Page 20 - Probability and Statistical Inference

Contents xv

4.5.1 The Student’s t Distribution 207
4.5.2 The F Distribution 209
4.5.3 The Beta Distribution 211
4.6 Special Continuous Multivariate Distributions 212
4.6.1 The Normal Distribution 212
4.6.2 The t Distribution 218
4.6.3 The F Distribution 219
4.7 Importance of Independence in Sampling Distributions 220
4.7.1 Reproductivity of Normal Distributions 220
4.7.2 Reproductivity of Chi-square Distributions 221
4.7.3 The Student’s t Distribution 223
4.7.4 The F Distribution 223
4.8 Selected Review in Matrices and Vectors 224
4.9 Exercises and Complements 227
5 Concepts of Stochastic Convergence 241
5.1 Introduction 241
5.2 Convergence in Probability 242
5.3 Convergence in Distribution 253
5.3.1 Combination of the Modes of Convergence 256
5.3.2 The Central Limit Theorems 257
5.4 Convergence of Chi-square, t, and F Distributions 264
5.4.1 The Chi-square Distribution 264
5.4.2 The Student’s t Distribution 264
5.4.3 The F Distribution 265
5.4.4 Convergence of the PDF and Percentage Points 265
5.5 Exercises and Complements 270
6 Sufficiency, Completeness, and Ancillarity 281
6.1 Introduction 281
6.2 Sufficiency 282
6.2.1 The Conditional Distribution Approach 284
6.2.2 The Neyman Factorization Theorem 288
6.3 Minimal Sufficiency 294
6.3.1 The Lehmann-Scheffé Approach 295
6.4 Information 300
6.4.1 One-parameter Situation 301
6.4.2 Multi-parameter Situation 304
6.5 Ancillarity 309
6.5.1 The Location, Scale, and Location-Scale Families 314
6.5.2 Its Role in the Recovery of Information 316
6.6 Completeness 318
6.6.1 Complete Sufficient Statistics 320
6.6.2 Basu’s Theorem 324
6.7 Exercises and Complements 327