This Volume II is the second half of a text for a course in statistics at the beginning graduate level. Statistics is a man-made science aiming at assisting humans in making decisions in the face of uncertainty. This science is built upon the rigorous theory of probability as described in Volume I. Thus, in studying this text,students should consult Volume I whenever needed.

As stated in the preface of Volume I, there are various reasons to write another text in statistics at the introductory level. An obvious reason is to make the topic of statistics pleasant for students!

Preface

1 An Invitation to Statistics

 1.1 A Motivating Example

 1.2 Generalities on Survey Sampling

 1.3 Statistical Data

 1.4 Statistical Models

 1.5 Some Computational Statistics

1.5.1 Generating uniform random variables

1.5.2 Generating non-uniform random variables

1.5.3 Monte Carlo methods

 1.6 Exercises

2 Sampling Distributions

 2.1 Sampling from a Bernoulli Population

2.1.1 The binomial distribution

2.1.2 The Poisson distribution

2.1.3 Other distributions related Bernoulli trials

 2.2 Sampling from a Normal Population

 2.3 Sampling from an Exponential Population

 2.4 Order Statistics

 2.5 Distributions of Quadratic Forms

 2.6 Exercises

3 Data Reduction

 3.1 Sufficient Statistics

 3.2 Complete Statistics

 3.3 Exponential and Location-scale Families

3.3.1 Exponential families

3.3.2 Location and scale families

 3.4 Exercises

4 Estimation

 4.1 Point Estimation

 4.2 The Best Unbiased Estimation

 4.3 Fisher Information and Efficiency

4.3.1 Fisher information

4.3.2 Cramer-Rao lower bound

 4.4 Two Methods of Finding Estimators

4.4.1 The method of moments

4.4.2 The method of maximum likelihood

4.4.3 Some properties of maximum likelihood estimators

 4.5 Confidence Sets

4.5.1 Pivotal quantities

4.5.2 Lengths of confidence intervals

 4.6 Bayes Estimation

4.6.1 Prior and posterior distributions

4.6.2 Bayes rules and minimax rules

4.6.3 Bayes and minimax estimators

4.6.4 Bayes intervals

 4.7 Exercises

5 Large Sample Estimation

 5.1 Consistency

5.1.1 Consistent estimators

5.1.2 Consistency of sample quartiles

5.1.3 Consistency of maximum likelihood estimators

 5.2 Asymptotic Normality

5.2.1 Univariate asymptotic distributions

5.2.2 The Delta method

5.2.3 Asymptotic distributions of the sample quartiles

5.2.4 Multivariate asymptotic distributions

 5.3 Asymptotic Normality of Maximum Likelihood Estimators

 5.4 Asymptotic Efficiency

 5.5 Large Sample Interval Estimation

5.5.1 Asymptotically pivotal quantities

5.5.2 Intervals based on maximum likelihood estimators

 5.6 Robust Estimation

5.6.1 The influence function

5.6.2 L-estimators

5.6.3 M-estimators

 5.7 Exercises

6 Tests of Statistical Hypotheses

 6.1 Introduction

 6.2 Basic Concepts in Hypothesis Testing

6.2.1 Two hypotheses

6.2.2 Two types of errors and the power function

6.2.3 The p-value

6.2.4 Randomized tests

 6.3 Most Powerful Tests

6.3.1 Neyman-Pearson lemma

 6.4 Uniformly Most Powerful Tests

6.4.1 Uniformly most powerful tests

6.4.2 Monotone likelihood ratio

6.4.3 Tests in one-parameter exponential family

 6.5 Unbiased Tests

 6.6 Tests and Confidence Sets

 6.7 Likelihood Ratio Tests

6.7.1 Likelihood ratio tests

6.7.2 Asymptotic distribution of the likelihood ratio

 6.8 Sequential Probability Ratio Tests

 6.9 Chi-Square Tests

6.9.1 Goodness-of-fit tests

6.9.2 Tests in contingency tables

 6.10 Bayes Tests

 6.11 Summary of Tests for Normal Populations

6.11.1 One sample test procedures

6.11.2 Two-sample test procedures

6.11.3 Tests in Bernoulli models for large samples

6.11.4 The paired t-test

 6.12 Exercises

7 Nonparametric Statistical Inference

 7.1 Inferences on Quartiles

7.1.1 Confidence intervals for quartiles

7.1.2 The sign test

7.1.3 Asymptotic relative efficiency

 7.2 The Wilcoxon Signed Rank Test

 7.3 The Mann-Whitney-Wilcoxon Test

 7.4 The Kolmogorov-Smirnov Test and Test of Normality

7.4.1 The Kolmogorov-Smirnov Test

7.4.2 Two-sample Kolmogorov-Smirnov test

7.4.3 Test of normality

 7.5 Exercises

Appendices

A Common Distributions

 A.1 Univariate Discrete Distributions

 A.2 Univariate Continuous Distributions

 A.3 Multivariate Distributions

B Some Common Statistical Tables

 B.1 The Standard Normal Distribution

 B.2 The Student's t Distribution

 B.3 The chi-Square Distribution

 B.4 The F Distribution

Bibliography

Index