Optimal Statistical Decisions

ISBN-10: 047168029X

ISBN-13: 9780471680291

Edition: 1970

List price: $154.00
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Book details

List price: $154.00
Copyright year: 1970
Publisher: John Wiley & Sons, Incorporated
Publication date: 4/30/2004
Binding: Paperback
Pages: 490
Size: 6.00" wide x 9.00" long x 1.00" tall
Weight: 1.540

Foreword
Preface
Survey of probability theory
Introduction
Experiments, Sample Spaces, and Probability
Experiments and Sample Spaces
Set Theory
Events and Probability
Conditional Probability
Binomial Coefficients
Exercises
Random Variables, Random Vectors, and Distribution Functions
Random Variables and Their Distributions
Multivariate Distributions
Sums and Integrals
Marginal Distributions and Independence
Vectors and Matrices
Expectations, Moments, and Characteristic Functions
Transformations of Random Variables
Conditional Distributions
Exercises
Some Special Univariate Distributions
Introduction
The Bernoulli Distribution
The Binomial Distribution
The Poisson Distribution
The Negative Binomial Distribution
The Hypergeometric Distribution
The Normal Distribution
The Gamma Distribution
The Beta Distribution
The Uniform Distribution
The Pareto Distribution
The t Distribution
The F Distribution
Exercises
Some Special Multivariate Distributions
Introduction
The Multinomial Distribution
The Dirichlet Distribution
The Multivariate Normal Distribution
The Wishart Distribution
The Multivariate t Distribution
The Bilateral Bivariate Pareto Distribution
Exercises
Subjective probability and utility
Subjective Probability
Introduction
Relative Likelihood
The Auxiliary Experiment
Construction of the Probability Distribution
Verification of the Properties of a Probability Distribution
Conditional Likelihoods
Exercises
Utility
Preferences among Rewards
Preferences among Probability Distributions
The Definition of a Utility Function
Some Properties of Utility Functions
The Utility of Monetary Rewards
Convex and Concave Utility Functions
The Axiomatic Development of Utility
Construction of the Utility Function
Verification of the Properties of a Utility Function
Extension of the Properties of a Utility Function to the Class P[subscript E]
Exercises
Statistical decision problems
Decision Problems
Elements of a Decision Problem
Bayes Risk and Bayes Decisions
Nonnegative Loss Functions
Concavity of the Bayes Risk
Randomization and Mixed Decisions
Convex Sets
Decision Problems in Which [similar]2 and D Are Finite
Decision Problems with Observations
Construction of Bayes Decision Functions
The Cost of Observation
Statistical Decision Problems in Which Both [Omega] and D Contain Two Points
Computation of the Posterior Distribution When the Observations Are Made in More Than One Stage
Exercises
Conjugate Prior Distributions
Sufficient Statistics
Conjugate Families of Distributions
Construction of the Conjugate Family
Conjugate Families for Samples from Various Standard Distributions
Conjugate Families for Samples from a Normal Distribution
Sampling from a Normal Distribution with Unknown Mean and Unknown Precision
Sampling from a Uniform Distribution
A Conjugate Family for Multinomial Observations
Conjugate Families for Samples from a Multivariate Normal Distribution
Multivariate Normal Distributions with Unknown Mean Vector and Unknown Precision Matrix
The Marginal Distribution of the Mean Vector
The Distribution of a Correlation
Precision Matrices Having an Unknown Factor
Exercises
Limiting Posterior Distributions
Improper Prior Distributions
Improper Prior Distributions for Samples from a Normal Distribution
Improper Prior Distributions for Samples from a Multivariate Normal Distribution
Precise Measurement
Convergence of Posterior Distributions
Supercontinuity
Solutions of the Likelihood Equation
Convergence of Supercontinuous Functions
Limiting Properties of the Likelihood Function
Normal Approximation to the Posterior Distribution
Approximations for Vector Parameters
Posterior Ratios
Exercises
Estimation, Testing Hypotheses, and Linear Statistical Models
Estimation
Quadratic Loss
Loss Proportional to the Absolute Value of the Error
Estimation of a Vector
Problems of Testing Hypotheses
Testing a Simple Hypothesis about the Mean of a Normal Distribution
Testing Hypotheses about the Mean of a Normal Distribution When the Precision Is Unknown
Deciding Whether a Parameter Is Smaller or Larger Than a Specified Value
Deciding Whether the Mean of a Normal Distribution Is Smaller or Larger Than a Specified Value
Linear Models
Testing Hypotheses in Linear Models
Investigating the Hypothesis That Certain Regression Coefficients Vanish
One-way Analysis of Variance
Exercises
Sequential decisions
Sequential Sampling
Gains from Sequential Sampling
Sequential Decision Procedures
The Risk of a Sequential Decision Procedure
Backward Induction
Optimal Bounded Sequential Decision Procedures
Illustrative Examples
Unbounded Sequential Decision Procedures
Regular Sequential Decision Procedures
Existence of an Optimal Procedure
Approximating an Optimal Procedure by Bounded Procedures
Regions for Continuing or Terminating Sampling
The Functional Equation
Approximations and Bounds for the Bayes Risk
The Sequential Probability-ratio Test
Characteristics of Sequential Probability-ratio Tests
Approximating the Expected Number of Observations
Exercises
Optimal Stopping
Introduction
The Statistician's Reward
Choice of the Utility Function
Sampling without Recall
Further Problems of Sampling with Recall and Sampling without Recall
Sampling without Recall from a Normal Distribution with Unknown Mean
Sampling with Recall from a Normal Distribution with Unknown Mean
Existence of Optimal Stopping Rules
Existence of Optimal Stopping Rules for Problems of Sampling with Recall and Sampling without Recall
Martingales
Stopping Rules for Martingales
Uniformly Integrable Sequences of Random Variables
Martingales Formed from Sums and Products of Random Variables
Regular Supermartingales
Supermartingales and General Problems of Optimal Stopping
Markov Processes
Stationary Stopping Rules for Markov Processes
Entrance-fee Problems
The Functional Equation for a Markov Process
Exercises
Sequential Choice of Experiments
Introduction
Markovian Decision Processes with a Finite Number of Stages
Markovian Decision Processes with an Infinite Number of Stages
Some Betting Problems
Two-armed-bandit Problems
Two-armed-bandit Problems When the Value of One Parameter Is Known
Two-armed-bandit Problems When the Parameters Are Dependent
Inventory Problems
Inventory Problems with an Infinite Number of Stages
Control Problems
Optimal Control When the Process Cannot Be Observed without Error
Multidimensional Control Problems
Control Problems with Actuation Errors
Search Problems
Search Problems with Equal Costs
Uncertainty Functions and Statistical Decision Problems
Sufficient Experiments
Examples of Sufficient Experiments
Exercises
References
Supplementary Bibliography
Name Index
Subject Index
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