| dc.description.abstract | 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.
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