Computational Statistics

  1. Resampling Methods
    1. The Bootstrap
      1. The Bootstrap Principle
        1. Resampling with Replacement
          1. Bootstrap Distribution
            1. Plug-in Principle
            2. Non-parametric Bootstrap
              1. Algorithm and Implementation
                1. Assumptions and Limitations
                2. Parametric Bootstrap
                  1. Model-Based Resampling
                    1. Parameter Estimation
                    2. Estimating Standard Errors and Bias
                      1. Bootstrap Standard Error
                        1. Bias Estimation
                          1. Bias Correction
                          2. Constructing Confidence Intervals
                            1. Percentile Method
                              1. Basic Bootstrap (Pivotal) Method
                                1. Bias-Corrected and Accelerated (BCa) Method
                                  1. Bootstrap-t Method
                                    1. Coverage Properties
                                    2. Applications in Regression Models
                                      1. Bootstrapping Regression Coefficients
                                        1. Residual Bootstrap
                                          1. Wild Bootstrap
                                            1. Model Selection with Bootstrap
                                            2. Bootstrap for Time Series
                                              1. Block Bootstrap
                                                1. Stationary Bootstrap
                                                  1. Model-Based Bootstrap
                                                2. The Jackknife
                                                  1. Jackknife Estimator of Bias
                                                    1. Delete-One Jackknife
                                                      1. Delete-d Jackknife
                                                      2. Jackknife Estimator of Variance
                                                        1. Variance Formula
                                                          1. Comparison with Other Methods
                                                          2. Leave-One-Out Resampling
                                                            1. Cross-Validation Connection
                                                              1. Computational Efficiency
                                                              2. Comparison with Bootstrap
                                                                1. Advantages and Disadvantages
                                                                  1. When to Use Each Method
                                                                2. Permutation Tests
                                                                  1. Rationale and Procedure
                                                                    1. Exchangeability Assumption
                                                                      1. Generating Permutation Distributions
                                                                        1. Calculating Test Statistics
                                                                        2. Comparison with Parametric Tests
                                                                          1. Assumptions and Robustness
                                                                            1. Power Considerations
                                                                            2. Applications in Hypothesis Testing
                                                                              1. Two-Sample Tests
                                                                                1. Correlation Tests
                                                                                  1. Regression Tests
                                                                                    1. ANOVA Applications
                                                                                  2. Cross-Validation
                                                                                    1. K-Fold Cross-Validation
                                                                                      1. Partitioning Data
                                                                                        1. Averaging Performance Metrics
                                                                                          1. Choice of K
                                                                                          2. Leave-One-Out Cross-Validation (LOOCV)
                                                                                            1. Computational Considerations
                                                                                              1. Bias-Variance Properties
                                                                                              2. Stratified Cross-Validation
                                                                                                1. Maintaining Class Proportions
                                                                                                  1. Applications in Classification
                                                                                                  2. Time Series Cross-Validation
                                                                                                    1. Forward Chaining
                                                                                                      1. Rolling Window Validation
                                                                                                      2. Model Selection and Assessment
                                                                                                        1. Choosing Tuning Parameters
                                                                                                          1. Preventing Overfitting
                                                                                                            1. Nested Cross-Validation

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                                                                                                        2. Monte Carlo Methods

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                                                                                                        4. Numerical Optimization in Statistics