Computational Statistics

  1. Monte Carlo Methods
    1. Core Principles of Monte Carlo Simulation
      1. The Law of Large Numbers
        1. Weak Law of Large Numbers
          1. Strong Law of Large Numbers
            1. Convergence of Sample Means
            2. The Central Limit Theorem
              1. Distribution of Sample Means
                1. Rate of Convergence
                  1. Applications to Monte Carlo
                  2. Estimating Error and Convergence
                    1. Monte Carlo Standard Error
                      1. Confidence Intervals for Estimates
                        1. Convergence Diagnostics
                          1. Sample Size Determination
                        2. Monte Carlo Integration
                          1. Basic Monte Carlo Estimator
                            1. Estimating Integrals via Sampling
                              1. Variance of the Estimator
                                1. Bias Analysis
                                2. Importance Sampling
                                  1. Motivation and Theory
                                    1. Choice of Importance Distribution
                                      1. Weight Calculation
                                        1. Variance Reduction in Importance Sampling
                                          1. Effective Sample Size
                                          2. Stratified Monte Carlo
                                            1. Stratification Strategies
                                              1. Variance Reduction Properties
                                              2. Quasi-Monte Carlo Methods
                                                1. Low-Discrepancy Sequences
                                                  1. Halton Sequences
                                                    1. Sobol Sequences
                                                  2. Variance Reduction Techniques
                                                    1. Antithetic Variates
                                                      1. Principle and Implementation
                                                        1. Correlation Structure
                                                          1. Efficiency Gains
                                                          2. Control Variates
                                                            1. Identifying Control Variates
                                                              1. Adjusting Estimates
                                                                1. Multiple Control Variates
                                                                2. Stratified Sampling
                                                                  1. Partitioning the Sample Space
                                                                    1. Allocation of Samples
                                                                      1. Proportional vs Optimal Allocation
                                                                      2. Common Random Numbers
                                                                        1. Synchronizing Randomness Across Simulations
                                                                          1. Applications in Comparison Studies
                                                                        2. Applications in Statistical Inference
                                                                          1. Simulating Null Distributions
                                                                            1. Permutation-Based Nulls
                                                                              1. Parametric Nulls
                                                                                1. Bootstrap-Based Nulls
                                                                                2. Calculating p-values
                                                                                  1. Empirical p-value Estimation
                                                                                    1. Accuracy and Precision
                                                                                    2. Power Analysis
                                                                                      1. Simulating Power Curves
                                                                                        1. Sample Size Determination
                                                                                          1. Effect Size Estimation
                                                                                          2. Confidence Interval Construction
                                                                                            1. Simulation-Based Intervals
                                                                                              1. Coverage Probability Assessment

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                                                                                          1. Foundations of Computational Statistics

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                                                                                          3. Resampling Methods