[内容简介]
《微分幾何在影響分析中的應用（英文版）》討論微分幾何在統計學影響分析中的應用，適合數學及統計學本科生或研究生閱讀。對于研習數學的學生，本書描述微分幾何在數學范疇以外的具體應用；對于研習統計的學生，本書則能幫助他們理解統計領域中的微分幾何概念。《微分幾何在影響分析中的應用（英文版）》要求讀者具備線性代數及向量微積分的基礎知識。書的第一部分圍繞法曲率、截面曲率和高斯曲率概念介紹了圖的幾何學知識；第二部分回顧了統計學的一些基本概念及模型，為理解影響分析提供必要的基礎知識；第三部分則集中討論上述幾何概念在局部影響分析中的應用，并探討如何有效地應用幾何概念以提高局部影響分析估計的效力。《微分幾何在影響分析中的應用（英文版）》為研習統計學或數學的學生架起了知識理解的橋梁，為數學與統計學的跨學科研究合作及相互推進發揮創新性的作用。
[目次]
Preface
Part I Geometry
1 Preliminaries
1.1 Linear algebra
1.1.1 Vectors and matrices
1.1.2 Symmetric bilinear forms
1.1.3 Vector subspaces
1.1.4 Linear maps from Rn to Rn
1.1.5 A convention
1.2 Vector calculus
1.2.1 Vector-valued functions and differentials
1.2.2 Taylor expansion and extrema
1.2.3 Extrema and Lagrange multiplier theorem
2 Euclidean Geometry
2.1 Orthogonal transformations
2.2 Rigid motions
2.3 Translation of vector subspaces
2.4 Conformal transformations
2.5 Orthonormal basis
2.6 Orthogonal projections
2.7 Areas and volumes
3 Geometry of Graphs
3.1 Graphs in Euclidean spaces
3.2 Normal sections
3.3 Cross sections in high dimension
3.4 First fundamental forms
4 Curvatures
4.1 Normal curvatures
4.1.1 Definition
4.1.2 Principal curvatures and principal directions
4.2 Sectional curvatures
5 Transformations and Invariance
5.1 Change of coordinates
5.2 Non-linear conformal transformations
5.3 Invariant curvatures Part II Statistics
6 Discrete Random Variables and Related Concepts
6.1 Preliminaries
6.2 Discrete random variables
6.2.1 Discrete random variables and probability function
6.2.2 Relative frequency histogram
6.2.3 Cumulative distribution function
6.3 Population parameters and sample statistics
6.3.1 Population mean and expected value
6.3.2 Sample statistic
6.3.3 Sample mean
6.3.4 Sample and population variances
6.4 Mathematical expectations
6.5 Maximum likelihood estimation
6.6Maximum likelihood estimation of the probability of a Bernoulliexperiment
7 Continuous Random Variables and Related Concepts
7.1 Continuous random variables
7.2 Mathematical expectation for continuous random variables
7.3 Mean and variance and their sample estimates
7.4 Basic properties of expectations
7.5 Normal distribution
7.6 Maximum likelihood estimation for continuous variables
7.7 Maximum likelihood estimation for the parameters of normaldistribution
7.8 Sampling distribution
8 Bivariate and Multivariate Distribution
9 Simple Linear Regression
10 Topics on Linear Regression Analysis
11 Basic Concepts
12 Measuring Local Influence
13 Relations Among Various Measures
14 Conformal Modifications
Appendix A Rank of Hat Matrix
Appendix B Ricci Curvature
Appendix C Cook-s Distance-Deleting Two Data Points
Bibliography
Index