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MATH235

Statistics

Matt Nunes

Lent Term 2017-18 Lancaster University

Contents:

1 Introduction
- 1.1 Outline Syllabus
- 1.2 Learning Outcomes
- 1.3 References
2 Statistical estimation and uncertainty
- 2.1 Statistical models and parameter estimation
  - 2.1.1 Estimates and estimators
  - 2.1.2 Estimation uncertainty
- 2.2 Properties of estimators
- 2.3 Summary
3 Hypothesis testing
- 3.1 Introduction
- 3.2 One sample tests
  - 3.2.1 Justification of the one-sample $t$ -test
- 3.3 Two sample tests
  - 3.3.1 Unpaired data
  - 3.3.2 Paired data
- 3.4 Summary
4 Confidence intervals
- 4.1 Bootstrap confidence intervals
- 4.2 Parametric confidence intervals
- 4.3 Summary
5 Analysis of Variance
- 5.1 Multiple $t$ -tests
- 5.2 One-way ANOVA
  - 5.2.1 ANOVA in R
- 5.3 Summary
6 Linear regression - motivation
- 6.1 Introduction
- 6.2 Simple linear regression
- 6.3 Simple linear regression
  - 6.4 Multiple linear regression
    - 6.4.1 Examples
      - 6.5 Summary
        
        7 Linear regression - fitting
        
        7.1 Estimation of regression coefficients $\beta$
        
        7.1.1 Examples
        
        7.2 Predicted values
        
        7.2.1 Estimation of residual variance $\sigma^{2}$
        
        7.3 Summary
        
        8 Sampling distribution of estimators
        
        8.1 Regression coefficients
        
        8.1.1 Expectation of least squares estimator
        
        8.1.2 Variance of least squares estimator
        
        8.2 Linear combinations of regression coefficients
        
        8.3 Residual error
        
        8.4 Hypothesis tests for the regression coefficients
6.4 Multiple linear regression
- 6.4.1 Examples
  - 6.5 Summary
    - 7 Linear regression - fitting
    - 7.1 Estimation of regression coefficients $\beta$
      - 7.1.1 Examples
        
        7.2 Predicted values
        
        7.2.1 Estimation of residual variance $\sigma^{2}$
        
        7.3 Summary
        
        8 Sampling distribution of estimators
        
        8.1 Regression coefficients
        
        8.1.1 Expectation of least squares estimator
        
        8.1.2 Variance of least squares estimator
        
        8.2 Linear combinations of regression coefficients
        
        8.3 Residual error
        
        8.4 Hypothesis tests for the regression coefficients
6.4.1 Examples
- 6.5 Summary
  - 7 Linear regression - fitting
  - 7.1 Estimation of regression coefficients $\beta$
    - 7.1.1 Examples
      - 7.2 Predicted values
        
        7.2.1 Estimation of residual variance $\sigma^{2}$
        
        7.3 Summary
        
        8 Sampling distribution of estimators
        
        8.1 Regression coefficients
        
        8.1.1 Expectation of least squares estimator
        
        8.1.2 Variance of least squares estimator
        
        8.2 Linear combinations of regression coefficients
        
        8.3 Residual error
        
        8.4 Hypothesis tests for the regression coefficients
7 Linear regression - fitting
7.1 Estimation of regression coefficients $\beta$
- 7.1.1 Examples
  - 7.2 Predicted values
    - 7.2.1 Estimation of residual variance $\sigma^{2}$
      - 7.3 Summary
        
        8 Sampling distribution of estimators
        
        8.1 Regression coefficients
        
        8.1.1 Expectation of least squares estimator
        
        8.1.2 Variance of least squares estimator
        
        8.2 Linear combinations of regression coefficients
        
        8.3 Residual error
        
        8.4 Hypothesis tests for the regression coefficients
8 Sampling distribution of estimators
8.1 Regression coefficients
- 8.1.1 Expectation of least squares estimator
- 8.1.2 Variance of least squares estimator
  - 8.2 Linear combinations of regression coefficients
    - 8.3 Residual error
      - 8.4 Hypothesis tests for the regression coefficients
10 Covariate selection
10.1 The F test
- 10.1.1 Where does the F-test come from?
10.2 Link to one-way ANOVA
10.3 Summary
- 11 Diagnostics
11 Diagnostics
11.1 Normality of residuals
11.2 Residuals vs. Fitted values
11.3 Residuals vs. Explanatory variables
11.4 Outliers
11.5 Influence
11.6 Diagnostics in R
11.7 Summary
- 12 Introduction to Likelihood Inference
12.2 What is likelihood?
12.3 Likelihood Examples: continuous parameters
12.4 Likelihood Examples: discrete parameters
12.5 Summary
- 13 Information and Sufficiency