Statistical Inference

• 0.1 Foundations

  • Probability Theory
  • Distribution Families
    • Discrete
    • Continuous
  • Fundamental Results
    • Law of Large Numbers
    • Central Limit Theorem
  • Frequentist vs Bayesian Approaches
  • Fundamentals of Statistics

• 0.2 Parametric Point Estimation

  • Properties of Estimators
    • Unbiased
    • Consistency - Converge in probability to true parameter
    • Efficiency
      • Cramér-Rao Lower Bound
    • Sufficiency
    • UMVUE etc (see 207 notes)
  • Estimation Methods
    • Method of Moments
    • Maximum Likelihood Estimation
      • MLE Asymptotic Normality
    • Bayesian Estimators (Posterior Mean, MAP)
  • Key Results
    • Fisher Information
    • Cramér-Rao Inequality
• Interval Estimation
  • Confidence Intervals
    • Construction via Pivotal Quantities
    • Normal Approximation via (Central Limit Theorem)
    • Bootstrap Confidence Intervals
  • Coverage Probability and Interpretations
• Hypothesis Testing
  • Fundamentals
    • Test Structures
    • Type I and Type II Errors
    • Test Power
  • Classical Hypothesis Tests
    • Z-Test
    • T-test
    • Chi-Squared Test
    • F-Test
  • Likelihood Ratio Tests
  • Neyman-Pearson Lemma (most powerful tests for simple hypotheses)
  • Asymptotic Results:
    • Wilks’ theorem (LRT asymptotic chi-square)
    • Wald and score tests
• Decision Theory
  • Fundamentals of Decision Theory
  • Bayes Risk Minimization
  • Minimax Principle
• Bayesian Inference
  • Prior, likelihood, posterior, predictive distribution
  • Conjugate priors and analytical results
  • Bayesian credible intervals vs. frequentist confidence intervals
  • Bayesian decision rules
• Resampling & Nonparametric Methods
  • Bootstrap (for bias/variance estimation, CI construction)
  • Permutation tests
  • Rank-based tests (Wilcoxon, Kruskal–Wallis)
• Asymptotics & Large-Sample Theory
  • Law of large numbers (LLN)
  • Central limit theorem (CLT)
  • Slutsky’s theorem, delta method
  • Consistency and asymptotic normality of MLE
  • Empirical distribution function and Glivenko–Cantelli theorem
• Model Selection & Information Criteria
  • AIC, BIC, likelihood cross-validation
  • Bias–variance tradeoff
  • Penalized likelihood (ridge, lasso in regression context)
• Advanced / Modern Topics
  • Multiple testing corrections (Bonferroni, FDR, Benjamini–Hochberg)
  • Nonparametric density estimation (kernel methods)
  • High-dimensional inference (regularization, sparsity)
  • Bayesian computational methods (MCMC, variational inference)

1 Backlinks

• Bayesian Inference
• Option Greeks