Probability and Statistical Inference

by Mukhopadhyay; Nitis

• ISBN13: 9780824703790
• ISBN10: 0824703790
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-03-22
• Publisher: CRC Press

Summary

This gracefully organized text presents 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 the basic ideas and techniques of probability theory and progressing to more rigorous topics, this treatment covers all of the topics typically addressed in a two-semester course in probability and statistical inference for graduate and upper-level undergraduate courses, including hypothesis testing, Bayesian analysis, and sample-size determination. The author reinforces important ideas and special techniques with drills and boxed summaries.

Table of Contents

Preface	v
Acknowledgments	xi
Notions of Probability	1	(64)
Introduction	1	(2)
About Sets	3	(3)
Axiomatic Development of Probability	6	(3)
The Conditional Probability and Independent Events	9	(9)
Calculus of Probability	12	(2)
Bayes's Theorem	14	(2)
Selected Counting Rules	16	(2)
Discrete Random Variables	18	(5)
Probability Mass and Distribution Functions	19	(4)
Continuous Random Variables	23	(9)
Probability Density and Distribution Functions	23	(5)
The Median of a Distribution	28	(1)
Selected Reviews from Mathematics	28	(4)
Some Standard Probability Distributions	32	(18)
Discrete Distributions	33	(4)
Continuous Distributions	37	(13)
Exercises and Complements	50	(15)
Expectations of Functions of Random Variables	65	(34)
Introduction	65	(1)
Expectation and Variance	65	(12)
The Bernoulli Distribution	71	(1)
The Binomial Distribution	72	(1)
The Poisson Distribution	73	(1)
The Uniform Distribution	73	(1)
The Normal Distribution	73	(3)
The Laplace Distribution	76	(1)
The Gamma Distribution	76	(1)
The Moments and Moment Generating Function	77	(9)
The Binomial Distribution	80	(1)
The Poisson Distribution	81	(1)
The Normal Distribution	82	(2)
The Gamma Distribution	84	(2)
Determination of a Distribution via MGF	86	(2)
The Probability Generating Function	88	(1)
Exercises and Complements	89	(10)
Multivariate Random Variables	99	(78)
Introduction	99	(1)
Discrete Distributions	100	(7)
The Joint, Marginal and Conditional Distributions	101	(2)
The Multinomial Distribution	103	(4)
Continuous Distributions	107	(12)
The Joint, Marginal and Conditional Distributions	107	(8)
Three and Higher Dimensions	115	(4)
Covariances and Correlation Coefficients	119	(6)
The Multinomial Case	124	(1)
Independence of Random Variables	125	(6)
The Bivariate Normal Distribution	131	(8)
Correlation Coefficient and Independence	139	(2)
The Exponential Family of Distributions	141	(4)
One-parameter Situation	141	(3)
Multi-parameter Situation	144	(1)
Some Standard Probability Inequalities	145	(14)
Markov and Bernstein-Chernoff Inequalities	145	(3)
Tchebysheff's Inequality	148	(1)
Cauchy-Schwarz and Covariance Inequalities	149	(3)
Jensen's and Lyapunov's Inequalities	152	(4)
Holder's Inequality	156	(1)
Bonferroni Inequality	157	(1)
Central Absolute Moment Inequality	158	(1)
Exercises and Complements	159	(18)
Functions of Random Variables and Sampling Distribution	177	(64)
Introduction	177	(2)
Using Distribution Functions	179	(11)
Discrete Cases	179	(2)
Continuous Cases	181	(1)
The Order Statistics	182	(3)
The Convolution	185	(2)
The Sampling Distribution	187	(3)
Using the Moment Generating Function	190	(2)
A General Approach with Transformations	192	(14)
Several Variable Situations	195	(11)
Special Sampling Distributions	206	(6)
The Student's t Distribution	207	(2)
The F Distribution	209	(2)
The Beta Distribution	211	(1)
Special Continuous Multivariate Distributions	212	(8)
The Normal Distribution	212	(6)
The t Distribution	218	(1)
The F Distribution	219	(1)
Importance of Independence in Sampling Distributions	220	(4)
Reproductivity of Normal Distributions	220	(1)
Reproductivity of Chi-square Distributions	221	(2)
The Student's t Distribution	223	(1)
The F Distribution	223	(1)
Selected Review in Matrices and Vectors	224	(3)
Exercises and Complements	227	(14)
Concepts of Stochastic Convergence	241	(40)
Introduction	241	(1)
Convergence in Probability	242	(11)
Convergence in Distribution	253	(11)
Combination of the Modes of Convergence	256	(1)
The Central Limit Theorems	257	(7)
Convergence of Chi-square, t, and F Distributions	264	(6)
The Chi-square Distribution	264	(1)
The Student's t Distribution	264	(1)
The F Distribution	265	(1)
Convergence of the PDF and Percentage Points	265	(5)
Exercises and Complements	270	(11)
Sufficiency, Completeness, and Ancillarity	281	(60)
Introduction	281	(1)
Sufficiency	282	(12)
The Conditional Distribution Approach	284	(4)
The Neyman Factorization Theorem	288	(6)
Minimal Sufficiency	294	(6)
The Lehmann-Scheffe Approach	295	(5)
Information	300	(9)
One-parameter Situation	301	(3)
Multi-parameter Situation	304	(5)
Ancillarity	309	(9)
The Location, Scale, and Location-Scale Families	314	(2)
Its Role in the Recovery of Information	316	(2)
Completeness	318	(9)
Complete Sufficient Statistics	320	(4)
Basu's Theorem	324	(3)
Exercises and Complements	327	(14)
Point Estimation	341	(54)
Introduction	341	(1)
Finding Estimators	342	(9)
The Method of Moments	342	(2)
The Method of Maximum Likelihood	344	(7)
Criteria to Compare Estimators	351	(7)
Unbiasedness, Variance and Mean Squared Error	351	(3)
Best Unbiased and Linear Unbiased Estimators	354	(4)
Improved Unbiased Estimator via Sufficiency	358	(7)
The Rao-Blackwell Theorem	358	(7)
Uniformly Minimum Variance Unbiased Estimator	365	(12)
The Cramer-Rao Inequality and UMVUE	366	(5)
The Lehmann-Scheffe Theorems and UMVUE	371	(3)
A Generalization of the Cramer-Rao Inequality	374	(1)
Evaluation of Conditional Expectations	375	(2)
Unbiased Estimation Under Incompleteness	377	(3)
Does the Rao-Blackwell Theorem Lead to UMVUE?	377	(3)
Consistent Estimators	380	(2)
Exercises and Complements	382	(13)
Tests of Hypotheses	395	(46)
Introduction	395	(1)
Error Probabilities and the Power Function	396	(5)
The Concept of a Best Test	399	(2)
Simple Null Versus Simple Alternative Hypotheses	401	(16)
Most Powerful Test via the Neyman-Pearson Lemma	401	(12)
Applications: No Parameters Are Involved	413	(3)
Applications: Observations Are Non-IID	416	(1)
One-Sided Composite Alternative Hypothesis	417	(8)
UMP Test via the Neyman-Pearson Lemma	417	(3)
Monotone Likelihood Ratio Property	420	(2)
UMP Test via MLR Property	422	(3)
Simple Null Versus Two-Sided Alternative Hypotheses	425	(4)
An Example Where UMP Test Does Not Exist	425	(1)
An Example Where UMP Test Exists	426	(2)
Unbiased and UMP Unbiased Tests	428	(1)
Exercises and Complements	429	(12)
Confidence Interval Estimation	441	(36)
Introduction	441	(2)
One-Sample Problems	443	(13)
Inversion of a Test Procedure	444	(2)
The Pivotal Approach	446	(5)
The Interpretation of a Confidence Coefficient	451	(1)
Ideas of Accuracy Measures	452	(3)
Using Confidence Intervals in the Tests of Hypothesis	455	(1)
Two-Sample Problems	456	(7)
Comparing the Location Parameters	456	(4)
Comparing the Scale Parameters	460	(3)
Multiple Comparisons	463	(6)
Estimating a Multivariate Normal Mean Vector	463	(2)
Comparing the Means	465	(2)
Comparing the Variances	467	(2)
Exercises and Complements	469	(8)
Bayesian Methods	477	(30)
Introduction	477	(2)
Prior and Posterior Distributions	479	(2)
The Conjugate Priors	481	(4)
Point Estimation	485	(3)
Credible Intervals	488	(5)
Highest Posterior Density	489	(3)
Contrasting with the Confidence Intervals	492	(1)
Tests of Hypotheses	493	(1)
Examples with Non-Conjugate Priors	494	(3)
Exercises and Complements	497	(10)
Likelihood Ratio and Other Tests	507	(32)
Introduction	507	(1)
One-Sample Problems	508	(7)
LR Test for the Mean	509	(3)
LR Test for the Variance	512	(3)
Two-Sample Problems	515	(7)
Comparing the Means	515	(4)
Comparing the Variances	519	(3)
Bivariate Normal Observations	522	(7)
Comparing the Means: The Paired Difference t Method	522	(3)
LR Test for the Correlation Coefficient	525	(3)
Tests for the Variances	528	(1)
Exercises and Complements	529	(10)
Large-Sample Inference	539	(30)
Introduction	539	(1)
The Maximum Likelihood Estimation	539	(3)
Confidence Intervals and Tests of Hypothesis	542	(13)
The Distribution-Free Population Mean	543	(5)
The Binomial Proportion	548	(5)
The Poisson Mean	553	(2)
The Variance Stabilizing Transformations	555	(8)
The Binomial Proportion	556	(3)
The Poisson Mean	559	(1)
The Correlation Coefficient	560	(3)
Exercises and Complements	563	(6)
Sample Size Determination: Two-Stage Procedures	569	(22)
Introduction	569	(4)
The Fixed-Width Confidence Interval	573	(6)
Stein's Sampling Methodology	573	(1)
Some Interesting Properties	574	(5)
The Bounded Risk Point Estimation	579	(5)
The Sampling Methodology	581	(1)
Some Interesting Properties	582	(2)
Exercises and Complements	584	(7)
Appendix	591	(42)
Abbreviations and Notation	591	(2)
A Celebration of Statistics: Selected Biographical Notes	593	(28)
Selected Statistical Tables	621	(12)
The Standard Normal Distribution Function	621	(5)
Percentage Points of the Chi-Square Distribution	626	(2)
Percentage Points of the Student's t Distribution	628	(2)
Percentage Points of the F Distribution	630	(3)
References	633	(16)
Index	649