An Introduction to Mathematical Statistics

Cover
Preface
Contents
0 Probability Preliminaries
	0.1 Between Zero and One
	0.2 Counting
	0.3 Sample Spaces, Events, and Some Simple Probabilities
	0.4 Some Very Brief Words About Independence and \"Disjointness\"
	0.5 A Brief Review of Random Variables, PDFs, and CDFs
	0.6 The Expected Value, Expectation, or Mean of a Random Variable or a Distribution
	0.7 The Variance of a Random Variable or a Distribution
	0.8 Indicator Notation and a Most Useful Appendix
	0.9 Joint PDFs, Marginals, and Independence
	0.10 Useful Properties of Expectation, Variance, and Covariance
1 MathStat Preliminaries: Four Important Tools for Mathematical Statistics
	1.1 Wait. Where are we going?
	1.2 \"Integrating Without Integrating\", A.K.A. The Best Trick You\'ll Learn Today
	1.3 Important Tool I: Finding Distributions of Transformations of Random Variables
	1.4 Important Tool II: Bivariate Transformations
	1.5 Important Tool II: Mins and Maxes
	1.6 Important Tool IV: Moment Generating Functions (MGFs)
2 Qualities of Estimators: Defining Good, Better, and Best
	2.1 Notation, Statistics, and Unbiasedness
	2.2 Mean Squared Error and Bias
	2.3 Convergence in Probability
	2.4 Convergence In Distribution
	2.5 The Central Limit Theorem
3 A \"Mostly Normal\" Introduction to Confidence Intervals
	3.1 A Confidence Interval for the Mean of a Normal Distribution: σ2 is Known
	3.2 An Approximate Confidence Interval for the Mean of Any Distribution
	3.3 A Confidence Interval for the Mean of a Normal Distribution: σ2 is Unknown and Sample Size is Small
	3.4 A Difference of Means
4 A \"Mostly Normal\" Introduction to Hypothesis Testing
	4.1 Errors in Hypothesis Testing
	4.2 Finding a Test: A Simple Null Hypothesis
	4.3 Finding a Test: A Composite Null Hypothesis
	4.4 Power
5 Estimation
	5.1 Method of Moments Estimators (MMEs)
	5.2 Maximum Likelihood Estimators (MLEs)
	5.3 The Cramer-Rao Lower Bound (CRLB)
	5.4 Asymptotic Properties of MLEs
	5.5 Uniformly Minimum Variance Unbiased Estimators (UMVUEs)
6 General Hypothesis Testing
	6.1 A Best Test
	6.2 Uniformly Most Powerful Tests (UMPs)
	6.3 Generalized Likelihood Ratio Tests (GLRTs)
A Tables of Distributions
B The Jacobian and a Change of Variables
	B.1 The Jacobian
	B.2 A Bivariate Transformation
	B.3 More Variables
C Standard Normal, t, X2 and F Tables
Symbols/Notation
Index