| 中文题名： | Wasserstein距离及其单样本拟合优度检验 |
| 姓名： | |
| 保密级别： | 公开 |
| 论文语种： | 中文 |
| 学科代码： | 070101 |
| 学科专业： | |
| 学生类型： | 学士 |
| 学位： | 理学学士 |
| 学位年度： | 2021 |
| 学校： | 北京师范大学 |
| 校区： | |
| 学院： | |
| 第一导师姓名： | |
| 第一导师单位： | |
| 提交日期： | 2021-06-10 |
| 答辩日期： | 2021-05-13 |
| 外文题名： | Wasserstein Distance and its goodness-of-fit test |
| 中文关键词： | |
| 外文关键词： | Optimal Transport ; Wasserstein Distance ; Goodness-of-fit test |
| 中文摘要： |
本文研究了 Wasserstein 距离及其在拟合优度检验方面的应用，分析了 Wasserstein 距离和 Wasserstein 统计量相对其他距离和统计量的优势特点。文 章主体分为四个章节，从最优运输理论引入 Wasserstein 距离；对 Wasserstein 距离的定义、性质、显示表达式做相应分析，通过举例说明的形式得出相比全 变差、范数距离、卡方距离、KL 散度、JS 散度等度量，具有自然度量离散连 续分布距离、有效度量支撑集不重合分布间的距离、反映分布转变路径等优点； 对基于 Wasserstein 距离的拟合优度检验统计量，如修正版统计量、标准化版 统计量、加权版统计量的构造与渐进分布结果做简要分析，通过与卡方统计量、 Kolmogorov统计量、Crame?r-von Mises统计量、Wilk-Shapiro统计量等的比较， 得出其拟合优度检验更精确、更稳健、应用更广泛的优点；在最后通过对两个 拟合优度检验问题的解决，进一步比较说明 Wasserstein 统计量优势。
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| 外文摘要： |
This paper studies Wasserstein distance and its application in goodness of fit test, and analyzes the advantages of Wasserstein distance and Wasserstein statistics over other distances and statistics. The main body of this paper is divided into four chapters. Firstly, introduces Wasserstein Distance from the optimal transportation theory including distribution problem, Monge problem and Kantorovich problem. Secondly, makes corresponding analysis on the definition, properties and display expression of Wasserstein distance. Through some examples, it is concluded that compared with total variation, norm distance, chi square distance, KL divergence, JS divergence and other measures, Wasserstein distance can naturally measure distance between discrete distribution and continuous distribution, can effectively measure distance between distributions supporting by non-overlapping sets, and can show the between distributions. Thirdly, the structure and asymptotic distribution of goodness of fit test statistics based on Wasserstein distance are briefly analyzed. By comparing with chi-square statistics, Kolmogorov statistics, Crame? r-von Mises statistics, Wilk-Shapiro statistics and so on, we find that the goodness of fit test is more accurate, more robust and more widely used. Finally, through solving the two goodness of fit test problems, we further understand the advantages of Wasserstein statistics.
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| 参考文献总数： | 12 |
| 插图总数： | 6 |
| 插表总数： | 5 |
| 馆藏号： | 本070101/21044 |
| 开放日期： | 2022-06-10 |
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