### Summary

Mathematical statistics typically represents one of the most difficult challenges in statistics, particularly for those with more applied, rather than mathematical, interests and backgrounds. Most textbooks on the subject provide little or no review of the advanced calculus topics upon which much of mathematical statistics relies and furthermore contain material that is wholly theoretical, thus presenting even greater challenges to those interested in applying advanced statistics to a specific area.

Mathematical Statistics with Applications presents the background concepts and builds the technical sophistication needed to move on to more advanced studies in multivariate analysis, decision theory, stochastic processes, or computational statistics. Applications embedded within theoretical discussions clearly demonstrate the utility of the theory in a useful and relevant field of application and allow readers to avoid sudden exposure to purely theoretical materials.

With its clear explanations and more than usual emphasis on applications and computation, this text reaches out to the many students and professionals more interested in the practical use of statistics to enrich their work in areas such as communications, computer science, economics, astronomy, and public health.

### Table of Contents

INTRODUCTION

REVIEW OF MATHEMATICS

Introduction

Combinatorics

Pascal's Triangle

Newton's Binomial Formula

Exponential Function

Stirling's Formula

Multinomial Theorem

Monotonic Functions

Convergence and Divergence

Taylor's Theorem

Differentiation and Summation

Some Properties of Integration

Integration by Parts

Region of Feasibility

Multiple Integration

Jacobian

Maxima and Minima

Lagrange Multiplier

L'Hôpital's Rule

Partial Fraction Expansion

Cauchy-Schwarz Inequality

Generating Functions

Difference Equations

Vectors, Matrices and Determinants

Real Numbers

PROBABILITY THEORY

Introduction

Subjective Probability, Relative Frequency and Empirical Probability

Sample Space

Decomposition of a Union of Events: Disjoint Events

Sigma Algebra and Probability Space

Rules and Axioms of Probability Theory

Conditional Probability

Law of Total Probability

Bayes Rule

Sampling With and Without Replacement

Probability and SIMULATION

Borel Sets

Measure Theory in Probability

Application of Probability Theory: Decision Analysis

RANDOM VARIABLES

Introduction

Discrete Random Variables

Cumulative Distribution Function

Continuous Random Variables

Joint Distributions

Independent Random Variables

Distribution of the Sum of Two Independent Random Variables

Moments, Expected Values and Variance

Covariance and Correlation

Distribution of a Function of a Random Variable

Multivariate Distributions and Marginal Densities

Conditional Expectations

Conditional Variance and Covariance

Moment Generating Functions

Characteristic Functions

Probability Generating Functions

DISCRETE DISTRIBUTIONS

Introduction

Bernoulli Distribution

Binomial Distribution

Multinomial Distribution

Hypergeometric Distribution

k-Variate Hypergeometric Distribution

Geometric Distribution

Negative Binomial Distribution

Negative Multinomial Distribution

Poisson Distribution

Discrete Uniform Distribution

Lesser Known Distributions

Joint Distributions

Convolutions

Compound Distributions

Branching Processes

Hierarchical Distributions

CONTINUOUS RANDOM VARIABLES

Location and Scale Parameters

Distribution of Functions of Random Variables

Uniform Distribution

Normal Distribution

Exponential Distribution

Poisson Process

Gamma Distribution

Beta Distribution

Chi-square Distribution

Student's t-Distribution

F-Distribution

Cauchy Distribution

Exponential Family

Hierarchical Models-Mixture Distributions

Other Distributions

Distributional Relationships

Additional Distributional Findings

DISTRIBUTIONS OF ORDER STATISTICS

Introduction

Rank Ordering

The Probability Integral Transformation

Distributions of Order Statistics in i.i.d. Samples

Expectations of Minimum and Maximum Order Statistics

Distributions of Single Order Statistics

Joint Distributions of Order Statistics

ASYMPTOTIC DISTRIBUTION THEORY

Introduction

Introducing Probability to the Limit Process

Introduction to Convergence in Distribution

Non-convergence

Introduction to Convergence in Probability

Convergence Almost Surely (with Probability One)

Convergence in rth Mean

Relationships Between Convergence Modalities

Application of Convergence in Distribution

Properties of Convergence in Probability

The Law of Large Numbers and Chebyshev's Inequality

The Central Limit Theorem

Proof of the Central Limit Theorem

The Delta Method

Convergence Almost Surely (with probability one)

POINT ESTIMATION

Introduction

Method of Moments Estimators

Maximum Likelihood Estimators

Bayes Estimators

Sufficient Statistics

Exponential Families

Other Estimators*

Criteria of a Good Point Estimator

HYPOTHESIS TESTING

Statistical Reasoning and Hypothesis Testing

Discovery, the Scientific Method, and Statistical Hypothesis Testing

Simple Hypothesis Testing

Statistical Significance

The Two Sample Test

Two Sided vs. One Sided Testing

Likelihood Ratios and the Neyman Pearson Lemma

One SampleTesting and the Normal Distribution

Two Sample Testing for the Normal Distribution

Likelihood Ratio Test and the Binomial Distribution

Likelihood Ratio Test and the Poisson Distribution

The Multiple Testing Issue

Nonparametric Testing

Goodness of Fit Testing

Fisher's Exact Test

Sample Size Computations

INTERVAL ESTIMATION

Introduction

Definition

Constructing Confidence Intervals

Bayesian Credible Intervals

Approximate Confidence Intervals and MLE Pivot

The Bootstrap Method*

Criteria of a Good Interval Estimator

Confidence Intervals and Hypothesis Tests

INTRODUCTION TO COMPUTATIONAL METHODS

The Newton-Raphson Method

The EM Algorithm

Simulation

Markov Chains

Markov Chain Monte Carlo Methods

INDEX

### Reviews

"Mathematical Statistics with Applications meets an unmet need in advanced undergraduate and graduate programs. It is remarkable in its coverage to modern statistical theory with the necessary rigor and its applications to practical problems. Another unique feature is the inclusion of mathematical results needed to understand statistical theory and modern computational tools for data analysis. All these features make the book a self contained and unique text for imparting a balanced knowledge of statistics to students aspiring to be in statistics profession or pursue a research career. It will also be a fine reference text for applied scientists who require the occasional use of mathematical statistics."

-C.R. Rao, Sc. D., F.R.S. Member, National Academy of Science, USA, Eberly Professor Emeritus of Statistics, Director of the Center for Multivariate Analysis, Penn State University, University Park, Pennsylvania, USA

"Each chapter has a number of exercises, in total about two hundred and fifty."

-N. D. C. Veraverbeke, Short Book Reviews of the ISI

"…would be a reasonable textbook for the introductory mathematical statistics sequence at the graduate level. The many applications will be an aid to learning and any theoretical deficiencies can be supplemented in the classroom."

-Patricia Pepple Williamson, Virginia Commonwealth University, Journal of the American Statistical Association