Statistical Theory A Concise Introduction

ISBN-10: 1439851840
ISBN-13: 9781439851845
Edition: 2013
List price: $74.95 Buy it from $52.19
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Description: This text presents a clear introduction to statistical theory for advanced undergraduate students taking a standard course in statistics. It details the main elements and basic concepts of statistical theory, including parameter estimation,  More...

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Book details

List price: $74.95
Copyright year: 2013
Publisher: CRC Press LLC
Publication date: 4/25/2013
Binding: Hardcover
Pages: 240
Size: 6.25" wide x 9.50" long x 0.75" tall
Weight: 1.452
Language: English

This text presents a clear introduction to statistical theory for advanced undergraduate students taking a standard course in statistics. It details the main elements and basic concepts of statistical theory, including parameter estimation, confidence intervals, hypothesis testing, Bayesian interference, and decision theory. The book takes an examples-based approach with clear exposition of key topics and just the right amount of mathematical formality. It also includes numerous exercises to enhance the students’ understanding of the topics discussed.

List of Figures
List of Tables
Preface
Introduction
Preamble
Likelihood
Sufficiency
*Minimal sufficiency
* Completeness
Exponential family of distributions
Exercises
Point Estimation
Introduction
Maximum likelihood estimation
Method of moments
Method of least squares
Goodness-of-estimation. Mean squared error
Unbiased estimation
Definition and main properties
Uniformly minimum variance unbiased estimators. The Cramer-Rao lower bound
*The Cramer-Rao lower bound for multivariate parameters
Rao-Blackwell theorem
*Lehmann-Scheff� theorem
Exercises
Confidence Intervals, Bounds, and Regions
Introduction
Quoting the estimation error
Confidence intervals
Confidence bounds
*Confidence regions
Exercises
Hypothesis Testing
Introduction
Simple hypotheses
Type I and Type II errors
Choice of a critical value
The p-value
Maximal power tests. Neyman-Pearson lemma
Composite hypotheses
Power function
Uniformly most powerful tests
Generalized likelihood ratio tests
Hypothesis testing and confidence intervals
Sequential testing
Exercises
Asymptotic Analysis
Introduction
Convergence and consistency in MSE
Convergence and consistency in probability
Convergence in distribution
The central limit theorem
Asymptotically normal consistency
Asymptotic confidence intervals
Asymptotically normal consistency of the MLE, Wald's confidence intervals, and tests
*Multiparameter case
Asymptotic distribution of the GLRT, Wilks' theorem
Exercises
Bayesian Inference
Introduction
Choice of priors
Conjugate priors
Noninformative (objective) priors
Point estimation
Interval estimation. Credible sets
Hypothesis testing
Simple hypotheses
Composite hypotheses
Testing a point null hypothesis
Exercises
*Elements of Statistical Decision Theory
Introduction and notations
Risk function and admissibility
Minimax risk and minimax rules
Bayes risk and Bayes rules
Posterior expected loss and Bayes actions
Admissibility and minimaxity of Bayes rules
Exercises
*Linear Models
Introduction
Definition and examples
Estimation of regression coefficients
Residuals. Estimation of the variance
Examples
Estimation of a normal mean
Comparison between the means of two independent normal samples with a common variance
Simple linear regression
Goodness-of-fit. Multiple correlation coefficient
Confidence intervals and regions for the coefficients
Hypothesis testing in linear models
Testing significance of a single predictor
Testing significance of a group of predictors
Testing a general linear hypothesis
Predictions
Analysis of variance
One-way ANOVA
Two-way ANOVA and beyond
Probabilistic Review
Introduction
Basic probabilistic laws
Random variables
Expected value and the variance
Chebyshev's and Markov's inequalities
Expectation of functions and the Jensen's inequality
Joint distribution
Covariance, correlation, and the Cauchy-Schwarz inequality
Expectation and variance of a sum of random variables
Conditional distribution and Bayes Theorem
Distributions of functions of random variables
Random vectors
Special families of distributions
Bernoulli and binomial distributions
Geometric and negative binomial distributions
Hypergeometric distribution
Poisson distribution
Uniform distribution
Exponential distribution
Weibull distribution
Gamma-distribution
Beta-distribution
Cauchy distribution
Normal distribution
Log-normal distribution
�<sup>2</sup> distribution
t-distribution
F-distribution
Multinormal distribution
Definition and main properties
Projections of normal vectors
Solutions of Selected Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Index

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