訓練學生基礎數理統計推導與證明能力

Train students’ basic mathematical statistics derivation and proof abilities

數理統計為統計系學生學習統計理論的核心課程，提供統計相關課程的理論根基，並且培養未來修習更高深統計相關課程的能力，課程內容涵概相當的廣度與深度，強調學生能夠了解基本的理論統計概念及在不同情況下的統計程序，課程主題包括機率理論與統計推論，範圍包含：
1. 機率：條件機率、隨機變數、分配函數、期望值、條件期望值
2. 尋找機率分配的技巧：變數變換、動差母函數
3. 分配：離散與連續型分配其特性及分配之間的關係、位置與尺度族、多變量常態分配、t分配和F分配、混合分配
4. 不等式：Chebyshev、Jensen、Hölder、Minkowski
5. 收歛概念與極限分配：不偏性、一致性、機率收斂、分配收斂、強收斂、中央極限定理、Delta方法
6. 隨機樣本：抽樣、單一樣本與兩樣本之信賴區間及假設檢定的常態理論及其相關的大樣本方法、順序統計量、生成隨機樣本技巧、拔靴法
7. 最大概似法及其漸進理論：Cramér-Rao不等式、有效性、最大概似估計量、最大概似估計量的漸近性質、EM演算法、評估點估計量的方法
8. 縮減資料：指數族、充分性、完備性、完備充分統計量、UMVUE、Rao-Blackwell定理、Basu定理
9. 最適假設檢定：Neyman-Pearson引理、MP檢定、UMP檢定、MLR族、UMPU檢定、LR檢定、sequential檢定
10. 區間估計：檢定統計量的轉換、樞紐、評估區間估計量的方法
11. 貝氏方法：決策、貝氏估計量、階層貝氏、經驗貝氏、馬可夫鍊蒙地卡羅法

Mathematical statistics is a core course for students in the Department of Statistics to learn statistical theory. It provides the theoretical foundation for statistics-related courses and cultivates the ability to take more advanced statistics-related courses in the future. The content of the course covers considerable breadth and depth, emphasizing that students can understand basic theoretical statistical concepts and statistical procedures in different situations. Course topics include probability theory and statistical inference, and the scope includes:
1. Probability: conditional probability, random variables, distribution function, expected value, conditional expected value
2. Techniques for finding probability distribution: variable transformation, dynamic difference generating function
3. Allocation: characteristics of discrete and continuous allocation and the relationship between allocation, position and scale family, multi-variable normal allocation, t allocation and F allocation, mixed allocation
4. Inequality: Chebyshev, Jensen, Hölder, Minkowski
5. Convergence concept and limit allocation: impartiality, consistency, probabilistic convergence, allocation convergence, strong convergence, central limit theorem, Delta method
6. Random samples: sampling, confidence intervals of single sample and two samples, normality theory of hypothesis testing and its related large sample methods, sequential statistics, techniques for generating random samples, and the boot method
7. Maximum approximate method and its asymptotic theory: Cramér-Rao inequality, effectiveness, maximum approximate estimator, asymptotic properties of maximum approximate estimator, EM algorithm, method of evaluating point estimators
8. Reduced information: exponential family, sufficiency, completeness, complete and sufficient statistics, UMVUE, Rao-Blackwell theorem, Basu theorem
9. Optimum hypothesis test: Neyman-Pearson lemma, MP test, UMP test, MLR family, UMPU test, LR test, sequential test
10. Interval Estimation: Conversion of test statistics, pivots, methods of evaluating interval estimators
11. Bayesian methods: decision-making, Bayesian estimator, hierarchical Bayesian, empirical Bayesian, Markov chain Monte Carlo method

Introduction to Mathematical Statistics.
By R.V. Hogg, J. McKean and A.T. Craig

Introduction to Mathematical Statistics.
By R.V. Hogg, J. McKean and A.T. Craig