A Course in Mathematical Statistics

by Roussas

• ISBN13: 9780125993159
• ISBN10: 0125993153
• Edition: 2nd
• Format: Hardcover
• Copyright: 1997-02-28
• Publisher: Elsevier Science

Summary

A Course in Mathematical Statistics, Second Edition, contains enough material for a year-long course in probability and statistics for advanced undergraduate or first-year graduate students, or it can be used independently for a one-semester (or even one-quarter) course in probability alone. It bridges the gap between high and intermediate level texts so students without a sophisticated mathematical background can assimilate a fairly broad spectrum of the theorems and results from mathematical statistics. The coverage is extensive, and consists of probability and distribution theory, and statistical inference. * Contains 25% new material * Includes the most complete coverage of sufficiency * Transformation of Random Vectors * Sufficiency / Completeness / Exponential Families * Order Statistics * Elements of Nonparametric Density Estimation * Analysis of Variance (ANOVA) * Regression Analysis * Linear Models

Table of Contents

Preface to the Second Edition	xv
Preface to the First Edition	xviii
Basic Concepts of Set Theory	1	(13)
Some Definitions and Notation	1	(7)
Exercises	5	(3)
Fields and σ-Fields	8	(6)
Some Probabilistic Concepts and Results	14	(39)
Probability Functions and Some Basic Properties and Results	14	(7)
Exercises	20	(1)
Conditional Probability	21	(6)
Exercises	25	(2)
Independence	27	(7)
Exercises	33	(1)
Combinatorial Results	34	(11)
Exercises	40	(5)
Product Probability Spaces	45	(2)
Exercises	47	(1)
The Probability of Matchings	47	(6)
Exercises	52	(1)
On Random Variables and Their Distributions	53	(32)
Some General Concepts	53	(2)
Discrete Random Variables (and Random Vectors)	55	(10)
Exercises	61	(4)
Continuous Random Variables (and Random Vectors)	65	(14)
Exercises	76	(3)
The Poisson Distribution as an Approximation to the Binomial Distribution and the Binomial Distribution as an Approximation to the Hypergeometric Distribution	79	(3)
Exercises	82	(1)
Random Variables as Measurable Functions and Related Results	82	(3)
Exercises	84	(1)
Distribution Functions, Probability Densities, and Their Relationship	85	(21)
The Cumulative Distribution Function (c.d.f. or d.f.) of a Random Vector---Basic Properties of the d.f. of a Random Variable	85	(6)
Exercises	89	(2)
The d.f. of a Random Vector and Its Properties---Marginal and Conditional d.f.'s and p.d.f.'s	91	(8)
Exercises	97	(2)
Quantiles and Modes of a Distribution	99	(3)
Exercises	102	(1)
Justification of Statements 1 and 2	102	(4)
Exercises	105	(1)
Moments of Random Variables---Some Moment and Probability Inequalities	106	(32)
Moments of Random Variables	106	(8)
Exercises	111	(3)
Expectations and Variances of Some R.V.'s	114	(8)
Exercises	119	(3)
Conditional Moments of Random Variables	122	(3)
Exercises	124	(1)
Some Important Applications: Probability and Moment Inequalities	125	(4)
Exercises	128	(1)
Covariance, Correlation Coefficient and Its Interpretation	129	(5)
Exercises	133	(1)
Justification of Relation (2) in Chapter 2	134	(4)
Characteristic Functions, Moment Generating Functions and Related Theorems	138	(26)
Preliminaries	138	(2)
Definitions and Basic Theorems---The One-Dimensional Case	140	(6)
Exercises	145	(1)
The Characteristic Functions of Some Random Variables	146	(4)
Exercises	149	(1)
Definitions and Basic Theorems---The Multidimensional Case	150	(3)
Exercises	152	(1)
The Moment Generating Function and Factorial Moment Generating Function of a Random Variable	153	(11)
Exercises	160	(4)
Stochastic Independence with Some Applications	164	(16)
Stochastic Independence: Criteria of Independence	164	(6)
Exercises	168	(2)
Proof of Lemma 2 and Related Results	170	(3)
Exercises	172	(1)
Some Consequences of Independence	173	(4)
Exercises	176	(1)
Independence of Classes of Events and Related Results	177	(3)
Exercise	179	(1)
Basic Limit Theorems	180	(32)
Some Modes of Convergence	180	(2)
Exercises	182	(1)
Relationships Among the Various Modes of Convergence	182	(5)
Exercises	187	(1)
The Central Limit Theorem	187	(9)
Exercises	194	(2)
Laws of Large Numbers	196	(3)
Exercises	198	(1)
Further Limit Theorems	199	(7)
Exercises	206	(1)
Polya's Lemma and Alternative Proof of the WLLN	206	(6)
Exercises	211	(1)
Transformations of Random Variables and Random Vectors	212	(33)
The Univariate Case	212	(7)
Exercises	218	(1)
The Multivariate Case	219	(16)
Exercises	233	(2)
Linear Transformations of Random Vectors	235	(7)
Exercises	240	(2)
The Probability Integral Transform	242	(3)
Exercise	244	(1)
Order Statistics and Related Theorems	245	(14)
Order Statistics and Related Distributions	245	(11)
Exercises	252	(4)
Further Distribution Theory: Probability of Coverage of a Population Quantile	256	(3)
Exercise	258	(1)
Sufficiency and Related Theorems	259	(25)
Sufficiency: Definition and Some Basic Results	260	(11)
Exercises	269	(2)
Completeness	271	(3)
Exercises	273	(1)
Unbiasedness---Uniqueness	274	(2)
Exercises	276	(1)
The Exponential Family of p.d.f.'s: One-Dimensional Parameter Case	276	(5)
Exercises	280	(1)
Some Multiparameter Generalizations	281	(3)
Exercises	282	(2)
Point Estimation	284	(43)
Introduction	284	(1)
Exercise	284	(1)
Criteria for Selecting an Estimator: Unbiasedness, Minimum Variance	285	(2)
Exercises	286	(1)
The Case of Availability of Complete Sufficient Statistics	287	(6)
Exercises	292	(1)
The Case Where Complete Sufficient Statistics Are Not Available or May Not Exist: Cramer-Rao Inequality	293	(9)
Exercises	301	(1)
Criteria for Selecting an Estimator: The Maximum Likelihood Principle	302	(7)
Exercises	308	(1)
Criteria for Selecting an Estimator: The Decision-Theoretic Approach	309	(3)
Finding Bayes Estimators	312	(6)
Exercises	317	(1)
Finding Minimax Estimators	318	(2)
Exercise	320	(1)
Other Methods of Estimation	320	(2)
Exercises	322	(1)
Asymptotically Optimal Properties of Estimators	322	(3)
Exercise	325	(1)
Closing Remarks	325	(2)
Exercises	326	(1)
Testing Hypotheses	327	(55)
General Concepts of the Neyman-Pearson Testing Hypotheses Theory	327	(2)
Exercise	329	(1)
Testing a Simple Hypothesis Against a Simple Alternative	329	(8)
Exercises	336	(1)
UMP Tests for Testing Certain Composite Hypotheses	337	(12)
Exercises	347	(2)
UMPU Tests for Testing Certain Composite Hypotheses	349	(4)
Exercises	353	(1)
Testing the Parameters of a Normal Distribution	353	(4)
Exercises	356	(1)
Comparing the Parameters of Two Normal Distributions	357	(4)
Exercises	360	(1)
Likelihood Ratio Tests	361	(9)
Exercises	369	(1)
Applications of LR Tests: Contingency Tables, Goodness-of-Fit Tests	370	(5)
Exercises	373	(2)
Decision-Theoretic Viewpoint of Testing Hypotheses	375	(7)
Sequential Procedures	382	(15)
Some Basic Theorems of Sequential Sampling	382	(6)
Exercises	388	(1)
Sequential Probability Ratio Test	388	(5)
Exercise	392	(1)
Optimality of the SPRT-Expected Sample Size	393	(1)
Exercises	394	(1)
Some Examples	394	(3)
Confidence Regions---Tolerance Intervals	397	(19)
Confidence Intervals	397	(1)
Exercise	398	(1)
Some Examples	398	(9)
Exercises	404	(3)
Confidence Intervals in the Presence of Nuisance Parameters	407	(3)
Exercise	410	(1)
Confidence Regions---Approximate Confidence Intervals	410	(3)
Exercises	412	(1)
Tolerance Intervals	413	(3)
The General Linear Hypothesis	416	(24)
Introduction of the Model	416	(2)
Least Square Estimators---Normal Equations	418	(6)
Canonical Reduction of the Linear Model---Estimation of σ2	424	(5)
Exercises	428	(1)
Testing Hypotheses About η = E(Y)	429	(4)
Exercises	433	(1)
Derivation of the Distribution of the ϝ Statistic	433	(7)
Exercises	436	(4)
Analysis of Variance	440	(23)
One-way Layout (or One-way Classification) with the Same Number of Observations Per Cell	440	(6)
Exercise	446	(1)
Two-way Layout (Classification) with One Observation Per Cell	446	(6)
Exercises	451	(1)
Two-way Layout (Classification) with K (≥ 2) Observations Per Cell	452	(6)
Exercises	457	(1)
A Multicomparison method	458	(5)
Exercises	462	(1)
The Multivariate Normal Distribution	463	(13)
Introduction	463	(4)
Exercises	466	(1)
Some Properties of Multivariate Normal Distributions	467	(2)
Exercise	469	(1)
Estimation of μ and Σ and a Test of Independence	469	(7)
Exercises	475	(1)
Quadratic Forms	476	(9)
Introduction	476	(1)
Some Theorems on Quadratic Forms	477	(8)
Exercises	483	(2)
Nonparametric Inference	485	(14)
Nonparametric Estimation	485	(2)
Nonparametric Estimation of a p.d.f.	487	(3)
Exercise	490	(1)
Some Nonparametric Tests	490	(3)
More About Nonparametric Tests: Rank Tests	493	(3)
Exercises	496	(1)
Sign Test	496	(2)
Relative Asymptotic Efficiency of Tests	498	(1)
Appendix I Topics from Vector and Matrix Algebra	499	(9)
I.1 Basic Definitions in Vector Spaces	499	(2)
I.2 Some Theorems on Vector Spaces	501	(1)
I.3 Basic Definitions About Matrices	502	(2)
I.4 Some Theorems About Matrices and Quadratic Forms	504	(4)
Appendix II Noncentral t-, X2-, and F-Distributions	508	(3)
II.1 Noncentral t-Distribution	508	(1)
II.2 Noncentral x2-Distribution	508	(1)
II.3 Noncentral F-Distribution	509	(2)
Appendix III Tables	511	(34)
1 The Cumulative Binomial Distribution	511	(9)
2 The Cumulative Poisson Distribution	520	(3)
3 The Normal Distribution	523	(3)
4 Critical Values for Student's t-Distribution	526	(3)
5 Critical Values for the Chi-Square Distribution	529	(3)
6 Critical Values for the F-Distribution	532	(10)
7 Table of Selected Discrete and Continuous Distributions and Some of Their Characteristics	542	(3)
Some Notation and Abbreviations	545	(2)
Answers to Selected Exercises	547	(14)
Index	561