An Introduction to Mathematical Statistics and Its Applications

Citation

Larsen, R. J., & Marx, M. L. (2018). An Introduction to Mathematical Statistics and Its Applications (6th ed.). Pearson.

Chapter Summary

Chapter 1: Introduction

The opening chapter places the subject in context with an overview, some classic examples of statistical analysis and a short history of statistics and probability.

Chapter 2: Probability

Sample spaces and the algebra of sets lead to the definition of probability, followed by conditional probability, independence and combinatorial methods for calculating the probability of complex events.

Chapter 3: Random Variables

Discrete and continuous random variables are introduced with their distributions, expected values and variances, and the effect of transformations and combinations of random variables.

Chapter 4: Special Distributions

This chapter examines the named distributions that recur in applications, with the Poisson, normal and binomial distributions as central examples.

Chapter 5: Estimation

Estimation covers the methods for obtaining estimators, notably maximum likelihood and the method of moments, and the properties used to judge them: unbiasedness, efficiency and consistency.

Chapter 6: Hypothesis Testing

The chapter sets out the logic of testing a null hypothesis against an alternative, including Type I and Type II errors and what they imply for a decision.

Chapter 7: Inferences Based on the Normal Distribution

Procedures for means and variances that rest on the normal distribution are developed here, including z and t procedures and the estimation of a variance.

Chapter 8: Types of Data: A Brief Overview

A short chapter that classifies kinds of data and relates each kind to an appropriate method of analysis and design.

Chapter 9: Two-Sample Inferences

Techniques for comparing two samples, with tests and intervals for means and variances (independent-sample t and F procedures) and the assumptions they need.

Chapter 10: Goodness-of-Fit Tests

The chapter asks how well a theoretical distribution describes observed data and develops the chi-square goodness-of-fit test.

Chapter 11: Regression

Linear regression is introduced through least-squares estimation, followed by ways of assessing how well a fitted model describes the data, including residual analysis.

Chapter 12: The Analysis of Variance

Analysis of variance extends the two-sample comparison to more than two groups, with multiple-comparison procedures and the assumptions behind the method.

Chapter 13: Randomized Block Designs

Randomized block designs show how to control extraneous variability among experimental units so that treatment effects can be compared more precisely.

Chapter 14: Nonparametric Statistics

The final chapter presents methods that do not assume a specific distribution for the data, including tests for matched pairs and independent samples such as the sign and Wilcoxon tests.

Key Concepts

Chapters 2–4: Probability, Random Variables, Special Distributions

• Conditional probability and independence: the basic tools for relating events to one another.
• Expected value and variance: the summary measures of a random variable's centre and spread.
• Special distributions: the binomial, Poisson and normal families used as models for data.

Chapters 5–7: Estimation, Hypothesis Testing, Inference from the Normal

• Maximum likelihood and method of moments: the two main ways the book obtains estimators.
• Unbiasedness, efficiency, consistency: criteria for comparing estimators.
• Type I and Type II errors: the two ways a test can go wrong.

Chapters 8–14: Applied Inference

• Goodness-of-fit test: a check of a hypothesised distribution against observed counts.
• Least squares: the fitting criterion behind linear regression.
• Analysis of variance and blocking: comparing several treatments, with or without controlling for known sources of variation.
• Nonparametric tests: distribution-free alternatives when parametric assumptions fail.

Critical Analysis

Strengths

• A clear target reader. The publisher presents the book as a calculus student's first exposure to mathematical statistics, written for students who have completed three or more semesters of calculus, and as substantial enough for a two-semester course.
• Statistics and experimental design together. Pearson describes the text as covering the principles of statistics and data modelling, and the contents confirm it: besides estimation and testing, three late chapters cover analysis of variance, randomized block designs and nonparametric methods.
• A full arc in one volume. Fourteen chapters run from probability through special distributions and inference to regression, designed experiments and distribution-free methods, so one text supports both the probability half and the inference half of a course.
• Computational exposure. The publisher's description says the book gives computationally minded students a realistic exposure to identifying data models.

Limitations

• Calculus is a real prerequisite. Readers without several semesters of calculus will find the probability and distribution chapters hard.
• Probability is covered briefly. Only three chapters precede estimation, so students who need a deeper foundation in probability will want a dedicated probability book alongside it.
• Bayesian and measure-theoretic material is not a feature. The contents list no chapter on Bayesian methods or on the measure-theoretic treatment of probability, so the book is a first course rather than a route to graduate theory.

Where It Fits in the Course

Within the probability and statistics list, this is the calculus-based first course in mathematical statistics. Blitzstein and Hwang's Introduction to Probability is a more thorough undergraduate treatment of probability alone, with story proofs and simulation in R, and is a good companion or prerequisite for Chapters 2 to 4. Illowsky and Dean's Introductory Statistics 2e and Spiegelhalter's The Art of Statistics teach statistical practice without the calculus, and make useful background for readers who need the ideas before the mathematics.

For deeper theory, Casella and Berger's Statistical Inference and Mavrakakis and Penzer's Probability and Statistical Inference cover similar ground at a graduate level, with Bayesian methods in the second. Williams's Probability with Martingales and Mitzenmacher and Upfal's Probability and Computing go further into probability, in the measure-theoretic and algorithmic directions respectively. Larsen and Marx is the point of entry to them, not a substitute.

How to Use This Book

The book is for mathematics, statistics, engineering and science students who have finished the calculus sequence and have not yet seen mathematical statistics. It can also serve as a self-study text for anyone who wants to see how the standard procedures are derived.

For a two-semester plan, take Chapters 1 to 7 in the first term (probability, random variables, special distributions, estimation, hypothesis testing and inference from the normal distribution) and Chapters 8 to 14 in the second (types of data, two-sample inference, goodness of fit, regression, analysis of variance, blocking and nonparametric methods). Students who have already taken a calculus-based probability course can read Chapters 2 to 4 as review and start in earnest with Chapter 5.

Work the exercises: this is where the derivations are practised. Compare a hypothesis-testing chapter with its estimation counterpart to see how intervals and tests answer the same questions.