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内容简介

　　Probability Theory、Probability Spaces and Random Elements、σ－fields and measures、Measurable functions and distributions、Integration and Differentiation、Integration、Radon.Nikodym derivative、Distributions and Their Characteristics、Distributions and probability densities、Moments and moment inequalities、Moment generating and characteristic functions、onditional Expectations、Conditional expectations、Independence、Conditional distributions、Markov chains and martingales、Asymptotic Theory、Convergence modes and stochastic orders等等。

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目录

Preface to the First Edition
Preface to the Second Edition
Chapter 1.Probability Theory
1.1 Probability Spaces and Random Elements
1.1.1σ－fields and measures
1.1.2 Measurable functions and distributions
1.2 Integration and Differentiation
1.2.1 Integration
1.2.2 Radon.Nikodym derivative
1.3 Distributions and Their Characteristics
1.3.1 Distributions and probability densities
1.3.2 Moments and moment inequalities
1.3.3 Moment generating and characteristic functions
1.4 Conditional Expectations
1.4.1 Conditional expectations
1.4.2 Independence
1.4.3 Conditional distributions
1.4.4 Markov chains and martingales
1.5 Asymptotic Theory
1.5.1 Convergence modes and stochastic orders
1.5.2 Weak convergence
1.5.3 Convergence of transformations
1.5.4 The law of large numbers
1.5.5 The central limit theorem
1.5.6 Edgeworth and Cornish-Fisher expansions
1.6 Exercises

Chapter 2. Fundamentals of Statistics
2.1 Populations，Samples，and Models
2.1.1 Populations and samples
2.1.2 Parametric and nonparametric models
2.1.3 Exponential and location.scale families
2.2 Statistics.Sufficiency，and Completeness
2.2.1 Statistics and their distributions
2.2.2 Sufficiency and minimal sufficiency
2.2.3 Complete statistics
2.3 Statistical Decision Theory
2.3.1 Decision rules，lOSS functions，and risks
2.3.2 Admissibility and optimality
2.4 Statistical Inference
2.4.1 P0il）t estimators
2.4.2 Hypothesis tests
2.4.3 Confidence sets
2.5 Asymptotic Criteria and Inference
2.5.1 Consistency
2.5.2 Asymptotic bias，variance，and mse
2.5.3 Asymptotic inference
2.6 Exercises

Chapter 3.Unbiased Estimation
3.1 The UMVUE
3.1.1 Sufficient and complete statistics
3.1.2 A necessary and.sufficient condition
3.1.3 Information inequality
3.1.4 Asymptotic properties of UMVUEs
3.2 U-Statistics
3.2.1 Some examples
3.2.2 Variances of U-statistics
3.2.3 The projection method
3.3 The LSE in Linear Models
3.3.1 The LSE and estimability
3.3.2 The UMVUE and BLUE
3.3.3 R0bustness of LSEs
3.3.4 Asymptotic properties of LSEs
3.4 Unbiased Estimators in Survey Problems
3.4.1 UMVUEs of population totals
3.4.2 Horvitz-Thompson estimators
3.5 Asymptotically Unbiased Estimators
3.5.1 Functions of unbiased estimators
3.5.2 The method ofmoments
3.5.3 V-statistics
3.5.4 The weighted LSE
3.6 Exercises

Chapter 4.Estimation in Parametric Models
4.1 Bayes Decisions and Estimators
4.1.1 Bayes actions
4.1.2 Empirical and hierarchical Bayes methods
4.1.3 Bayes rules and estimators
4.1.4 Markov chain Mollte Carlo
4.2 Invariance......
4.2.1 One-parameter location families
4.2.2 One-parameter seale families
4.2.3 General location-scale families
4.3 Minimaxity and Admissibility
4.3.1 Estimators with constant risks
4.3.2 Results in one-parameter exponential families
4.3.3 Simultaneous estimation and shrinkage estimators
4.4 The Method of Maximum Likelihood
4.4.1 The likelihood function and MLEs
4.4.2 MLEs in generalized linear models
4.4.3 Quasi-likelihoods and conditional likelihoods
4.5 Asymptotically Efficient Estimation
4.5.1 Asymptotic optimality
4.5.2 Asymptotic efficiency of MLEs and RLEs
4.5.3 Other asymptotically efficient estimators
4.6 Exercises

Chapter 5.Estimation in Nonparametric Models
5.1 Distribution Estimators
5.1.1 Empirical C.d.f.s in i.i.d.cases
5.1.2 Empirical likelihoods
5.1.3 Density estimation
5.1.4 Semi-parametric methods
5.2 Statistical Functionals
5.2.1 Differentiability and asymptotic normality
5.2.2 L-.M-.and R-estimators and rank statistics
5.3 Linear Functions of Order Statistics
5.3.1 Sample quantiles
5.3.2 R0bustness and efficiency
5.3.3 L-estimators in linear models
5.4 Generalized Estimating Equations
5.4.1 The GEE method and its relationship with others
5.4.2 Consistency of GEE estimators
5.4.3 Asymptotic normality of GEE estimators
5.5 Variance Estimation
5.5.1 The substitution.method
5.5.2 The jackknife
5.5.3 The bootstrap
5.6 Exercises

Chapter 6.Hypothesis Tests
6.1 UMP Tests
6.1.1 The Neyman-Pearson lemma
6.1.2 Monotone likelihood ratio
6.1.3 UMP tests for two-sided hypotheses
6.2 UMP Unbiased Tests
6.2.1 Unbiasedness，similarity，and Neyman structure
6.2.2 UMPU tests in exponential families
6.2.3 UMPU tests in normal families
……
Chapter 7 Confidence Sets
References
List of Notation
List of Abbreviations
Index of Definitions，Main Results，and Examples
Author Index
Subject Index

前言/序言

　　This book is intended for a course entitled Mathematical Statistics offered at the Department of Statistics，University of Wisconsin．Madison．This course，taught in a mathematically rigorous fashion，covers essential materials in statistical theory that a first or second year graduate student typicallY needs to learn as preparation for work on a Ph．D．degree in statistics．The course is designed for two 15-week semesters．with three lecture hours and two discussion hours in each week. Students in this course are assumed to have a good knowledge of advanced calgulus．A course in real analy．sis or measure theory prior to this course is often recommended．Chapter 1 provides a quick overview of important concepts and results in measure-theoretic probability theory that are used as tools in mathematical statistics．Chapter 2 introduces some fundamental concepts in statistics，including statistical models．the principle of SUfIlciency in data reduction，and two statistical approaches adopted throughout the book： statistical decision theory and statistical inference．
　　Each of Chapters 3 through 7 provides a detailed study of an important topic in statistical decision theory and inference：Chapter 3 introduces the theory of unbiased estimation；Chapter 4 studies theory and methods in point estimation ander parametric models；Chapter 5 covers point estimation in nonparametric settings；Chapter 6 focuses on hypothesis testing；and Chapter 7 discusses interval estimation and confidence sets．The classical frequentist approach is adopted in this book．although the Bayesian approach is also introduced (§2．3．2，§4．1，§6．4．4，and§7．1．3)．Asymptotic(1arge sample)theory，a crucial part of statistical inference，is studied throughout the book，rather than in a separate chapter．
　　About 85％of the book covers classical results in statistical theory that are typically found in textbooks of a similar level．These materials are in the Statistics Department’S Ph．D．qualifying examination syllabus．

数理统计（第2版）（英文版） [Mathematical Statistics(Second Edition)] 电子书 下载 mobi epub pdf txt

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译者序前言第1章 概率1 1.1 引言1 1.2 样本空间1 1.3 概率测度3 1.4 概率计算：计数方法5 1.4.1 乘法原理 6 1.4.2 排列与组合 7 1.5 条件概率12 1.6 独立性17 1.7 结束语19 1.8 习题20第2章 随机变量26 2.1 离散随机变量26 2.1.1 伯努利随机变量27 2.1.2 二项分布28 2.1.3 几何分布和负二项分布29 2.1.4 超几何分布 30 2.1.5 泊松分布31 2.2 连续随机变量34 2.2.1 指数密度36 2.2.2 伽马密度38 2.2.3 正态分布39 2.2.4 贝塔密度41 2.3 随机变量的函数42 2.4 结束语45 2.5 习题46第3章 联合分布51 3.1 引言51 3.2 离散随机变量52 3.3 连续随机变量53 3.4 独立随机变量60 3.5 条件分布61 3.5.1 离散情形61 3.5.2 连续情形62 3.6 联合分布随机变量函数67 3.6.1 和与商68 3.6.2 一般情形70 3.7 极值和顺序统计量73 3.8 习题75第4章 期望82 4.1 随机变量的期望82 4.1.1 随机变量函数的期望85 4.1.2 随机变量线性组合的期望 87 4.2 方差和标准差91 4.2.1 测量误差模型94 4.3 协方差和相关96 4.4 条件期望和预测102 4.4.1 定义和例子102 4.4.2 预测106 4.5 矩生成函数108 4.6 近似方法112 4.7 习题116第5章 极限定理123 5.1 引言123 5.2 大数定律123 5.3 依分布收敛和中心极限定理125 5.4 习题130第6章 正态分布的导出分布133 6.1 引言133 6.2 x2分布、t分布和F分布 133 6.3 样本均值和样本方差134 6.4 习题136第7章 抽样调查138 7.1 引言138 7.2 总体参数138 7.3 简单随机抽样140 7.3.1 样本均值的期望和方差140 7.3.2 总体方差的估计 145 7.3.3 X 抽样分布的正态近似 148 7.4 比率估计152 7.5 分层随机抽样157 7.5.1 引言和记号157 7.5.2 分层估计的性质 157 7.5.3 分配方法 160 7.6 结束语163 7.7 习题164第8章 参数估计和概率分布拟合176 8.1 引言176 8.2 粒子排放量的泊松分布拟合176 8.3 参数估计177 8.4 矩方法179 8.5 最大似然方法184 8.5.1 多项单元概率的最大似然估计187 8.5.2 最大似然估计的大样本理论189 8.5.3 最大似然估计的置信区间 193 8.6 参数估计的贝叶斯方法197 8.6.1 先验的进一步注释204 8.6.2 后验的大样本正态近似205 8.6.3 计算问题 206 8.7 效率和克拉默{拉奥下界207 8.7.1 例子：负二项分布210 8.8 充分性212 8.8.1 因子分解定理212 8.8.2 拉奥{布莱克韦尔定理215 8.9 结束语216 8.10 习题217第9章 假设检验和拟合优度评估228 9.1 引言228 9.2 奈曼{皮尔逊范式229 9.2.1 显著性水平的设定和p 值概念 232 9.2.2 原假设232 9.2.3 一致最优势检验 233 9.3 置信区间和假设检验的对偶性233 9.4 广义似然比检验235 9.5 多项分布的似然比检验236 9.6 泊松散布度检验240 9.7 悬挂根图242 9.8 概率图244 9.9 正态性检验248 9.10 结束语249 9.11 习题250第10章 数据汇总260 10.1 引言260 10.2 基于累积分布函数的方法 260 10.2.1 经验累积分布函数 260 10.2.2 生存函数262 10.2.3 分位数{分位数图266 10.3 直方图、密度曲线和茎叶图268 10.4 位置度量270 10.4.1 算术平均271 10.4.2 中位数 272 10.4.3 截尾均值274 10.4.4 M 估计274 10.4.5 位置估计的比较275 10.4.6 自助法评估位置度量的变异性 275 10.5 散度度量277 10.6 箱形图278 10.7 利用散点图探索关系279 10.8 结束语281 10.9 习题281第11章 两样本比较 289 11.1 引言289 11.2 两独立样本比较289 11.2.1 基于正态分布的方法289 11.2.2 势298 11.2.3 非参数方法：曼恩{惠特尼检验299 11.2.4 贝叶斯方法305 11.3 配对样本比较306 11.3.1 基于正态分布的方法307 11.3.2 非参数方法：符号秩检验308 11.3.3 例子：测量鱼的汞水平310 11.4 试验设计311 11.4.1 乳腺动脉结扎术311 11.4.2 安慰剂效应312 11.4.3 拉纳克郡牛奶试验 312 11.4.4 门腔分术313 11.4.5 FD&C Red No.40313 11.4.6 关于随机化的进一步评注314 11.4.7 研究生招生的观测研究、混杂和偏见315 11.4.8 审前调查315 11.5 结束语316 11.6 习题317第12章 方差分析328 12.1 引言328 12.2 单因子试验设计328 12.2.1 正态理论和 F 检验329 12.2.2 多重比较问题 333 12.2.3 非参数方法：克鲁斯卡尔{沃利斯检验335 12.3 二因子试验设计336 12.3.1 可加性参数化 337 12.3.2 二因子试验设计的正态理论339 12.3.3 随机化区组设计344 12.3.4 非参数方法：弗里德曼检验346 12.4 结束语347 12.5 习题348第13章 分类数据分析354 13.1 引言354 13.2 费舍尔精确检验354 13.3 卡方齐性检验355 13.4 卡方独立性检验358 13.5 配对设计360 13.6 优势比362 13.7 结束语36

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