Modern Concepts and Theorems of Mathematical Statistics

1 Fundamentals of Mathematical Statistics.- 1 Basic Definitions, Concepts, Results, and Theorems.- §1.1. Probability Concepts.- §1.2. Random Samples.- §1.3. Moments.- §1.4. Some Inequalities Involving Probabilities and Moments.- §1.5. Characteristic Functions.- §1.6. Moment Generating Functions.- §1.7. Determination of a Distribution from Its Moments.- §1.8. Probability Integral Transform.- §1.9. Unbiased and Asymptotically Unbiased Estimators.- §1.10. Uniformly Minimum Variance Unbiased Estimators.- §1.11. Consistency of an Estimator.- §1.12. M-Estimators.- §1.13. L-Estimators and the ?-Trimmed Mean.- §1.14. R-Estimators.- §1.15. Hodges—Lehmann Estimator.- §1.16. U-Statistics.- §1.17. Cramér—Rao—Frêchet Lower Bound.- §1.18. Sufficient Statistics.- §1.19. Fisher—Neyman Factorization Theorem for Sufficient Statistics.- §1.20. Rao—Blackwell Theorem.- §1.21. Completeness of Statistics and Their Families of Distributions.- §1.22. Theorem on Completeness of Statistics with Sampling from theExponential Family.- §1.23. Lehmann—Scheffé Uniqueness Theorem27.- §1.24. Efficiency, Relative Efficiency, and Asymptotic Efficiency of Estimators.- §1.25. Estimation by the Method of Moments.- §1.26. Confidence Intervals.- §1.27. Tolerance Intervals.- §1.28. Simple and Composite Hypotheses, Type-I and Type-II Errors, Level of Significance or Size, Power of a Test and Consistency.- §1.29. Randomized and Nonrandomized Test Functions.- §1.30. Uniformly Most Powerful (UMP), Most Powerful (MP), Unbiased and Uniformly Most Powerful Unbiased (UMPU) Tests.- §1.31. Neyman—Pearson Fundamental Lemma.- §1.32. Monotone Likelihood Ratio Property of Family of Distributions and Related Theorems for UMP and UMPU Tests for Composite Hypotheses.- §1.33. Locally Most Powerful Tests.- §1.34. Locally Most Powerful Unbiased Tests.- §1.35. Likelihood Ratio Test.- §1.36. Theorems on Unbiasedness of Tests.- §1.37. Relative Efficiency of Tests.- §1.38. Sequential Probability Ratio Test (SPRT).- §1.39. Bayes and Decision-Theoretic Approach.- §1.40. The Linear Hypothesis.- §1.41. The Bootstrap and the Jackknife.- §1.42. Robustness.- §1.43. Pitman—Fisher Randomization Methods.- §1.44. Nonparametric Methods.- 2 Fundamental Limit Theorems.- §2.1. Modes of Convergence of Random Variables.- §2.2. Slutsky’s Theorem.- §2.3. Dominated Convergence Theorem.- §2.4. Limits and Differentiation Under Expected Values with Respect to a Parameter t.- §2.5. Helly—Bray Theorem.- §2.6. Levy—Cramér Theorem.- §2.7. Functions of a Sequence of Random Variables.- §2.8. Weak Laws of Large Numbers.- §2.9. Strong Laws of Large Numbers.- §2.10. Berry—Esséen Inequality.- §2.11. de Moivre—Laplace Theorem.- §2.12. Lindeberg—Lévy Theorem.- §2.13. Liapounov Theorem.- §2.14. Kendall—Rao Theorem.- §2.15. Limit Theorems for Moments and Functions of Moments.- §2.16. Edgeworth Expansions.- §2.17. Quantiles.- §2.18. Probability Integral Transform with Unknown Location and/or Scale Parameters.- §2.19. ?-Trimmed Mean.- §2.20. Borel’s Theorem.- §2.21. Glivenko—Cantelli Theorem.- §2.22. Kolmogorov—Smirnov Limit Theorems.- §2.23. Chi-Square Test of Fit.- §2.24. Maximum Likelihood Estimators.- §2.25. M-Estimators.- §2.26. Likelihood Ratio Statistic.- §2.27. On Some Consistency Problems of Tests.- §2.28. Pitman Asymptotic Efficiency.- §2.29. Hodges—Lehmann Estimators.- §2.30. Hoeffding’s Theorems for U-Statistics.- §2.31. Wald—Wolfowitz Theorem.- §2.32. Chernoff-Savage’s for R-Statistics.- §2.33. Miller’s for Jackknife Statistics.- 2 Statistical Distributions.- 3 Distributions.- §3.1. Binomial.- §3.2. Multinomial.- §3.3. Geometric.- §3.4. Pascal Negative Binomial.- §3.5. Hypergeometric.- §3.6 Poisson.- §3.7. Wilcoxon’s Null (One-Sample).- §3.8. Wilcoxon—(Mann—Whitney)’s Null (Two-Sample).- §3.9. Runs.- §3.10. Pitman—Fisher Randomization (One-Sample).- §3.11. Pitman’s Permutation Test of the Correlation Coefficient.- §3.12. Pitman’s Randomization (Two-Sample).- §3.13. Pitman’s Randomization (k-Sample).- §3.14. Kolmogorov—Smirnov’s Null (One-Sample).- §3.15. Kolmogorov—Smirnov’s Null (Two-Sample).- §3.16. Uniform (Rectangular).- §3.17. Triangular.- §3.18. Pareto.- §3.19. Exponential.- §3.20. Erlang and Gamma.- §3.21. Weibull and Rayleigh.- §3.22. Beta.- §3.23. Half-Normal.- §3.24. Normal (Gauss).- §3.25. Cauchy.- §3.26. Lognormal.- §3.27. Logistic.- §3.28. Double-Exponential.- §3.29. Hyperbolic-Secant.- §3.30. Slash.- §3.31. Tukey’s Lambda.- §3.32. Exponential Family.- §3.33. Exponential Power.- §3.34. Pearson Types.- §3.35. Chi-Square ?2.- §3.36. Student’s T.- §3.37. Fisher’s F.- §3.38. Noncentral Chi-Square.- §3.39. Noncentral Student.- §3.40. Noncentral Fisher’s F.- §3.41. Order Statistics.- §3.42. Sample Range.- §3.43. Median of a Sample.- §3.44. Extremes of a Sample.- §3.45. Studenized Range.- §3.46. Probability Integral Transform.- §3.47. $$\bar X,\bar X - \bar Y$$.- §3.48. S12,S12/S22 and Bartlett’s M.- §3.49. Bivariate Normal.- §3.50. Sample Correlation Coefficient.- §3.51. Multivariate Normal.- §3.52. Wishart.- §3.53. Hotelling’s T2.- §3.54. Dirichlet.- 4 Some Relations Between Distributions.- §4.1. Binomial and Binomial.- §4.2. Binomial and Multinomial.- §4.3. Binomial and Beta.- §4.4. Binomial and Fisher’s F.- §4.5. Binomial and Hypergeometric.- §4.6. Binomial and Poisson.- §4.7. Binomial and Normal.- §4.8. Geometric and Pascal.- §4.9. Beta and Beta.- §4.10. Beta and Fisher’s F.- §4.11. Beta and Chi-Square.- §4.12. Beta and Uniform.- §4.13. Poisson and Poisson.- §4.14. Poisson and Chi-Square.- §4.15. Poisson and Exponential.- §4.16. Poisson and Normal.- §4.17. Exponential and Exponential.- §4.18. Exponential and Erlang.- §4.19. Exponential and Weibull.- §4.20. Exponential and Uniform.- §4.21. Cauchy and Normal.- §4.22. Cauchy and Cauchy.- §4.23. Normal and Lognormal.- §4.24. Normal and Normal.- §4.25. Normal and Chi-Square.- §4.26. Normal and Multivariate Normal.- §4.27. Normal and Other Distributions.- §4.28. Exponential Family and Other Distributions.- §4.29. Exponential Power and Other Distributions.- §4.30. Pearson Types and Other Distributions.- §4.31. Chi-Square and Chi-Square.- §4.32. Chi-Square and Gamma.- §4.33. Chi-Square and Fisher’s F.- §4.34. Student, Normal, and Chi-Square.- §4.35. Student and Cauchy.- §4.36. Student and Hyperbolic-Secant.- §4.37. Student and Fisher’s F.- §4.38. Student and Normal.- §4.39. Student and Beta.- §4.40. Student and Sample Correlation Coefficient.- §4.41. Fisher’s F and Logistic.- §4.42. Fisher’s F and Fisher’s Z-Transform.- §4.43. Noncentral Chi-Square and Normal.- §4.44. Noncentral Chi-Square and Noncentral Chi-Square.- §4.45. Noncentral Student, Normal, and Chi-Square.- §4.46. Noncentral Fisher’s F, Noncentral Chi-Square, and Chi-Square.- §4.47. Multivariate Normal and Multivariate Normal.- §4.48. Multivariate Normal and Chi-Square.- §4.49. Multivariate Normal and Noncentral Chi-Square.- §4.50. Multivariate Normal and Fisher’s F.- §4.51. Multivariate Normal and Noncentral Fisher’s F.- §4.52. Dirichlet and Dirichlet.- §4.53. Dirichlet and Beta.- Author Index.