Probability and statistics for economists / Bruce E. Hansen.

Includes bibliographical references and index.

1. Basic Probability Theory -- 1.1. Introduction -- 1.2. Outcomes and Events -- 1.3. Probability Function -- 1.4. Properties of the Probability Function -- 1.5. Equally Likely Outcomes -- 1.6. Joint Events -- 1.7. Conditional Probability -- 1.8. Independence -- 1.9. Law of Total Probability -- 1.10. Bayes Rule -- 1.11. Permutations and Combinations -- 1.12. Sampling with and without Replacement -- 1.13. Poker Hands -- 1.14. Sigma Fields* -- 1.15. Technical Proofs* -- 1.16. Exercises -- -- 2. Random Variables -- 2.1. Introduction -- 2.2. Random Variables -- 2.3. Discrete Random Variables -- 2.4. Transformations -- 2.5. Expectation -- 2.6. Finiteness of Expectations -- 2.7. Distribution Function -- 2.8. Continuous Random Variables -- 2.9. Quantiles -- 2.10. Density Functions -- 2.11. Transformations of Continuous Random Variables -- 2.12. Non-Monotonic Transformations -- 2.13. Expectation of Continuous Random Variables -- 2.14. Finiteness of Expectations -- 2.15. Unifying Notation -- 2.16. Mean and Variance -- 2.17. Moments -- 2.18. Jensen's Inequality -- 2.19. Applications of Jensen's Inequality* -- 2.20. Symmetric Distributions -- 2.21. Truncated Distributions -- 2.22. Censored Distributions -- 2.23. Moment Generating Function -- 2.24. Cumulants -- 2.25. Characteristic Function -- 2.26. Expectation: Mathematical Details* -- 2.27. Exercises -- -- 3. Parametric Distributions -- 3.1. Introduction -- 3.2. Bernoulli Distribution -- 3.3. Rademacher Distribution -- 3.4. Binomial Distribution -- 3.5. Multinomial Distribution -- 3.6. Poisson Distribution -- 3.7. Negative Binomial Distribution -- 3.8. Uniform Distribution -- 3.9. Exponential Distribution -- 3.10. Double Exponential Distribution -- 3.11. Generalized Exponential Distribution -- 3.12. Normal Distribution -- 3.13. Cauchy Distribution -- 3.14. Student t Distribution -- 3.15. Logistic Distribution -- 3.16. Chi-Square Distribution -- 3.17. Gamma Distribution -- 3.18. F Distribution -- 3.19. Non-Central Chi-Square -- 3.20. Beta Distribution -- 3.21. Pareto Distribution -- 3.22. Lognormal Distribution -- 3.23. Weibull Distribution -- 3.24. Extreme Value Distribution -- 3.25. Mixtures of Normals -- 3.26. Technical Proofs* -- 3.27. Exercises -- -- 4. Multivariate Distributions -- 4.1. Introduction -- 4.2. Bivariate Random Variables -- 4.3. Bivariate Distribution Functions -- 4.4. Probability Mass Function -- 4.5. Probability Density Function -- 4.6. Marginal Distribution -- 4.7. Bivariate Expectation -- 4.8. Conditional Distribution for Discrete X -- 4.9. Conditional Distribution for Continuous X -- 4.10. Visualizing Conditional Densities -- 4.11. Independence -- 4.12. Covariance and Correlation -- 4.13. Cauchy-Schwarz Inequality -- 4.14. Conditional Expectation -- 4.15. Law of Iterated Expectations -- 4.16. Conditional Variance -- 4.17. H ölder's and Minkowski's Inequalities* -- 4.18. Vector Notation -- 4.19. Triangle Inequalities* -- 4.20. Multivariate Random Vectors -- 4.21. Pairs of Multivariate Vectors -- 4.22. Multivariate Transformations -- 4.23. Convolutions -- 4.24. Hierarchical Distributions -- 4.25. Existence and Uniqueness of the Conditional Expectation* -- 4.26. Identification -- 4.27. Exercises -- -- 5. Normal and Related Distributions -- 5.1. Introduction -- 5.2. Univariate Normal -- 5.3. Moments of the Normal Distribution -- 5.4. Normal Cumulants -- 5.5. Normal Quantiles -- 5.6. Truncated and Censored Normal Distributions -- 5.7. Multivariate Normal -- 5.8. Properties of the Multivariate Normal -- 5.9. Chi-Square, t,F , and Cauchy Distributions -- 5.10. Hermite Polynomials* -- 5.11. Technical Proofs* -- 5.12. Exercises -- -- 6. Sampling -- 6.1. Introduction -- 6.2. Samples -- 6.3. Empirical Illustration -- 6.4. Statistics, Parameters, and Estimators -- 6.5. Sample Mean -- 6.6. Expected Value of Transformations -- 6.7. Functions of Parameters -- 6.8. Sampling Distribution -- 6.9. Estimation Bias -- 6.10. Estimation Variance -- 6.11. Mean Squared Error -- 6.12. Best Unbiased Estimator -- 6.13. Estimation of Variance -- 6.14. Standard Error -- 6.15. Multivariate Means -- 6.16. Order Statistics∗ -- 6.17. Higher Moments of Sample Mean* -- 6.18. Normal Sampling Model -- 6.19. Normal Residuals -- 6.20. Normal Variance Estimation -- 6.21. Studentized Ratio -- 6.22. Multivariate Normal Sampling -- 6.23. Exercises -- -- 7. Law of Large Numbers -- 7.1. Introduction -- 7.2. Asymptotic Limits -- 7.3. Convergence in Probability -- 7.4. Chebyshev's Inequality -- 7.5. Weak Law of Large Numbers -- 7.6. Counterexamples -- 7.7. Examples -- 7.8. Illustrating Chebyshev's Inequality -- 7.9. Vector-Valued Moments -- 7.10. Continuous Mapping Theorem -- 7.11. Examples -- 7.12. Uniformity Over Distributions* -- 7.13. Almost Sure Convergence and the Strong Law* -- 7.14. Technical Proofs* -- 7.15. Exercises -- -- 8. Central Limit Theory -- 8.1. Introduction -- 8.2. Convergence in Distribution -- 8.3. Sample Mean -- 8.4. A Moment Investigation -- 8.5. Convergence of the Moment Generating Function -- 8.6. Central Limit Theorem -- 8.7. Applying the Central Limit Theorem -- 8.8. Multivariate Central Limit Theorem -- 8.9. Delta Method -- 8.10. Examples -- 8.11. Asymptotic Distribution for Plug-In Estimator -- 8.12. Covariance Matrix Estimation -- 8.13. t -Ratios -- 8.14. Stochastic Order Symbols -- 8.15. Technical Proofs* -- 8.16. Exercises -- --

9. Advanced Asymptotic Theory* -- 9.1. Introduction -- 9.2. Heterogeneous Central Limit Theory -- 9.3. Multivariate Heterogeneous Central Limit Theory -- 9.4. Uniform Central Limit Theory -- 9.5. Uniform Integrability -- 9.6. Uniform Stochastic Bounds -- 9.7. Convergence of Moments -- 9.8. Edgeworth Expansion for the Sample Mean -- 9.9. Edgeworth Expansion for Smooth Function Model -- 9.10. Cornish-Fisher Expansions -- 9.11. Technical Proofs* -- -- 10. Maximum Likelihood Estimation -- 10.1. Introduction -- 10.2. Parametric Model -- 10.3. Likelihood -- 10.4. Likelihood Analog Principle -- 10.5. Invariance Property -- 10.6. Examples -- 10.7. Score, Hessian, and Information -- 10.8. Examples -- 10.9. Cram ér-Rao Lower Bound -- 10.10. Examples -- 10.11. Cram ér-Rao Bound for Functions of Parameters -- 10.12. Consistent Estimation -- 10.13. Asymptotic Normality -- 10.14. Asymptotic Cram ér-Rao Efficiency -- 10.15. Variance Estimation -- 10.16. Kullback-Leibler Divergence -- 10.17. Approximating Models -- 10.18. Distribution of the MLE under Misspecification -- 10.19. Variance Estimation under Misspecification -- 10.20. Technical Proofs* -- 10.21. Exercises -- -- 11. Method of Moments -- 11.1. Introduction -- 11.2. Multivariate Means -- 11.3. Moments -- 11.4. Smooth Functions -- 11.5. Central Moments -- 11.6. Best Unbiased Estimation -- 11.7. Parametric Models -- 11.8. Examples of Parametric Models -- 11.9. Moment Equations -- 11.10. Asymptotic Distribution for Moment Equations -- 11.11. Example: Euler Equation -- 11.12. Empirical Distribution Function -- 11.13. Sample Quantiles -- 11.14. Robust Variance Estimation -- 11.15. Technical Proofs* -- 11.16. Exercises -- -- 12. Numerical Optimization -- 12.1. Introduction -- 12.2. Numerical Function Evaluation and Differentiation -- 12.3. Root Finding -- 12.4. Minimization in One Dimension -- 12.5. Failures of Minimization -- 12.6. Minimization in Multiple Dimensions -- 12.7. Constrained Optimization -- 12.8. Nested Minimization -- 12.9. Tips and Tricks -- 12.10. Exercises -- -- 13. Hypothesis Testing -- 13.1. Introduction -- 13.2. Hypotheses -- 13.3. Acceptance and Rejection -- 13.4. Type I and Type II Errors -- 13.5. One-Sided Tests -- 13.6. Two-Sided Tests -- 13.7. What Does "Accept ℍ0" Mean about ℍ0? -- 13.8. t Test with Normal Sampling -- 13.9. Asymptotic t Test -- 13.10. Likelihood Ratio Test for Simple Hypotheses -- 13.11. Neyman-Pearson Lemma -- 13.12. Likelihood Ratio Test against Composite Alternatives -- 13.13. Likelihood Ratio and t Tests -- 13.14. Statistical Significance -- 13.15. p-Value -- 13.16. Composite Null Hypothesis -- 13.17. Asymptotic Uniformity -- 13.18. Summary -- 13.19. Exercises -- -- 14. Confidence Intervals -- 14.1. Introduction -- 14.2. Definitions -- 14.3. Simple Confidence Intervals -- 14.4. Confidence Intervals for the Sample Mean under Normal Sampling -- 14.5. Confidence Intervals for the Sample Mean under Non-Normal Sampling -- 14.6. Confidence Intervals for Estimated Parameters -- 14.7. Confidence Interval for the Variance -- 14.8. Confidence Intervals by Test Inversion -- 14.9. Use of Confidence Intervals -- 14.10. Uniform Confidence Intervals -- 14.11. Exercises -- -- 15. Shrinkage Estimation -- 15.1. Introduction -- 15.2. Mean Squared Error -- 15.3. Shrinkage -- 15.4. James-Stein Shrinkage Estimator -- 15.5. Numerical Calculation -- 15.6. Interpretation of the Stein Effect -- 15.7. Positive-Part Estimator -- 15.8. Summary -- 15.9. Technical Proofs* -- 15.10. Exercises -- -- 16. Bayesian Methods -- 16.1. Introduction -- 16.2. Bayesian Probability Model -- 16.3. Posterior Density -- 16.4. Bayesian Estimation -- 16.5. Parametric Priors -- 16.6. Normal-Gamma Distribution -- 16.7. Conjugate Prior -- 16.8. Bernoulli Sampling -- 16.9. Normal Sampling -- 16.10. Credible Sets -- 16.11. Bayesian Hypothesis Testing -- 16.12. Sampling Properties in the Normal Model -- 16.13. Asymptotic Distribution -- 16.14. Technical Proofs* -- 16.15. Exercises -- -- 17. Nonparametric Density Estimation -- 17.1. Introduction -- 17.2. Histogram Density Estimation -- 17.3. Kernel Density Estimator -- 17.4. Bias of Density Estimator -- 17.5. Variance of Density Estimator -- 17.6. Variance Estimation and Standard Errors -- 17.7. Integrated Mean Squared Error of Density Estimator -- 17.8. Optimal Kernel -- 17.9. Reference Bandwidth -- 17.10. Sheather-Jones Bandwidth* -- 17.11. Recommendations for Bandwidth Selection -- 17.12. Practical Issues in Density Estimation -- 17.13. Computation -- 17.14. Asymptotic Distribution -- 17.15. Undersmoothing -- 17.16. Technical Proofs* -- 17.17. Exercises -- -- 18. Empirical Process Theory -- 18.1. Introduction -- 18.2. Framework -- 18.3. Glivenko-Cantelli Theorem -- 18.4. Packing, Covering, and Bracketing Numbers -- 18.5. Uniform Law of Large Numbers -- 18.6. Functional Central Limit Theory -- 18.7. Conditions for Asymptotic Equicontinuity -- 18.8. Donsker's Theorem -- 18.9. Technical Proofs* -- 18.10. Exercises -- -- Appendix 1. Limits -- Appendix 2. Series -- Appendix 3. Factorials -- Appendix 4. Exponentials -- Appendix 5. Logarithms -- Appendix 6. Differentiation -- Appendix 7. Mean Value Theorem -- Appendix 8. Integration -- Appendix 9. Gaussian Integral -- Appendix 10. Gamma Function -- Appendix 11. Matrix Algebra.

"A comprehensive introduction to probability and statistics through the perspective of applying it to economic analysis"-- Provided by publisher.