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Mathematical Primer for Social Statistics 2nd Revised edition [Mīkstie vāki]

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(McMaster University, Canada)
  • Formāts: Paperback / softback, 256 pages, height x width: 215x139 mm, weight: 320 g
  • Sērija : Quantitative Applications in the Social Sciences
  • Izdošanas datums: 02-Mar-2021
  • Izdevniecība: SAGE Publications Inc
  • ISBN-10: 1071833200
  • ISBN-13: 9781071833209
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Mathematical Primer for Social Statistics 2nd Revised editionPalielināt
  • Formāts: Paperback / softback, 256 pages, height x width: 215x139 mm, weight: 320 g
  • Sērija : Quantitative Applications in the Social Sciences
  • Izdošanas datums: 02-Mar-2021
  • Izdevniecība: SAGE Publications Inc
  • ISBN-10: 1071833200
  • ISBN-13: 9781071833209
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781071833209.html
Keywords:

A Mathematical Primer for Social Statistics, Second Edition presents mathematics central to learning and understanding statistical methods beyond the introductory level: the basic "language" of matrices and linear algebra and its visual representation, vector geometry; differential and integral calculus; probability theory; common probability distributions; statistical estimation and inference, including likelihood-based and Bayesian methods. The volume concludes by applying mathematical concepts and operations to a familiar case, linear least-squares regression. The Second Edition pays more attention to visualization, including the elliptical geometry of quadratic forms and its application to statistics. It also covers some new topics, such as an introduction to Markov-Chain Monte Carlo methods, which are important in modern Bayesian statistics. A companion website includes materials that enable readers to use the R statistical computing environment to reproduce and explore computations and visualizations presented in the text. The book is an excellent companion to a "math camp" or a course designed to provide foundational mathematics needed to understand relatively advanced statistical methods.

 

Series Editor's Introduction xi
Acknowledgments xiii
Preface xv
What's New in the Second Edition xvii
Notation xvii
Recommended Reading xx
Website xxii
About the Author xxiii
1 Matrices, Linear Algebra, and Vector Geometry: The Basics
1(41)
1.1 Matrices
1(17)
1.1.1 Introducing the Actors: Definitions
1(4)
1.1.2 Simple Matrix Arithmetic
5(6)
1.1.3 Matrix Inverses
11(4)
1.1.4 Determinants
15(1)
1.1.5 The Kronecker Product
16(2)
1.2 Basic Vector Geometry
18(2)
1.3 Vector Spaces and Subspaces
20(11)
1.3.1 Orthogonality and Orthogonal Projections
25(6)
1.4 Matrix Rank and the Solution of Linear Simultaneous Equations
31(11)
1.4.1 Rank
31(2)
1.4.2 Linear Simultaneous Equations
33(5)
1.4.3 Generalized Inverses
38(4)
2 Matrix Decompositions and Quadratic Forms
42(19)
2.1 Eigenvalues and Eigenvectors
42(5)
2.1.1 Generalized Eigenvalues and Eigenvectors
46(1)
2.1.2 The Singular-Value Decomposition
46(1)
2.2 Quadratic Forms and Positive-Definite Matrices
47(9)
2.2.1 The Elliptical Geometry of Quadratic Forms
48(7)
2.2.2 The Cholesky Decomposition
55(1)
2.3 The QR Decomposition
56(5)
2.3.1 Using the QR Decomposition to Compute Eigenvalues and Eigenvectors
60(1)
3 An Introduction to Calculus
61(45)
3.1 Review
61(8)
3.1.1 Numbers
61(1)
3.1.2 Lines and Planes
62(2)
3.1.3 Polynomials
64(1)
3.1.4 Logarithms and Exponentials
65(2)
3.1.5 Basic Trigonometric Functions
67(2)
3.2 Limits
69(4)
3.2.1 The "Epsilon-Delta" Definition of a Limit
69(3)
3.2.2 Finding a Limit: An Example
72(1)
3.2.3 Rules for Manipulating Limits
72(1)
3.3 The Derivative of a Function
73(8)
3.3.1 The Derivative as the Limit of the Difference Quotient: An Example
75(1)
3.3.2 Derivatives of Powers
76(1)
3.3.3 Rules for Manipulating Derivatives
77(2)
3.3.4 Derivatives of Logs and Exponentials
79(1)
3.3.5 Derivatives of the Basic Trigonometric Functions
80(1)
3.3.6 Second-Order and Higher-Order Derivatives
80(1)
3.4 Optimization
81(5)
3.4.1 Optimization: An Example
83(3)
3.5 Multivariable and Matrix Differential Calculus
86(11)
3.5.1 Partial Derivatives
86(2)
3.5.2 Lagrange Multipliers for Constrained Optimization
88(2)
3.5.3 Differential Calculus in Matrix Form
90(3)
3.5.4 Numerical Optimization
93(4)
3.6 Taylor Series
97(1)
3.7 Essential Ideas of Integral Calculus
98(8)
3.7.1 Areas: Definite Integrals
98(2)
3.7.2 Indefinite Integrals
100(1)
3.7.3 The Fundamental Theorem of Calculus
101(3)
3.7.4 Multivariable Integral Calculus
104(2)
4 Elementary Probability Theory
106(16)
4.1 Probability Basics
106(5)
4.1.1 Axioms of Probability
107(1)
4.1.2 Relations Among Events, Conditional Probability, and Independence
108(2)
4.1.3 Bonferroni Inequalities
110(1)
4.2 Random Variables
111(8)
4.2.1 Expectation and Variance
114(1)
4.2.2 Joint and Conditional Probability Distributions
115(2)
4.2.3 Independence, Dependence, and Covariance
117(1)
4.2.4 Vector Random Variables
118(1)
4.3 Transformations of Random Variables
119(3)
4.3.1 Transformations of Vector Random Variables
120(2)
5 Common Probability Distributions
122(23)
5.1 Some Discrete Probability Distributions
122(4)
5.1.1 The Binomial and Bernoulli Distributions
122(2)
5.1.2 The Multinomial Distributions
124(1)
5.1.3 The Poisson Distributions
125(1)
5.1.4 The Negative Binomial Distributions
126(1)
5.2 Some Continuous Distributions
126(16)
5.2.1 The Normal Distributions
127(1)
5.2.2 The Chi-Square (Χ2) Distributions
128(2)
5.2.3 Student's t-Distributions
130(2)
5.2.4 The F-Distributions
132(1)
5.2.5 The Multivariate-Normal Distributions
132(5)
5.2.6 The Exponential Distributions
137(1)
5.2.7 The Inverse-Gaussian Distributions
137(1)
5.2.8 The Gamma Distributions
138(1)
5.2.9 The Beta Distributions
139(2)
5.2.10 The Wishart Distributions
141(1)
5.3 Exponential Families of Distributions
142(3)
5.3.1 The Binomial Family
144(1)
5.3.2 The Normal Family
144(1)
5.3.3 The Multinomial Family
144(1)
6 An Introduction to Statistical Theory
145(53)
6.1 Asymptotic Distribution Theory
145(6)
6.1.1 Probability Limits
145(2)
6.1.2 Asymptotic Expectation and Variance
147(2)
6.1.3 Asymptotic Distribution
149(1)
6.1.4 Vector and Matrix Random Variables
149(2)
6.2 Properties of Estimators
151(12)
6.2.1 Bias and Unbias
151(1)
6.2.2 Mean-Squared Error and Efficiency
151(2)
6.2.3 Consistency
153(1)
6.2.4 Sufficiency
154(1)
6.2.5 Robustness
154(9)
6.3 Maximum-Likelihood Estimation
163(15)
6.3.1 Preliminary Example
164(3)
6.3.2 Properties of Maximum-Likelihood Estimators
167(2)
6.3.3 Wald, Likelihood-Ratio, and Score Tests
169(4)
6.3.4 Several Parameters
173(3)
6.3.5 The Delta Method
176(2)
6.4 Introduction to Bayesian Inference
178(20)
6.4.1 Bayes's Theorem
178(3)
6.4.2 Extending Bayes's Theorem
181(2)
6.4.3 An Example of Bayesian Inference
183(2)
6.4.4 Bayesian Interval Estimates
185(1)
6.4.5 Bayesian Inference for Several Parameters
186(1)
6.4.6 Markov-Chain Monte Carlo
186(12)
7 Putting the Math to Work: Linear Least-Squares Regression
198(21)
7.1 Least-Squares Fit
198(5)
7.1.1 Computing the Least-Squares Solution by the QR and SVD Decompositions
201(2)
7.2 A Statistical Model for Linear Regression
203(1)
7.3 The Least-Squares Coefficients as Estimators
204(1)
7.4 Statistical Inference for the Regression Model
205(3)
7.5 Maximum-Likelihood Estimation of the Regression Model
208(1)
7.6 Random Xs
209(3)
7.7 The Elliptical Geometry of Linear Least-Squares Regression
212(7)
7.7.1 Simple Regression
212(1)
7.7.2 Multiple Regression
213(6)
References 219(2)
Index 221
John Fox received a BA from the City College of New York and a PhD from the University of Michigan, both in Sociology. He is Professor Emeritus of Sociology at McMaster University in Hamilton, Ontario, Canada, where he was previously the Senator William McMaster Professor of Social Statistics. Prior to coming to McMaster, he was Professor of Sociology, Professor of Mathematics and Statistics, and Coordinator of the Statistical Consulting Service at York University in Toronto. Professor Fox is the author of many articles and books on applied statistics, including \emph{Applied Regression Analysis and Generalized Linear Models, Third Edition} (Sage, 2016). He is an elected member of the R Foundation, an associate editor of the Journal of Statistical Software, a prior editor of R News and its successor the R Journal, and a prior editor of the Sage Quantitative Applications in the Social Sciences monograph series.