本书是以作者在加利福尼亚大学伯克利分校统计学系给高年级本科生和研究生授课的教学讲义为基础写成的，前半部分为概率，后半部分为统计。书中的主要内容包括概率、随机变量及其分布、期望连续及离散模型、独立性、条件概率分布、密度函数及期望、线性分析，线性回归、泊松分布、逻辑回归及泊松回归等。
尽管本书的重点是使读者对主要概念有个全面的理解，但它同时还向读者介绍了实际数据分析的方方面面。本书适合作为高等院校数学及相关专业高年级本科生或研究生概率统计课程的教材，同时也可作为相关领域科技人员的参考资料。
全新的教学方法与现代数值逼近方法和数值分析的其他方面相适应加强统计理论。方法论及应用之间的联系在线性模型及广义线性模型的处理中，省略重复的矩阵运算，强调概念理解。
Charles J．Stone，斯坦福大学统计学博士，现为加利福尼亚大学伯克利分校统计系教授，主要研究方向是非参数统计模型、统计软件。
The writing of this textbook began a decade ago with my realization that, in this age of computers, the core of a first course in theoretical statistics should no longer consist of the classical material on mathematical statistics involving sufficiency,unbiasedness, efficiency, optimality, and decision theory.
At that time ! decided to write a textbook for a one-year upper division or graduate level course on probability and statistics, in which the first half would serve as a textbook for a one-semester course in probability and the second half would serve as a textbook for a one-semester course in statistics having probability as a prerequisite. The probability portion would provide the proper background for the treatment of statistics, and the statistics portion in turn would apply and reinforce most of the probability material. The level of the book would be compatible with having two years of calculus, including a modicum of linear algebra (properties of vectors and matrices that are summarized in Appendix A), as its prerequisite.
The decision was made to avoid the convention of requiting random variables to be real-valued. In the more general treatment of random variables given here,it is not necessary to introduce "w" or to distinguish between probability spaces and distributions of random variables. By this means and by making use of vectors and matrices (with which the students have been surprisingly comfortable), we get a more thorough and systematic treatment of mean vectors, variance--covariance matrices, multivariate transformations, and the multivariate normal distribution in Chapter 5 than is usual for a book at this level. Moreover, by treating independence directly and thereby delaying the introduction of conditional probability, we get a
more thorough treatment of conditioning in Chapter 6.
Rather than having the statistics portion of the book deal with a.wide variety of models, each on a superficial basis, I decided to concentrate on two Closely related major topics: (i) normal models and the corresponding linear models; and (ii) bino-mial and Poisson models and the corresponding generalized linear models logistic regression and Poisson regression. The first of these two major topics, treated in Chapters 7-11, covers confidence intervals based on the t distribution, t tests, F tests, the least-squares method, and the design and analysis of experiments. The sec-ond topic, treated in Chapters 12 and 13, covers inference for binomial and Poisson distributions as exponential families, the maximum-likelihood method, iteratively-reweighted least squares, normal and multivariate normal approximation to the distribution of maximum-likelihood estimates, Wald tests, and likelihood-ratio tests.
Although the emphasis is on developing a thorough conceptual understanding of the material, both the text and supplementary projects introduce the students to various aspects of the analysis of real data.
While writing this textbook, my research has involved the functional approach to statistical modeling--specifically, the use of polynomial splines to model main effects and their tensor products to model interactions. This approach has been successfully applied to model regression functions and the logarithms of density and conditional density functions, hazard and conditional hazard functions (in survival analysis), and spectral density functions (in the analysis of stationary time series).
The model fitting involves least-squares and maximum-likelihood estimation. The theory involves rates of convergence, and the corresponding practical methodology involves stepwise addition of basis functions using Rao statistics, stepwise deletion using Wald statistics, and a variety of approaches to final model selection.
After a few years of working on the textbook, I decided to switch from the purely parametric approach to linear and generalized linear modeling to one motivated by my research interests. Thus the treatment presented here involves an unknown function that is assumed to be a member of a specified finite-dimensional linear space of functions. Once we choose a basis of the space, we get the usual formulation in terms of unknown regression coefficients. Nevertheless, this innovative approach has a number of important advantages:
It is compatible with the modern approach to numerical approximation and other aspects of numerical analysis.
It strengthens the connections among statistical theory, methodology, and appli-cations.
It emphasizes conceptual understanding over obscure matrix manipulations in the treatment of linear and generalized linear models.
It facilitates the proper interpretation of regression coefficients and their linear combinations, especially in the context of logistic regression and models that are not linear in the covariates.
It provides a smooth transition from the usual ANOVA-type analysis of frac-tional factorial experiments and other orthogonal arrays to the use of quadratic polynomials in response-surface exploration.
It provides a common framework for using linear and quadratic functions of factor-level combinations in analyzing small-scale experiments and using poly-nomial splines or other flexible function spaces in analyzing large observational studies (a topic for a more advanced tex0.
The book has been developed through the experience of using preliminary ver-sions in teaching upper-division and graduate-level one-year introductory courses in probability and statistics in the Statistics Department at the University of Califor-nia at Berkeley. The students have been drawn from a variety of majors——including statistics, mathematics, engineering, economics, business administration, and bio- statistics. The course format has invariably involved three one-hour (50-minute) lectures and one 1-2 hour discussion section conducted by teaching assistants per week. The discussion sections have mainly been devoted to going over problems and exams, preparations for exams, and preparation for the four computer projects per semester. [Current versions of these projects, which are interfaced to the statis-tical package S, along with the relevant data sets for the statistics projects, can beobtained from the author (stone@stat.berkeley.edu) or publisher.
The sections in the book are in one-to-one correspondence with individual lec-tures. There are 72 such sections: 36 sections in the first six chapters——on proba-bility; and 36 in the last seven chapters——on statistics. In particular, Section 1 in Chapter 1 and Section 1 in Chapter 7 are designed to complement introductory lectures at the beginning of the two semesters. Typically there have been 43 class meetings per semester, with 36 being devoted to primary lectures on the various sections, 5 to reviewing this material, 2 to midterm exams.
This textbook can be used for first or second courses for students at various levels, from a first course at the upper-division level for relatively strong students to a course at the graduate level. For a first course at the undergraduate level, the lectures and discussion sections should focus on the motivation, intuition, basic concepts and applications. Let the students read the derivations on their own and do not hold them responsible for this material on exams. Also, it would be appropriate to have open book exams, containing tasks similar to those in the problems in the text.
For a second course for graduate-level students, on the other hand, the lectures and discussion sections should put more emphasis on the derivations, and the students should be held responsible for some such derivations on the exams, which should be at least partly closed book. Naturally, in either case, it would be helpful to make hard copies of the instructor's lectures available to the students as a cohesive, but individualized, treatment intermediate between the full treatment in the book and the summary at the end of the book.
It is my pleasure to thank my students and teaching assistants over the years for their numerous corrections, comments, and suggestions, which have led to substan-tial improvements in the quality of this textbook.
CHAPTER 1 Random Variables and Their Distributions 1
1.1 Introduction 1
1.2 Sample Distributions 5
1.3 Distributions 14
1.4 Random Variables 23
1.5 Probability Functions and Density Functions 33
1.6 Distribution Functions and Quantiles 45
1.7 Univariate Transformations 60
1.8 Independence 69
CHAPTER 2 Expectation 81
2.1 Introduction 81
2.2 Properties of Expectation 91
2.3 Variance 99
2.4 Weak Law of Large Numbers 110
2.5 Simulation and the Monte Carlo Method 121
CHAPTER 3 Special Continuous Models 134
3.1 Gamma and Beta Distributions 134
3.2 The Normal Distribution 145
3.3 Normal Approximation and the Central Limit Theorem
CHAPTER 4 Special Discrete Models 162
4.1 Combinatorics ' 162
4.2 The Binomial Distribution 172
4.3 The Multinomial Distribution 188
4.4 The Poisson Distribution 195
4.5 The Poisson Process 204
CHAPTER 5 Dependence 209
5.1 Covariance, Linear Prediction, and Correlation 209
5.2 Multivariate Expectation 219
5.3 Covariance and Variance-Covariance Matrices 225
5.4 Multiple Linear Prediction 236
5.5 Multivariate Density Functions 242
5.6 Invertible Transformations 252
5.7 The Multivariate Normal Distribution 263
CHAPTER 6 Conditioning 274
6.1 Conditional Distributions 274
6.2 Sampling Without Replacement 285
6.3 Hypergeometric Distribution 292
6.4 Conditional Density Functions 300
6.5 Conditional Expectation 307
6.6 Prediction 316
6.7 Conditioning and the Multivariate Normal Distribution 322
6.8 Random Parameters 330
CHAPTER 7 Normal Models 338
7.1 Introduction 338
7.2 Chi-Square, t, and F Distributions 344
7.3 Confidence Intervals 353
7.4 The t Test of an Inequality 365
7.5 The t Test of an Equality 375
7.6 The F Test 388
CHAPTER 8 Introduction to Linear Regression 396
8.1 The Method of Least Squares 396
8.2 Factorial Experiments 407
8.3 Input-Response and Experimental Models 415
CHAPTER 9 Linear Analysis 427
9.1 Linear Spaces 427
9.2 Identifiability 438
9.3 Saturated Spaces 447
9.4 Inner Products 454
9.5 Orthogonal Projections 470
9.6 Normal Equations 485
CHAPTER 10 Linear Regression 494
10.1 Least-Squares Estimation 494
10.2 Sums of Squares 506
10.3 Distribution Theory 515
10.4 Sugar Beet Experiment 526
10.5 Lube Oil Experiment 538
10.6 The t Test 552
10.7 Submodels 560
10.8 The F Test 568
CHAPTER 11 Orthogonal Arrays 579
11.1 Main Effects 579
11.2 Interactions 595
11.3 Experiments with Factors Having Three Levels' 611
11.4 Randomization, Blocking, and Covariates 620
CHAPTER 12 Binomial and Poisson Models 635
12.1 Nominal Confidence Intervals and Tests 636
12.2 Exact P-Values 651
12.3 One-Parameter Exponential Families 662
CHAPTER 13 Logistic Regression and Poisson Regression 673
13.1 Input-Response and Experimental Models 675
13.2 Maximum-Likelihood Estimation 686
13.3 Existence and Uniqueness of the Maximum-Likelihood Estimate 699
13.4 Iteratively Reweighted Least-Squares Method 709
13.5 Normal Approximation 723
13.6 The Likelihood-Ratio Test 736
APPENDIX A Properties of Vectors and Matrices 751
APPENDIX B Summary of Probability 760
B.1 Random Variables and Their Distributions 760
B.2 Random Vectors 769
APPENDIX C Summary of Statistics 774
C.1 Normal Models 774
C.2 Linear Regression 779
C.3 Binomial and Poisson Models 785
C.4 Logistic Regression and Poisson Regression 787
APPENDIX D Hints and Answers 798
APPENDIX E Tables 828
Index 833
