Optimal Statistical Decisions
Optimal Statistical Decisions
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DOI：
10.2307/2987329
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发表时间：
1970-06
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影响因子：
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通讯作者：
M. Degroot
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文献类型：
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作者：
M. Degroot
Foreword.Preface.PART。概率论概论。第1章。介绍。第二章。实验、样本空间与概率。2.1实验与样本空间。2.2集合论。2.3事件与概率。2.4条件概率。2.5二项系数。习题。第3章。随机变量、随机向量与分布函数。3.1随机变量及其分布。3.2多元分布。3.3和与积分。3.4边际分布与独立性。3.5向量与矩阵。3.6期望、矩与特征函数。3.7随机变量的变换。3.8条件分布。习题。第4章。一些特殊的单变量分布。4.1简介。4.2伯努利分布。4.3二项分布。4.4泊松分布。4.5负二项分布。4.6超几何分布。4.7正态分布。4.8伽玛分布。4.9 Beta分布。4.10均匀分布。4.11帕累托分布。4.12 t分布。4.13 F分布。习题。第五章。一些特殊的多元分布。5.1简介。5.2多项分布。5.3狄利克雷分布。5.4多元正态分布。5.5 Wishart分布。5.6多元t分布。5.7双边双变量帕累托分布。练习题。第二部分。主观概率和效用。第六章。主观概率。6.1介绍。6.2相对似然。6.3辅助实验。6.4概率分布的构造。6.5概率分布性质的验证。6.6条件似然。习题。第七章。效用。7.1奖励中的偏好。7.2概率分布中的偏好。7.3效用函数的定义。7.4效用函数的一些性质。7.5货币奖励的效用。7.6凸效用函数和凹效用函数。7.7效用的焦虑发展。7.8效用函数的构造。7.9效用函数性质的验证。7.10效用函数性质向类的推广。第三部分。统计决策问题。第八章。决策问题。8.1决策问题的要素。8.2贝叶斯风险与贝叶斯决策。8.3非负损失函数。8.4贝叶斯风险的凹凸性。8.5随机化与混合决策。8.6凸集。8.7 ~2和D为有限的决策问题。8.8有观测值的决策问题。8.9贝叶斯决策函数的构造。8.10观测值的代价。8.11两者都有的统计决策问题。8.12在多个阶段进行观测时后验分布的计算。练习题。共轭先验分布。9.1充分统计量。9.2分布的共轭族。9.3共轭族的构造。9.4各种标准分布样本的共轭族。9.5正态分布样本的共轭族。9.6均值和精度未知的正态分布抽样。9.7均匀分布抽样。9.8多项式观测值的共轭族。9.9多元样本的共轭族正态分布。9.10具有未知平均向量和未知精度矩阵的多元正态分布。9.11平均向量的边际分布。9.12相关分布。9.13具有未知因子的精度矩阵。练习题，第10章。限制后验分布。10.1不当先验分布。10.2正态分布样本的不当先验分布。10.3多元正态分布样本的不当先验分布。10.4精确测量。10.5后验分布的收敛性。10.6超连续性。10.7似然方程的解。10.8超连续函数的收敛性。10.9似然函数的极限性质。10.10后验的正态近似分布。10.11向量参数的近似。10.12后验比。练习题。第11章。估计，检验假设，和线性统计模型。11.1估计。11.2二次损失。11.3与误差绝对值成比例的损失。11.4向量的估计。11.5检验假设的问题。11.6检验一个关于正态分布均值的简单假设。11.7检验一个关于正态分布均值的假设。11.8确定一个参数是大于还是小于一个特定的值。11.9确定一个正态分布的均值是大于还是小于一个特定的值值。11.10线性模型。11.11检验线性模型中的假设。11.12调查某些回归系数消失的假设。11.13单因素方差分析。练习题。第四部分。序贯决策。第十二章。顺序抽样。12.1顺序抽样的收益。12.2顺序决策过程。12.3顺序决策过程的风险。12.4逆向归纳。12.5最优有界顺序决策过程。12.6举例说明。12.7无界顺序决策过程。12.8正则顺序决策过程。12.9最优过程的存在性。12.10用有界过程逼近最优过程。12.11继续或终止抽样的区域。12.12泛函方程。12.13贝叶斯风险的逼近与界。12.14序列概率比检验。12.15序列概率比检验的特征。12.16逼近期望观测数。习题。第13章。最优停止。13.1介绍。13.2统计学家的奖励。13.3效用函数的选择。13.4无召回抽样。13.5有召回抽样和无召回抽样的进一步问题。13.6均值未知正态分布的无召回抽样。13.7均值未知正态分布的有召回抽样。13.8最优停止规则的存在性。13.9有召回抽样和无召回抽样问题的最优停止规则的存在性13.10鞅。13.11鞅的停止规则。13.12随机变量的一致可积序列。13.13由随机变量的和与积构成的鞅。13.14正则上鞅。13.15上鞅和最优停止的一般问题。13.16马尔可夫过程。13.17马尔可夫过程的平稳停止规则。13.18入门费问题。13.19马尔可夫过程的泛函方程。习题。第14章。实验的顺序选择。14.1简介。14.2有限阶段的马尔可夫决策过程。14.3无限阶段的马尔可夫决策过程。14.4一些投注问题。14.5双臂盗匪问题。14.6一个参数值已知的双臂盗匪问题。14.7参数相关的双臂盗匪问题。14.8库存问题。14.9无限阶段的库存问题。14.10控制问题。14.11最优不能准确观察过程时的控制。14.12多维控制问题。14.13有驱动错误的控制问题。14.14搜索问题。14.15等代价搜索问题。14.16不确定性函数和统计决策问题。14.17充分实验。14.18充分实验的例子。习题。参考文献。补充参考书目。索引名称。主题索引。
Foreword.Preface.PART ONE. SURVEY OF PROBABILITY THEORY.Chapter 1. Introduction.Chapter 2. Experiments, Sample Spaces, and Probability.2.1 Experiments and Sample Spaces.2.2 Set Theory.2.3 Events and Probability.2.4 Conditional Probability.2.5 Binomial Coefficients.Exercises.Chapter 3. Random Variables, Random Vectors, and Distributions Functions.3.1 Random Variables and Their Distributions.3.2 Multivariate Distributions.3.3 Sums and Integrals.3.4 Marginal Distributions and Independence.3.5 Vectors and Matrices.3.6 Expectations, Moments, and Characteristic Functions.3.7 Transformations of Random Variables.3.8 Conditional Distributions.Exercises.Chapter 4. Some Special Univariate Distributions.4.1 Introduction.4.2 The Bernoulli Distributions.4.3 The Binomial Distribution.4.4 The Poisson Distribution.4.5 The Negative Binomial Distribution.4.6 The Hypergeometric Distribution.4.7 The Normal Distribution.4.8 The Gamma Distribution.4.9 The Beta Distribution.4.10 The Uniform Distribution.4.11 The Pareto Distribution.4.12 The t Distribution.4.13 The F Distribution.Exercises.Chapter 5. Some Special Multivariate Distributions.5.1 Introduction.5.2 The Multinomial Distribution.5.3 The Dirichlet Distribution.5.4 The Multivariate Normal Distribution.5.5 The Wishart Distribution.5.6 The Multivariate t Distribution.5.7 The Bilateral Bivariate Pareto Distribution.Exercises.PART TWO. SUBJECTIVE PROBABILITY AND UTILITY.Chapter 6. Subjective Probability.6.1 Introduction.6.2 Relative Likelihood.6.3 The Auxiliary Experiment.6.4 Construction of the Probability Distribution.6.5 Verification of the Properties of a Probability Distribution.6.6 Conditional Likelihoods.Exercises.Chapter 7. Utility.7.1 Preferences Among Rewards.7.2 Preferences Among Probability Distributions.7.3 The Definitions of a Utility Function.7.4 Some Properties of Utility Functions.7.5 The Utility of Monetary Rewards.7.6 Convex and Concave Utility Functions.7.7 The Anxiomatic Development of Utility.7.8 Construction of the Utility Function.7.9 Verification of the Properties of a Utility Function.7.10 Extension of the Properties of a Utility Function to the Class ?E.Exercises.PART THREE. STATISTICAL DECISION PROBLEMS.Chapter 8. Decision Problems.8.1 Elements of a Decision Problem.8.2 Bayes Risk and Bayes Decisions.8.3 Nonnegative Loss Functions.8.4 Concavity of the Bayes Risk.8.5 Randomization and Mixed Decisions.8.6 Convex Sets.8.7 Decision Problems in Which ~2 and D Are Finite.8.8 Decision Problems with Observations.8.9 Construction of Bayes Decision Functions.8.10 The Cost of Observation.8.11 Statistical Decision Problems in Which Both ? and D contains Two Points.8.12 Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage.Exercises.Chapter 9. Conjugate Prior Distributions.9.1 Sufficient Statistics.9.2 Conjugate Families of Distributions.9.3 Construction of the Conjugate Family.9.4 Conjugate Families for Samples from Various Standard Distributions.9.5 Conjugate Families for Samples from a Normal Distribution.9.6 Sampling from a Normal Distribution with Unknown Mean and Unknown Precision.9.7 Sampling from a Uniform Distribution.9.8 A Conjugate Family for Multinomial Observations.9.9 Conjugate Families for Samples from a Multivariate Normal Distribution.9.10 Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision matrix.9.11 The Marginal Distribution of the Mean Vector.9.12 The Distribution of a Correlation.9.13 Precision Matrices Having an Unknown Factor.Exercises.Chapter 10. Limiting Posterior Distributions.10.1 Improper Prior Distributions.10.2 Improper Prior Distributions for Samples from a Normal Distribution.10.3 Improper Prior Distributions for Samples from a Multivariate Normal Distribution.10.4 Precise Measurement.10.5 Convergence of Posterior Distributions.10.6 Supercontinuity.10.7 Solutions of the Likelihood Equation.10.8 Convergence of Supercontinuous Functions.10.9 Limiting Properties of the Likelihood Function.10.10 Normal Approximation to the Posterior Distribution.10.11 Approximation for Vector Parameters.10.12 Posterior Ratios.Exercises.Chapter 11. Estimation, Testing Hypotheses, and linear Statistical Models.11.1 Estimation.11.2 Quadratic Loss.11.3 Loss Proportional to the Absolute Value of the Error.11.4 Estimation of a Vector.11.5 Problems of Testing Hypotheses.11.6 Testing a Simple Hypothesis About the Mean of a Normal Distribution.11.7 Testing Hypotheses about the Mean of a Normal Distribution.11.8 Deciding Whether a Parameter Is Smaller or larger Than a Specific Value.11.9 Deciding Whether the Mean of a Normal Distribution Is Smaller or larger Than a Specific Value.11.10 Linear Models.11.11 Testing Hypotheses in Linear Models.11.12 Investigating the Hypothesis That Certain Regression Coefficients Vanish.11.13 One-Way Analysis of Variance.Exercises.PART FOUR. SEQUENTIAL DECISIONS.Chapter 12. Sequential Sampling.12.1 Gains from Sequential Sampling.12.2 Sequential Decision Procedures.12.3 The Risk of a Sequential Decision Procedure.12.4 Backward Induction.12.5 Optimal Bounded Sequential Decision procedures.12.6 Illustrative Examples.12.7 Unbounded Sequential Decision Procedures.12.8 Regular Sequential Decision Procedures.12.9 Existence of an Optimal Procedure.12.10 Approximating an Optimal Procedure by Bounded Procedures.12.11 Regions for Continuing or Terminating Sampling.12.12 The Functional Equation.12.13 Approximations and Bounds for the Bayes Risk.12.14 The Sequential Probability-ratio Test.12.15 Characteristics of Sequential Probability-ratio Tests.12.16 Approximating the Expected Number of Observations.Exercises.Chapter 13. Optimal Stopping.13.1 Introduction.13.2 The Statistician's Reward.13.3 Choice of the Utility Function.13.4 Sampling Without Recall.13.5 Further Problems of Sampling with Recall and Sampling without Recall.13.6 Sampling without Recall from a Normal Distribution with Unknown Mean.13.7 Sampling with Recall from a Normal Distribution with Unknown Mean.13.8 Existence of Optimal Stopping Rules.13.9 Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall.13.10 Martingales.13.11 Stopping Rules for Martingales.13.12 Uniformly Integrable Sequences of Random Variables.13.13 Martingales Formed from Sums and Products of Random Variables.13.14 Regular Supermartingales.13.15 Supermartingales and General Problems of Optimal Stopping.13.16 Markov Processes.13.17 Stationary Stopping Rules for Markov Processes.13.18 Entrance-fee Problems.13.19 The Functional Equation for a Markov Process.Exercises.Chapter 14. Sequential Choice of Experiments.14.1 Introduction.14.2 Markovian Decision Processes with a Finite Number of Stages.14.3 Markovian Decision Processes with an Infinite Number of Stages.14.4 Some Betting Problems.14.5 Two-armed-bandit Problems.14.6 Two-armed-bandit Problems When the Value of One Parameter Is Known.14.7 Two-armed-bandit Problems When the Parameters Are Dependent.14.8 Inventory Problems.14.9 Inventory Problems with an Infinite Number of Stages.14.10 Control Problems.14.11 Optimal Control When the Process Cannot Be Observed without Error.14.12 Multidimensional Control Problems.14.13 Control Problems with Actuation Errors.14.14 Search Problems.14.15 Search Problems with Equal Costs.14.16 Uncertainty Functions and Statistical Decision Problems.14.17 Sufficient Experiments.14.18 Examples of Sufficient Experiments.Exercises.References.Supplementary Bibliography.Name Index.Subject Index.