Copula-based statistical inferences for a common mean vector and correlation ratios using bivariate data

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施嘉翰(Jia-Han Shih) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

(Copula-based statistical inferences for a common mean vector and correlation ratios using bivariate data)

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摘要(中)

本篇論文致力於發展使用二元資料對共同平均向量以及相關比進行基於聯結函數之統計推論。在文獻中，二元共同平均向量的估計是一個經典的問題，然而，現行的統計方法只局限於二元常態模型，這促使我們去研究可以包含廣泛的相依結構之聯結函數模型。另一方面，在文獻中也有許多相關係數被提出以描述兩個隨機變數之關係，其中包含皮爾森相關係數、斯皮爾曼相關係數以及肯德爾相關係數，然而，這些相關係數大多無法描述兩個隨機變數間之不對稱關係，這也促使我們去研究相關比之估計和理論
在本篇論文的第一部分，我們提出了一個基於一般性聯結函數的方法來對二元平均向量進行估計，為此，我們定義了在已知共變異矩陣下之一般性聯結函數模型，並且在此模型下，我們推導了一個二元平均向量的最大概似估計量以及費雪情報矩陣之一般性型式，基於獨立但不同分布之理論，我們研究了提出的最大概似估計量之大樣本理論。我們設計了模擬研究來確認提出方法的表現，並且分析了一筆真實資料做示範，我們也為R軟體使用者在CommonMean.Copula套件中提供了計算程式。
在本篇論文的第二部分，我們研究了聯結函數之相關比，首先，我們為Sungur (2005) 所定義的聯結函數相關比推導了一個新的表示法，藉由使用由 Darsow et al. (1992) 所定義的 -product 算子，我們證明了聯結函數相關比等於兩個聯結函數的 -product 之斯皮爾曼相關係數，並且，我們的新表示法也很自然地指出了一個推廣聯結函數相關比的方法，那就是將斯皮爾曼相關係數換置為其他任意的相關係數。此外，我們也研究了聯結函數相關比的理論性質，其中包含兩個相關比的差異以及不連續性，對於多元聯結函數，我們也定義了相關比矩陣並證明了其不變性。
除了理論結果，我們提出了一個無母數方法對聯結函數相關比進行推論，我們的基本工具為由 Segers et al. (2017) 所提出的經驗貝塔聯結函數，且是經驗伯恩斯坦聯結函數的一個特例。我們證明了經驗貝塔聯結函數的聯結函數相關比具有封閉表示法，根據我們得到的新表示法，我們提出了聯結函數相關比以及差異測數的新估計量，此外，我們也使用了重新抽樣的方法來建立新估計量的信賴區間。我們設計了模擬研究來確認提出方法的表現，並且分析了一筆真實資料做示範，最後，我們分析了額外兩組資料來帶出未來可能的研究方向。

摘要(英)

This thesis is devoted to develop copula-based methods for making statistical inferences on 1) a common mean vector, and 2) correlation ratios using bivariate data. The estimation of a bivariate common mean vector is a classical problem in the literature. However, the existing statistical methods are limited to the bivariate normal models. This motivates us to explore alternative models that allow a broad class of dependence structures modeled by copulas. On the other hand, many measures of association have been proposed to describe association between two random variables such as Pearson’s correlation coefficient, Spearman’s rho, and Kendall’s tau. However, many of them cannot capture the asymmetric relationship between two random variables. This motives us to study the theory and estimation method for correlation ratios, which are useful for analyzing data having asymmetric association patterns.
In the first part of this thesis, we propose a general copula-based approach for estimating a bivariate common mean vector. We define a general copula model for estimating the bivariate common mean vector under known covariance matrices. Then, under the bivariate copula model, we derive a maximum likelihood estimator (MLE) for the common mean vector. In addition, we derive a general form for the Fisher information matrix. Based on the theory of independent but not identically distributed samples, we study the asymptotic properties of the proposed MLE. Simulation studies are conducted to examine the performance of the proposed method, and a real dataset is analyzed for illustration. The computational programs are made available for R users in our R package CommonMean.Copula.
In the second part of this thesis, we study the copula correlation ratio. We first derive a new expression of the copula correlation ratio that was defined by Sungur (2005). By utilizing the -product operator defined by Darsow et al. (1992), we show that the copula correlation ratio is equal to Spearman’s rho of the -product of two copulas. In addition, our new expression also suggests a natural generalization of the copula correlation ratio by allowing Spearman’s rho to be replaced by any other measure of association. Theoretical properties of the copula correlation ratios are investigated, including difference and discontinuity. For multivariate copulas, we also define the copula correlation ratio matrix with showing its invariance property.
In addition to the theoretical results, we propose a nonparametric inference method for the copula correlation ratio. Our fundamental tool is the so-called empirical beta copula that was proposed by Segers et al. (2017) and is a special case of the empirical Bernstein copula. We will show that the copula correlation ratio of the empirical beta copula has a closed-form expression. Based on our newly obtained expression, we propose new estimators for the copula correlation ratio and difference measure. Resampling technique of the empirical beta copula is employed to construct confidence interval for our new estimators. Simulations studies are conducted to examine the performance of the proposed method, and a real dataset is analyzed for illustration. Lastly, we use two additional datasets to raise potential future works.

關鍵字(中)

★ 漸進理論
★ 相關係數
★ 方向性相關
★ 費雪情報
★ 可逆聯結函數
★ 肯德爾相關係數
★ 馬可夫積
★ 最大概似估計法
★ 迴歸性相關
★ 斯皮爾曼相關係數
★ 斯坦等式

關鍵字(英)

★ Asymptotic theory
★ Correlation coefficient
★ Directional association
★ Fisher information
★ Invertible copula
★ Kendall′s tau
★ Markov product
★ Maximum likelihood estimation
★ Regression association
★ Spearman′s rho
★ Stein′s identity

論文目次

Chapter 1 Introduction ..... 1
1.1 Bivariate copulas: A brief history ..... 1
1.2 Estimation of a bivariate common mean vector ..... 3
1.3 Estimation of association ..... 6
Chapter 2 Background on copulas ..... 9
2.1 Copulas ..... 9
2.2 Survival and semisurvival copulas ..... 13
2.3 Spearman’s rho and Kendall’s tau ..... 15
2.4 The -product ..... 17
Chapter 3 General copula-based estimator of a bivariate common mean vector ..... 21
3.1 General copula model ..... 21
3.2 Maximum likelihood estimation ..... 28
3.3 Fisher information matrix ..... 30
3.3.1 Approximate Fisher information matrix ..... 35
3.4 Asymptotic theory ..... 37
3.5 Confidence set ..... 40
3.6 Software ..... 42
3.7 Simulation ..... 44
3.7.1 Simulation design ..... 44
3.7.2 Simulation results ..... 45
3.8 Real data revisit: Bivariate entrance exam scores ..... 49
Chapter 4 Copula correlation ratio: Theory ..... 54
4.1 Copula correlation ratio ..... 54
4.1.1 Alternative expression ..... 56
4.1.2 Generalization ..... 61
4.2 Theoretical properties ..... 64
4.2.1 Difference ..... 64
4.2.2 Discontinuity ..... 67
4.3 The copula correlation ratio matrix ..... 68
Chapter 5 Copula correlation ratio: Estimation ..... 72
5.1 Parametric estimation ..... 73
5.2 Nonparametric inference ..... 75
5.2.1 Empirical beta copula ..... 75
5.2.2 Point estimation ..... 77
5.2.3 Testing inequality ..... 78
5.3 Simulation ..... 80
5.3.1 Simulation design ..... 80
5.3.2 Simulation results ..... 82
5.4 Real data revisit: Bivariate income ..... 85
5.5 Additional data analysis ..... 87
5.5.1 The gross income data ..... 87
5.5.2 The ovarian cancer data ..... 89
Chapter 6 Discussions and future works ..... 92
6.1 Discussions of Chapter 3 ..... 92
6.2 Discussions of Chapter 4 ..... 93
6.3 Discussions of Chapter 5 ..... 95
Appendix A Proofs for Chapter 3 ..... 97
Appendix A1 Proof of Lemma 3.3 ..... 97
Appendix A2 Proof of Theorem 3.3 ..... 99
Appendix B Proof for Chapter 4 ..... 108
Appendix B1 Proof of Theorem 4.2 ..... 108
Appendix C Proof for Chapter 6 ..... 111
Appendix C1 Derivations of the Fisher information matrix under the bivariate common mean Clayton copula model with exponential margins ..... 111
References ..... 117

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指導教授

江村剛志(Takeshi Emura)

審核日期

2020-3-16

推文