the actual distribution of the underline asset return have Fat-Tail, extreme value events, character. So under normal distribution assumption, the VaR value is improper compared with the actual losses.
The paper discuss three methods. Historical Simulated method - Variance-Covariance method and Mixture Normal .simulating those asset, return and VaR by given proper confidence level. About the Mixture Normal Distribution, we use both EM algorithm and Quasi-Bayesian MLE calculating its parameters. Finally, we use tree VaR testing methods, Back test、Forward tes and Binomial test -----comparing its VaR loss probability
We find the following results:
1.Under 1% left-tail critical probability, asset return distribution has significant Fat-tail character.
2.Using Mixture Normal distribution we can catch more Fat-tail character precisely than the other two methods.
3.The kurtosis of Mixture Normal is close to the actual kurtosis, this means that the Mixture Normal distribution can catch the Leptokurtosis phenomenon.
Key words: Value at Risk、VaR、Fat tail、Historical simulation method、 Variance-Covariance method、Mixture Normal distribution、Quasi-Bayesian MLE、EM algorithm、Back test、 Forward test、 Leptokurtosis
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