A Concise Introduction to Statistical Inference

1.Statistical Inference

2.Theory and Calculus of Probability
2.1.Probability models
2.2.Random variables and their distributions
2.3.Moments of random variables
2.3.1.Expectation
2.3.2.Higher order moments
2.3.3.Variance
2.4.Multivariate distributions
2.4.1.Association between two random variables
2.5.Normal distribution
2.5.1.Bivariate normal distribution
2.6.Limit theorems for the sample mean
2.7.Chapter summary
2.8.Exercises and problems

3.From Probability to Statistics
3.1.A first stab at statistical inference
3.2.Sampling distributions
3.3.Chapter summary
Contents note continued: 3.4.Exercises and problems

4.Statistical Inference for the Mean based on a Large Sample
4.1.Simple statistical model for the mean
4.2.Confidence intervals
4.3.Hypothesis tests
4.3.1.The p-value
4.3.2.Errors and power
4.4.Chapter summary
4.5.Exercises and problems

5.Statistical Models and Sampling Distributions
5.1.Statistical models
5.2.Some examples of statistical models
5.3.Statistics
5.4.Sampling distributions
5.5.Chapter summary
5.6.Exercises and problems

6.Estimation of Parameters
6.1.Introduction
6.2.Maximum likelihood estimators
6.3.Comparing estimators
6.3.1.Unbiased estimators
6.3.2.Mean squared error
6.4.Method of moments
6.5.A useful asymptotic result
6.6.Chapter summary
6.7.Exercises and problems

7.Confidence Intervals
7.1.Introduction
7.2.Basic idea
7.3.Confidence intervals for means
7.3.1.Mean of a normal population with variance known
Contents note continued: 7.3.2.Mean of a normal population with variance unknown
7.3.3.Mean of an unknown distribution based on a large sample
7.4.Confidence interval for any parameter based on a large sample
7.5.Differences between populations based on large samples
7.5.1.Difference between population means
7.5.2.Difference between proportions
7.6.Confidence intervals for the variance of a normal sample
7.7.Chapter summary
7.8.Exercises and problems

8.Hypothesis Testing
8.1.Introduction
8.2.Hypotheses, decisions, and errors
8.3.What test statistic should we use?
8.4.Examples of commonly used tests
8.4.1.Test of a normal mean with known variance
8.4.2.Test of a normal mean with unknown variance
8.4.3.Test of a mean based on a large sample
8.4.4.Test of a proportion based on a large sample
8.4.5.Test of the difference between two means based on a large sample
Contents note continued: 8.4.6.Test of the difference between two proportions in a large sample
8.5.p-value
8.6.Statistical significance
8.7.Chapter summary
8.8.Exercises, problems, and discussion

9.Linear Regression
9.1.Basic ideas
9.2.Estimation: Least squares method
9.3.Decomposition of errors
9.4.Inferences based on the OLS estimator
9.5.Linear regression, causation, and correlation
9.6.Chapter summary
9.7.Exercises, problems, and discussion

10.Bayesian Inference
10.1.Introduction
10.2.Basic concepts
10.3.A simple statistical model from a Bayesian perspective
10.4.Highest density regions
10.5.Hypothesis testing
10.6.Evidence and the likelihood principle
10.7.Decision theory
10.8.An economic decision problem with statistical information
10.9.Linear regression
10.10.Chapter summary
10.11.Exercises and problems
Appendices
A.Commonly used discrete distributions
Contents note continued: B.Commonly used continuous distributions
C.Special distributions
C.1.Χ2 distribution
C.2.Student's t-distribution
C.3.F-distribution
D.The Greek alphabet
E.Mathematical notation
F.Summation and product notation
G.Exponentials and (natural) logarithms
H.Subjective probability.