Probability Theory

1. Statistical Inference
   1. Point Estimation
      1. Properties of Estimators
         1. Unbiasedness
         2. Consistency
         3. Efficiency
         4. Sufficiency
      2. Methods of Point Estimation
         1. Method of Moments
         2. Maximum Likelihood Estimation (MLE)
            1. Properties of MLE
            2. Log-likelihood Functions
         3. Bayes Estimators
            1. Posterior Mean
            2. Posterior Mode
      3. Applications of Point Estimation
         1. Parameter Estimation
         2. Population Mean and Proportion Estimation
   2. Interval Estimation
      1. Confidence Intervals
         1. Interpretation and Usage
         2. Calculation Methods
            1. Normal Approximation Intervals
            2. t-Distribution for Small Samples
            3. Bootstrap Methods
         3. Confidence Level and Interval Width
      2. Credibility Intervals in Bayesian Inference
         1. Credible Sets
         2. Highest Posterior Density Interval
      3. Prediction Intervals for Future Observations
      4. Robustness of Interval Estimates
   3. Hypothesis Testing
      1. Formulation of Hypotheses
         1. Null Hypothesis (H0)
         2. Alternative Hypothesis (H1 or H2)
      2. Test Statistics
         1. Z-tests
         2. t-tests
         3. Chi-square Tests
         4. F-tests
      3. Significance Level (α)
         1. Choice and Implications
         2. One-tailed vs Two-tailed Tests
      4. p-values
         1. Calculation and Meaning
         2. Comparison with Significance Level
      5. Errors in Testing
         1. Type I Error (False Positive)
            1. Probability and Consequences
         2. Type II Error (False Negative)
            1. Probability and Consequences
         3. Power of a Test
            1. Definition
            2. Factors Affecting Power
               1. Sample Size
               2. Effect Size
               3. Variability
      6. Likelihood Ratio Tests
         1. Non-nested Model Comparisons
         2. Composite Hypotheses
      7. Non-parametric Tests
         1. Wilcoxon Rank-Sum Test
         2. Kruskal-Wallis Test
         3. Mann-Whitney U Test
   4. Advanced Concepts in Statistical Inference
      1. Asymptotic Properties
         1. Asymptotic Normality
         2. Asymptotic Efficiency
      2. Bootstrap Techniques
         1. Resampling Methods
         2. Applications in Bias Reduction
      3. Decision Theoretic Approaches
         1. Loss Functions
         2. Risk and Decision Rules
      4. Robust Inference
         1. Handling Outliers
         2. Robust Estimation Techniques