Probability and statistical inference

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Title:

Personal Author:

Mukhopadhyay, Nitis, 1950-

Series:

Statistics, textbooks and monographs ; 162

Statistics, textbooks and monographs ; v. 162

Publication Information:

London : CRC, c2000

Physical Description:

xviii, 665 p. : ill. ; 24 cm.

ISBN:

9780824703790

Subject Term:

Probabilities

Mathematical statistics

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010316651	QA273 M854 2000	Open Access Book	Book	Searching... Unknown

Priced very competitively compared with other textbooks at this level!
This gracefully organized textbook reveals the rigorous theory of probability and statistical inference in the style of a tutorial, using worked examples, exercises, numerous figures and tables, and computer simulations to develop and illustrate concepts.

Beginning with an introduction to the basic ideas and techniques in probability theory and progressing to more rigorous topics, Probability and Statistical Inference
studies the Helmert transformation for normal distributions and the waiting time between failures for exponential distributions
develops notions of convergence in probability and distribution
spotlights the central limit theorem (CLT) for the sample variance
introduces sampling distributions and the Cornish-Fisher expansions
concentrates on the fundamentals of sufficiency, information, completeness, and ancillarity
explains Basu's Theorem as well as location, scale, and location-scale families of distributions
covers moment estimators, maximum likelihood estimators (MLE), Rao-Blackwellization, and the Cramér-Rao inequality
discusses uniformly minimum variance unbiased estimators (UMVUE) and Lehmann-Scheffé Theorems
focuses on the Neyman-Pearson theory of most powerful (MP) and uniformly most powerful (UMP) tests of hypotheses, as well as confidence intervals
includes the likelihood ratio (LR) tests for the mean, variance, and correlation coefficient
summarizes Bayesian methods
describes the monotone likelihood ratio (MLR) property
handles variance stabilizing transformations
provides a historical context for statistics and statistical discoveries
showcases great statisticians through biographical notes

Employing over 1400 equations to reinforce its subject matter, Probability and Statistical Inference is a groundbreaking text for first-year graduate and upper-level undergraduate courses in probability and statistical inference who have completed a calculus prerequisite, as well as a supplemental text for classes in Advanced Statistical Inference or Decision Theory.

Preface	p. v
Acknowledgments	p. xi
1 Notions of Probability	p. 1
1.1 Introduction	p. 1
1.2 About Sets	p. 3
1.3 Axiomatic Development of Probability	p. 6
1.4 The Conditional Probability and Independent Events	p. 9
1.4.1 Calculus of Probability	p. 12
1.4.2 Bayes's Theorem	p. 14
1.4.3 Selected Counting Rules	p. 16
1.5 Discrete Random Variables	p. 18
1.5.1 Probability Mass and Distribution Functions	p. 19
1.6 Continuous Random Variables	p. 23
1.6.1 Probability Density and Distribution Functions	p. 23
1.6.2 The Median of a Distribution	p. 28
1.6.3 Selected Reviews from Mathematics	p. 28
1.7 Some Standard Probability Distributions	p. 32
1.7.1 Discrete Distributions	p. 33
1.7.2 Continuous Distributions	p. 37
1.8 Exercises and Complements	p. 50
2 Expectations of Functions of Random Variables	p. 65
2.1 Introduction	p. 65
2.2 Expectation and Variance	p. 65
2.2.1 The Bernoulli Distribution	p. 71
2.2.2 The Binomial Distribution	p. 72
2.2.3 The Poisson Distribution	p. 73
2.2.4 The Uniform Distribution	p. 73
2.2.5 The Normal Distribution	p. 73
2.2.6 The Laplace Distribution	p. 76
2.2.7 The Gamma Distribution	p. 76
2.3 The Moments and Moment Generating Function	p. 77
2.3.1 The Binomial Distribution	p. 80
2.3.2 The Poisson Distribution	p. 81
2.3.3 The Normal Distribution	p. 82
2.3.4 The Gamma Distribution	p. 84
2.4 Determination of a Distribution via MGF	p. 86
2.5 The Probability Generating Function	p. 88
2.6 Exercises and Complements	p. 89
3 Multivariate Random Variables	p. 99
3.1 Introduction	p. 99
3.2 Discrete Distributions	p. 100
3.2.1 The Joint, Marginal and Conditional Distributions	p. 101
3.2.2 The Multinomial Distribution	p. 103
3.3 Continuous Distributions	p. 107
3.3.1 The Joint, Marginal and Conditional Distributions	p. 107
3.3.2 Three and Higher Dimensions	p. 115
3.4 Covariances and Correlation Coefficients	p. 119
3.4.1 The Multinomial Case	p. 124
3.5 Independence of Random Variables	p. 125
3.6 The Bivariate Normal Distribution	p. 131
3.7 Correlation Coefficient and Independence	p. 139
3.8 The Exponential Family of Distributions	p. 141
3.8.1 One-parameter Situation	p. 141
3.8.2 Multi-parameter Situation	p. 144
3.9 Some Standard Probability Inequalities	p. 145
3.9.1 Markov and Bernstein-Chernoff Inequalities	p. 145
3.9.2 Tchebysheff's Inequality	p. 148
3.9.3 Cauchy-Schwarz and Covariance Inequalities	p. 149
3.9.4 Jensen's and Lyapunov's Inequalities	p. 152
3.9.5 Holder's Inequality	p. 156
3.9.6 Bonferroni Inequality	p. 157
3.9.7 Central Absolute Moment Inequality	p. 158
3.10 Exercises and Complements	p. 159
4 Functions of Random Variables and Sampling Distribution	p. 177
4.1 Introduction	p. 177
4.2 Using Distribution Functions	p. 179
4.2.1 Discrete Cases	p. 179
4.2.2 Continuous Cases	p. 181
4.2.3 The Order Statistics	p. 182
4.2.4 The Convolution	p. 185
4.2.5 The Sampling Distribution	p. 187
4.3 Using the Moment Generating Function	p. 190
4.4 A General Approach with Transformations	p. 192
4.4.1 Several Variable Situations	p. 195
4.5 Special Sampling Distributions	p. 206
4.5.1 The Student's t Distribution	p. 207
4.5.2 The F Distribution	p. 209
4.5.3 The Beta Distribution	p. 211
4.6 Special Continuous Multivariate Distributions	p. 212
4.6.1 The Normal Distribution	p. 212
4.6.2 The t Distribution	p. 218
4.6.3 The F Distribution	p. 219
4.7 Importance of Independence in Sampling Distributions	p. 220
4.7.1 Reproductivity of Normal Distributions	p. 220
4.7.2 Reproductivity of Chi-square Distributions	p. 221
4.7.3 The Student's t Distribution	p. 223
4.7.4 The F Distribution	p. 223
4.8 Selected Review in Matrices and Vectors	p. 224
4.9 Exercises and Complements	p. 227
5 Concepts of Stochastic Convergence	p. 241
5.1 Introduction	p. 241
5.2 Convergence in Probability	p. 242
5.3 Convergence in Distribution	p. 253
5.3.1 Combination of the Modes of Convergence	p. 256
5.3.2 The Central Limit Theorems	p. 257
5.4 Convergence of Chi-square, t, and F Distributions	p. 264
5.4.1 The Chi-square Distribution	p. 264
5.4.2 The Student's t Distribution	p. 264
5.4.3 The F Distribution	p. 265
5.4.4 Convergence of the PDF and Percentage Points	p. 265
5.5 Exercises and Complements	p. 270
6 Sufficiency, Completeness, and Ancillarity	p. 281
6.1 Introduction	p. 281
6.2 Sufficiency	p. 282
6.2.1 The Conditional Distribution Approach	p. 284
6.2.2 The Neyman Factorization Theorem	p. 288
6.3 Minimal Sufficiency	p. 294
6.3.1 The Lehmann-Scheffe Approach	p. 295
6.4 Information	p. 300
6.4.1 One-parameter Situation	p. 301
6.4.2 Multi-parameter Situation	p. 304
6.5 Ancillarity	p. 309
6.5.1 The Location, Scale, and Location-Scale Families	p. 314
6.5.2 Its Role in the Recovery of Information	p. 316
6.6 Completeness	p. 318
6.6.1 Complete Sufficient Statistics	p. 320
6.6.2 Basu's Theorem	p. 324
6.7 Exercises and Complements	p. 327
7 Point Estimation	p. 341
7.1 Introduction	p. 341
7.2 Finding Estimators	p. 342
7.2.1 The Method of Moments	p. 342
7.2.2 The Method of Maximum Likelihood	p. 344
7.3 Criteria to Compare Estimators	p. 351
7.3.1 Unbiasedness, Variance and Mean Squared Error	p. 351
7.3.2 Best Unbiased and Linear Unbiased Estimators	p. 354
7.4 Improved Unbiased Estimator via Sufficiency	p. 358
7.4.1 The Rao-Blackwell Theorem	p. 358
7.5 Uniformly Minimum Variance Unbiased Estimator	p. 365
7.5.1 The Cramer-Rao Inequality and UMVUE	p. 366
7.5.2 The Lehmann-Scheffe Theorems and UMVUE	p. 371
7.5.3 A Generalization of the Cramer-Rao Inequality	p. 374
7.5.4 Evaluation of Conditional Expectations	p. 375
7.6 Unbiased Estimation Under Incompleteness	p. 377
7.6.1 Does the Rao-Blackwell Theorem Lead to UMVUE?	p. 377
7.7 Consistent Estimators	p. 380
7.8 Exercises and Complements	p. 382
8 Tests of Hypotheses	p. 395
8.1 Introduction	p. 395
8.2 Error Probabilities and the Power Function	p. 396
8.2.1 The Concept of a Best Test	p. 399
8.3 Simple Null Versus Simple Alternative Hypotheses	p. 401
8.3.1 Most Powerful Test via the Neyman-Pearson Lemma	p. 401
8.3.2 Applications: No Parameters Are Involved	p. 413
8.3.3 Applications: Observations Are Non-IID	p. 416
8.4 One-Sided Composite Alternative Hypothesis	p. 417
8.4.1 UMP Test via the Neyman-Pearson Lemma	p. 417
8.4.2 Monotone Likelihood Ratio Property	p. 420
8.4.3 UMP Test via MLR Property	p. 422
8.5 Simple Null Versus Two-Sided Alternative Hypotheses	p. 425
8.5.1 An Example Where UMP Test Does Not Exist	p. 425
8.5.2 An Example Where UMP Test Exists	p. 426
8.5.3 Unbiased and UMP Unbiased Tests	p. 428
8.6 Exercises and Complements	p. 429
9 Confidence Interval Estimation	p. 441
9.1 Introduction	p. 441
9.2 One-Sample Problems	p. 443
9.2.1 Inversion of a Test Procedure	p. 444
9.2.2 The Pivotal Approach	p. 446
9.2.3 The Interpretation of a Confidence Coefficient	p. 451
9.2.4 Ideas of Accuracy Measures	p. 452
9.2.5 Using Confidence Intervals in the Tests of Hypothesis	p. 455
9.3 Two-Sample Problems	p. 456
9.3.1 Comparing the Location Parameters	p. 456
9.3.2 Comparing the Scale Parameters	p. 460
9.4 Multiple Comparisons	p. 463
9.4.1 Estimating a Multivariate Normal Mean Vector	p. 463
9.4.2 Comparing the Means	p. 465
9.4.3 Comparing the Variances	p. 467
9.5 Exercises and Complements	p. 469
10 Bayesian Methods	p. 477
10.1 Introduction	p. 477
10.2 Prior and Posterior Distributions	p. 479
10.3 The Conjugate Priors	p. 481
10.4 Point Estimation	p. 485
10.5 Credible Intervals	p. 488
10.5.1 Highest Posterior Density	p. 489
10.5.2 Contrasting with the Confidence Intervals	p. 492
10.6 Tests of Hypotheses	p. 493
10.7 Examples with Non-Conjugate Priors	p. 494
10.8 Exercises and Complements	p. 497
11 Likelihood Ratio and Other Tests	p. 507
11.1 Introduction	p. 507
11.2 One-Sample Problems	p. 508
11.2.1 LR Test for the Mean	p. 509
11.2.2 LR Test for the Variance	p. 512
11.3 Two-Sample Problems	p. 515
11.3.1 Comparing the Means	p. 515
11.3.2 Comparing the Variances	p. 519
11.4 Bivariate Normal Observations	p. 522
11.4.1 Comparing the Means: The Paired Difference t Method	p. 522
11.4.2 LR Test for the Correlation Coefficient	p. 525
11.4.3 Tests for the Variances	p. 528
11.5 Exercises and Complements	p. 529
12 Large-Sample Inference	p. 539
12.1 Introduction	p. 539
12.2 The Maximum Likelihood Estimation	p. 539
12.3 Confidence Intervals and Tests of Hypothesis	p. 542
12.3.1 The Distribution-Free Population Mean	p. 543
12.3.2 The Binomial Proportion	p. 548
12.3.3 The Poisson Mean	p. 553
12.4 The Variance Stabilizing Transformations	p. 555
12.4.1 The Binomial Proportion	p. 556
12.4.2 The Poisson Mean	p. 559
12.4.3 The Correlation Coefficient	p. 560
12.5 Exercises and Complements	p. 563
13 Sample Size Determination: Two-Stage Procedures	p. 569
13.1 Introduction	p. 569
13.2 The Fixed-Width Confidence Interval	p. 573
13.2.1 Stein's Sampling Methodology	p. 573
13.2.2 Some Interesting Properties	p. 574
13.3 The Bounded Risk Point Estimation	p. 579
13.3.1 The Sampling Methodology	p. 581
13.3.2 Some Interesting Properties	p. 582
13.4 Exercises and Complements	p. 584
14 Appendix	p. 591
14.1 Abbreviations and Notation	p. 591
14.2 A Celebration of Statistics: Selected Biographical Notes	p. 593
14.3 Selected Statistical Tables	p. 621
14.3.1 The Standard Normal Distribution Function	p. 621
14.3.2 Percentage Points of the Chi-Square Distribution	p. 626
14.3.3 Percentage Points of the Student's t Distribution	p. 628
14.3.4 Percentage Points of the F Distribution	p. 630
References	p. 633
Index	p. 649

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