Mathematical Statistics and Data Analysis

by RICE

• ISBN13: 9780534209346
• ISBN10: 0534209343
• Edition: 2nd
• Format: Paperback
• Copyright: 1994-06-01
• Publisher: Duxbury Press

Summary

1. PROBABILITY Introduction / Sample Spaces / Probability Measures / Computer Probabilities: Counting Methods / Conditional Probability / Independence / Concluding Remarks / Problems 2. RANDOM VARIABLES Discrete Random Variables / Continuous Random Variables / Functions of a Random Variable / Concluding Remarks / Problems 3. JOINT DISTRIBUTIONS Introduction / Discrete Random Variables / Continuous Random Variables / Independent Random Variables / Conditional Distributions / Functions of Jointly Distributed Random Variables / Extreme and Order Statistics / Problems 4. EXPECTED VALUES The Expected Value of a Random Variable / Variance and Standard Deviation / Covariance and Correlation / Conditional Expectation and Prediction / The Moment-Generating Function / Approximate Methods / Problems 5. LIMIT THEOREMS Introduction / The Law of Large Number / Convergence in Distribution and the Central Limit Theorem / Problems 6. DISTRIBUTIONS DERIVED FROM THE NORMAL DISTRIBUTION Introduction / Chi-Squared, t, and F Distributions / The Sample Mean and Sample Variance / Problems 7. SURVEY SAMPLING Introduction / Production Parameters / Simple Random Sampling / Estimation of a Ratio / Stratified Random Sampling / Concluding Remarks / Problems 8. ESTIMATION OF PARAMETERS AND FITTING OF PROBABILITY DISTRIBUTIONS Introduction / Fitting the Poisson Distribution to Emissions of Alpha Particles / Parameter Estimation / The Method of Moments / The Method of Maximum Likelihood / Efficiency and the Cramer-Rao Lower Bound / Sufficiency / Concluding Remarks / Problems 9. TESTING HYPOTHESES AND ASSESSING GOODNESS OF FIT Introduction / The Neyman-Pearson Paradigm / Optimal Tests: The Neyman-Pearson Lemma / The Duality of Confidence Intervals and Hypothesis Tests / Generalized Likelihood Ratio Tests / Likelihood Ratio Tests for the Multinomial Distribution / The Poisson Dispersion Test / Hanging Rootograms / Probability Plots / Tests for Normality / Concluding Remarks / Problems 10. SUMMARIZING DATA Introduction / Methods Based on the Cumulative Distribution Function / Histograms, Density Curves, and Stem-and-Leaf Plots / Measures of Location / Measures of Dispersion / Boxplots / Concluding Remarks / Problems 11. COMPARING TWO SAMPLES Introduction / Comparing Two Independent Samples / Comparing Paired Samples / Experimental Design / Concluding Remarks / Problems 12. THE ANALYSIS OF VARIANCE Introduction / The One-Way Layout / The Two-Way Layout / Concluding Remarks / Problems 13. THE ANALYSIS OF CATEGORICAL DATA Introduction / Fisher's Exact Test / The Chi-Square Test of Homogeneity / The Chi-Square Test of Independence / Matched-Pair Designs / Odds Ratios / Concluding Remarks / Problems 14. LINEAR LEAST SQUARES Introduction / Simple Linear Regression / The Matrix Approach to Linear Least Square / Statistical Properties of Least Squares Estimates / Multiple Linear Regression--An Example / Conditional Inference, Unconditional Inference, and the Bootstrap / Concluding Remarks / Problems 15. DECISION THEORY AND BAYESIAN INFERENCE Introduction / Decision Theory / The Subjectivist Point of View / Concluding Remarks / Problems / APPENDIXES / A. COMMON DISTRIBUTIONS / B. TABLES / BIBLIOGRAPHY / ANSWERS TO SELECTED PROBLEMS / AUTHOR INDEX / INDEX TO DATA SETS / SUBJECT INDEX

Table of Contents

Probability	1	(32)
Introduction	1	(1)
Sample Spaces	2	(2)
Probability Measures	4	(3)
Computing Probabilities: Counting Methods	7	(8)
The Multiplication Principle	8	(1)
Permutations and Combinations	9	(6)
Conditional Probability	15	(6)
Independence	21	(3)
Concluding Remarks	24	(1)
Problems	24	(9)
Random Variables	33	(36)
Discrete Random Variables	33	(13)
Bernoulli Random Variables	35	(1)
The Binomial Distribution	36	(2)
The Geometric and Negative Binomial Distributions	38	(1)
The Hypergeometric Distribution	39	(2)
The Poisson Distribution	41	(5)
Continuous Random Variables	46	(11)
The Exponential Density	48	(2)
The Gamma Density	50	(3)
The Normal Distribution	53	(4)
Functions of a Random Variable	57	(4)
Concluding Remarks	61	(1)
Problems	62	(7)
Joint Distributions	69	(42)
Introduction	69	(2)
Discrete Random Variables	71	(2)
Continuous Random Variables	73	(10)
Independent Random Variables	83	(2)
Conditional Distributions	85	(7)
The Discrete Case	85	(1)
The Continuous Case	86	(6)
Functions of Jointly Distributed Random Variables	92	(8)
Sums and Quotients	92	(3)
The General Case	95	(5)
Extrema and Order Statistics	100	(3)
Problems	103	(8)
Expected Values	111	(52)
The Expected Value of a Random Variable	111	(11)
Expectations of Functions of Random Variables	116	(3)
Expectations of Linear Combinations of Random Variables	119	(3)
Variance and Standard Deviation	122	(7)
A Model for Measurement Error	126	(3)
Covariance and Correlation	129	(6)
Conditional Expectation and Prediction	135	(7)
Definitions and Examples	135	(5)
Prediction	140	(2)
The Moment-Generating Function	142	(7)
Approximate Methods	149	(5)
Problems	154	(9)
Limit Theorems	163	(14)
Introduction	163	(1)
The Law of Large Numbers	163	(3)
Convergence in Distribution and the Central Limit Theorem	166	(7)
Problems	173	(4)
Distributions Derived from the Normal Distribution	177	(8)
Introduction	177	(1)
X2,t, and F Distributions	177	(2)
The Sample Mean and the Sample Variance	179	(3)
Problems	182	(3)
Survey Sampling	185	(54)
Introduction	185	(1)
Population Parameters	186	(2)
Simple Random Sampling	188	(18)
The Expectation and Variance of the Sample Mean	189	(7)
Estimation of the Population Variance	196	(3)
The Normal Approximation to the Sampling Distribution of X	199	(7)
Estimation of a Ratio	206	(7)
Stratified Random Sampling	213	(10)
Introduction and Notation	213	(1)
Properties of Stratified Estimates	214	(4)
Methods of Allocation	218	(5)
Concluding Remarks	223	(2)
Problems	225	(14)
Estimation of Parameters and Fitting of Probability Distributions	239	(60)
Introduction	239	(1)
Fitting the Poisson Distribution to Emissions of Alpha Particles	239	(4)
Parameter Estimation	243	(3)
The Method of Moments	246	(7)
The Method of Maximum Likelihood	253	(20)
Maximum Likelihood Estimates of Multinomial Cell Probabilities	259	(2)
Large Sample Theory for Maximum Likelihood Estimates	261	(5)
Confidence Intervals for Maximum Likelihood Estimates	266	(7)
Efficiency and the Cramer-Rao Lower Bound	273	(7)
An Example: The Negative Binomial Distribution	277	(3)
Sufficiency	280	(5)
A Factorization theorem	281	(3)
The Rao-Blackwell Theorem	284	(1)
Concluding Remarks	285	(1)
Problems	286	(13)
Testing Hypotheses and Assessing Goodness of Fit	299	(46)
Introduction	299	(1)
The Neyman-Pearson Paradigm	300	(3)
Optimal Tests: The Neyman-Pearson Lemma	303	(3)
The Duality of Confidence Intervals and Hypothesis Tests	306	(2)
Generalized Likelihood Ratio Tests	308	(2)
Likelihood Ratio Tests for the Multinomial Distribution	310	(6)
The Poisson Dispersion Test	316	(2)
Hanging Rootograms	318	(3)
Probability Plots	321	(6)
Tests for Normality	327	(3)
Concluding Remarks	330	(1)
Problems	331	(14)
Summarizing Data	345	(42)
Introduction	345	(1)
Methods Based on the Cumulative Distribution Function	346	(11)
The Empirical Cumulative Distribution Function	346	(2)
The Survival Function	348	(5)
Quantile-Quantile Plots	353	(4)
Histograms, Density Curves, and Stem-and-Leaf Plots	357	(4)
Measures of Location	361	(9)
The Arithmetic Mean	361	(3)
The Median	364	(1)
The Trimmed Mean	365	(1)
M Estimates	366	(1)
Comparison of Location Estimates	367	(1)
Estimating Variability of Location Estimates by the Bootstrap	367	(3)
Measures of Dispersion	370	(2)
Boxplots	372	(2)
Concluding remarks	374	(1)
Problems	375	(12)
Comparing Two samples	387	(56)
Introduction	387	(1)
Comparing Two Independent Samples	388	(22)
Methods Based on the Normal Distribution	388	(8)
An Example---A Study of Iron Retention	396	(4)
Power	400	(2)
A Nonparametric Method---the Mann-Whitney Test	402	(8)
Comparing Paired Samples	410	(7)
Methods Based on the Normal distribution	411	(2)
A Nonparametric method---the signed Rank Test	413	(2)
An Example---Measuring Mercury Levels in fish	415	(2)
Experimental Design	417	(7)
Mammary Artery Ligation	417	(1)
The Placebo Effect	418	(1)
The Lanarkshire Milk Experiment	419	(1)
The Portocaval shunt	419	(1)
FD&C Red No. 40	420	(1)
Further Remarks on Randomization	421	(1)
Observational Studies, Confounding, and Bias in Graduate Admissions	422	(1)
Fishing Expeditions	423	(1)
Concluding Remark	424	(1)
Problems	425	(18)
The Analysis of Variance	443	(40)
Introduction	443	(1)
The One-Way Layout	443	(12)
Normal Theory; the F Test	445	(6)
The Problem of Multiple Comparisons	451	(1)
Tukey's Method	451	(2)
The Bonferroni Method	453	(1)
A Nonparametric Method---The Kruskal-Wallis Test	453	(2)
The Two-Way Layout	455	(16)
Additive Parametrization	455	(3)
Normal theory for the Two-way Layout	458	(7)
Randomized Block Designs	465	(4)
A Nonparametric Method---Friedman's Test	469	(2)
Concluding Remarks	471	(1)
Problems	472	(11)
The Analysis of Categorical Data	483	(24)
Introduction	483	(1)
Fisher's Exact Test	483	(2)
The Chi-Square Test of Homogeneity	485	(4)
The Chi-Square Test of Independence	489	(3)
Matched-Pairs Designs	492	(2)
Odds Ratios	494	(4)
Concluding Remarks	498	(1)
Problems	498	(9)
Linear Least Squares	507	(64)
Introduction	507	(4)
Simple Linear Regression	511	(18)
Statistical Properties of the Estimated Slope and Intercept	511	(4)
Assessing the Fit	515	(11)
Correlation and Regression	526	(3)
The Matrix Approach to Linear Least Squares	529	(3)
Statistical Properties of Least Squares Estimates	532	(12)
Vector-Valued Random Variables	532	(5)
Mean and Covariance of Least Squares Estimates	537	(2)
Estimation of σ2	539	(2)
Residuals and Standardized Residuals	541	(1)
Inference about β	542	(2)
Multiple Linear Regressions---An Example	544	(4)
Conditional Inference, Unconditional Inference, and the Bootstrap	548	(3)
Concluding Remarks	551	(1)
Problems	552	(19)
Decision Theory and Bayesian Inference	571	(26)
Introduction	571	(1)
Decision theory	571	(16)
Bayes Rules and Minimax Rules	573	(5)
Posterior Analysis	578	(2)
Classification and Hypothesis Testing	580	(4)
Estimation	584	(3)
The Subjectivist Point of View	587	(10)
Bayesian Inference for the Normal Distribution	589	(3)
Bayesian Analysis for the Binomial Distribution	592	(5)
Concluding Remarks	597	(1)
Problems	597
Appendix A Common Distributions	A1
Appendix B Tables	A4
Bibliography	A25
Answers to Selected Problems	A31
Author Index	A43
Index to Data Sets	A45
Subject to Index	A46