An Introduction to Mathematical Statistics and Its Applications Paperback – 1 Jun 2005
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Just one drawback: when is the sequel?
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The second half of this book is devoted to hypothesis testing and regression. There is an excellent explanation of the mathematical presuppositions of the various classical experimental methodologies ranging from chi-square to t-tests to generalized likelihood ratio testing. It contains a very nicely organized chapter on general regression analysis, concentrating on the common least squares case under the usual transforms (e.g. exponential, logistic, etc.).
Like many books in mathematics, this introduction starts from first principles in the topic it's introducing, but assumes some "mathematical sophistication". In this case, it assumes you're comfortable with basic definition-example-theorem style and that you understand the basics of multivariate differential equations. I was a math and computer science undergrad who did much better in abstract algebra and set theory than analysis and diff eqs, but I found this book extremely readable. I couldn't have derived the proofs, but I could follow them because they were written as clearly as anything I've ever read in mathematics. I found the explanation of the central limit theorem and the numerous normal approximation theorems for sampling to be exceptionally clear.
The examples were both illuminating and entertaining. One of the beauties of statistics is that the examples are almost always interesting real-world problems, in this case ranging from biological (e.g. significance testing for cancer clusters) to man-made (e.g. Poisson models of football scoring) to physical (e.g. loaded dice). The examples tied directly to the techniques being explored. The exercises were more exercise-like in this book than in some math books where they're a dumping ground for material that wouldn't fit into the body of the text. This book has clearly been tuned over many years of classroom use with real students.
I read this book because I found I couldn't understand the applied statistics I was reading in machine learning and Bayesian data analysis research papers in my field (computational linguistics). In paticular, I wanted the background to be able to tackle books such as Hastie et al.'s "Elements of Statistical Learning" or Gelman et al.'s "Bayesian Data Analysis", both of which pretty much assume a good grounding in the topics covered in this book by Larsen and make excellent follow-on reading.
The book is readable and well-written and I'll probably use it again if and when I teach the sequence. The authors, as Jay I. Simon pointed out in an earlier review, have a sense of humor. For example, a random walk problem begins with the following sentence: "A somewhat inebriated conventioneer finds himself in the embarrassing predicament of being unable to predetermine whether his next step will be forward or backward." There are several other examples of humor: for example, the authors discuss an airline known as Doomsday Airlines.
The reason that I give the book only four stars is that the rigor is on occasion illusory, as Glitzer pointed out in another review. Here is a chapter-by-chapter review.
Preface: The authors claim that the first 7 chapters can easily be covered in one semester. I don't agree with this statement. We covered the first four chapters and part of the fifth, and very few of my students suggested that I was going slowly.
Chapter 1: This is an historical introduction. I don't know about the accuracy of the history (although I believe that the history is accurate), but the authors tell a good story. The treatment of the golden ratio is problematic, since their definition inverts one of the ratios and so their definition is the reciprocal of the usual golden ratio. This is not that problematic in itself, but the continued fraction representation converges to the usual golden ratio.
Chapter 2: This introduces elementary probability and combinatorics. It is one of the best chapters in the text with excellent examples and a good introduction to the Kolmogorov axiomatic framework which does not get bogged down in measure theoretic details.
Chapter 3: Random variables are introduced in this chapter, the longest in the book. Much of the material is well-done in this chapter, but the introduction of continuous random variables is a mess. They initially define continuous sample spaces to be those that are uncountable blatantly disregarding the possibility of mixed distributions. They then define a continuous real-valued random variable to be a function between two subsets of the real numbers and assert without justification that a probability density function. The `definition' is in any case simultaneously too restrictive (the input space need not be real) and too general (the observation space of a binomial random variable is a subset of the real numbers). In the discussion of the relationship between a cdf and a pdf, the authors blatantly misapply the fundamental theorem of calculus since there is no reason to assume that a pdf is continuous. This disregard of basic regularity issues permeates the chapter, usually without comment from the authors. Although there were other factors (many due to me), the confused treatment of continuous random variables was a contributor to the fact that most of my class never had a clear idea of what a random variable was. However, despite these issues, the chapter is still fairly good. The examples and exercises are well done, and not all of them are routine.
Chapter 4: This chapter is devoted to a discussion of some of the more important distributions. The material is generally of high quality. The central limit theorem is stated and the proof is deferred to an appendix. The appendix starts off by stating that the full proof is beyond the level of the text. While I agree with this, I do not understand why one would devote an appendix to `a proof of the central limit theorem' without giving a proof. This is an example of the illusory nature of the apparent rigor of the text.
Chapter 5: This is a very hard chapter on estimation. The key sections are on maximum likelihood estimators, confidence intervals, unbiasedness, and (perhaps) efficiency. Given the difficulty of the notion of sufficiency, I thought that the authors did an excellent job with it. The optional section on Bayesian estimation is also well done.
Chapter 6: Hypothesis testing is introduced here. The authors routinely state hypotheses tests as theorems starting in this chapter. This seems to be an abuse of the term, and when they `prove' the theorems, they typically show that the hypothesis test is at least approximately a generalized likelihood ratio test (GLRT); which is not the same thing at all. Saying that, the basic idea of what an hypothesis test actually is and how to perform one is explained well.
Chapter 7: The basic t and chi-square tests are introduced here. Note that the hypotheses tests, by the time they are actually stated, are pretty obvious which makes it strange that appendices are devoted to their proofs. As noted above, the tests are shown in the appendices to be (at least approximately) GLRTs. I did like the derivation of the various sampling distributions.
Chapter 8: This chapter discusses how to classify data. Although it comes at an appropriate place in the discussion, it might be better to have it earlier so that more students have a chance to consider it in a classroom setting.
Chapter 9: This chapter discusses two-sample data. It's pretty vanilla.
Chapter 10: Here we look at goodness-of-fit tests. The discussion is nice, although I think more attention should have been paid to the categorical distribution rather than simply leaping to the general multinomial distribution.
Chapter 11: At this point, the examples and exercises become much more computationally intensive. For this chapter discusses regression, covariance, and the bivariate normal distribution. I think this is one of the better chapters in the text, although a linear algebraic point of view for the multivariate normal distribution would have made an elegant addition.
Chapter 12: ANOVA is now introduced. Given the complexity of the setup, the authors give a very nice exposition.
We did not have time to discuss chapters 13 (randomized block design) or 14 (non-parametric statistics). My impression is that they are less rigorous but give a good overall view of the basic ideas.
All in all, I would recommend this book to other instructors, and will recommend (actually require) it for future prob/stat students. The book appears to be at about the right level and is superior to the competition. That is despite the confused treatment of continuous random variables and the insistence on stating hypotheses tests as theorems.
As for the reviewer below, from Wharton: I give you credit for taking STAT-430 and not 101, but do you really believe the book is quite that bad?
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