Customer Reviews

An Introduction to Mathematical Statistics and Its Applications (3rd Edition)

5 star:		(7)
4 star:		(3)
3 star:		(1)
2 star:		(3)
1 star:		(3)

 

 

The first half of the book begins with basic discrete and continuous probability theory. It continues with thorough overviews of the basic distributions (normal, Poisson, binomial, multinomial, chi-squared and student-T). The focus is on basic probability and variance analysis, though it briefly covers higher-order moments.

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.

I used this as the text in a sequence on probability and statistics I taught recently, and I soon came to regret this choice. The authors are obviously quite confused about basic concepts. Here are some examples: the "definition" of the median ignores obvious problems with existence and uniqueness; the "proof" of the central limit theorem is thoroughly incomplete; the "theorems" on the tests in Sect. 9.2, 9.3 summarize previous discussions, but the "proofs" of these theorems (we are even referred to an appendix - no small surprise when the statements seem obvious) establish something entirely different; finally, to conclude this (very incomplete) selection, there is the delightful claim that the golden ratio is a transcendental number (which just proves that the authors don't have the slightest idea what a transcendental number really is, but then it might have been wise to avoid the use of the term).

In addition to these blatant problems, the authors' treatment frequently misses the point and/or is confusing.

This book manages to stay focused on the main ideas all the way through. It uses no more math than what is necessary to derive the proofs of most theorems (although some are omitted). The main ideas of each chapter is introduced before the details are worked out, and summarized at the end of each chapter. The examples and case-studies are usually interesting (sometimes thought-provoking), instead of solely being based on urns and coloured balls. And the exercises range from trivial to interesting...

In short, this is about as good as a textbook gets...

I am surprised by the number of negative reviews for what I consider to be a nicely written, well thought out, and logically presented introductory course on mathematical statistics. Yes, a working knowledge of elementary calculus is a prerequisite. But the mathematics invoked in the exposition of concepts and theorems are kept as simple as possible while maintaining that modest level of rigor appropriate for a introductory exposition. If you do not have the minimal mathematical prerequisites (such as freshman calculus), blame your instructor or your school for selecting an inappropriate text. But don't blame the authors! I thought the examples and problems were appropriate in their level of difficulty (mostly not so hard) and the relation to the material just covered. There are plenty of poorly written, impossibly dry, inpenetrable texts on statistics out there - this is not one of them. In addition, the book is attractively packaged, the paper quality is excellent, the visuals are informative and clearly presented - that also should not be taken for granted. Lastly the authors have a wicked entertaining sense of humor that spice the presentation throughout. I consider this book to be a welcome addition to the set of modern textbooks available to the curious serious student of probability and statistics.

This text is a good introduction to probability theory and mathematical statistics for the mathematically mature. It presumes a working knowledge of multivariate calculus and good facility with symbolic manipulation. The major shortcoming, though, is the small amount of sample problems and exercises after each section. As this book has not been revised since the mid-1980's, perhaps the authors will take this into account for the next edition (I await). Two particularly nice features of this book are the historical backgrounds of probability theory and the dry, dark humor that the authors interject. These make the book a bit more entertaining.

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?

I have just finished teaching a year of probability and statistics out of the fourth edition of this text. As I was teaching the course, it became clear just how difficult it is to write a mathematically rigorous undergraduate text in mathematical statistics. I selected this book because it seemed to be the best of the department recommended texts. For example, it is a bit more rigorous than Wackerly, Mendenhall, and SchaefferMathematical Statistics with Applications; significantly more rigorous than DevoreProbability and Statistics for Engineering and the Sciences; and just as rigorous but apparently more popular than Hogg and TannisProbability and Statistical Inference (8th Edition).
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.

I found the book to have clear explanations of many statistical concepts. I am a computer science graduate student with no formal background in statistics. I have looked at several statistics and probability theory textbooks and have found this one to be the easiest to understand and clearly written. That said, I am not using this book for a course, so I am not doing practice problems. The small number of problems for each section may be a shortcoming for those who are using this book for a course. Also, the proofs can be a little terse, although I have found this to be the case with other stat books.

The text presents all relevant information, but does so in such a confusing and poorly explained fashion as to prompt the reader to wonder if the authors have ever met anyone who hasn't known all subtleties of probability since the womb. There is no avenue for the student who does not understand, no pedagogy whatsoever. Everything is presented at lightning pace with blisteringly difficult proofs and, often, no meaningful explanation of the physical meaning of the concepts explained. A very solid background in calculus is an absolute necessity, to the point where many problems in the text are more challenging in evaluating integrals than they are in actually applying concepts. This is a serious problem that recurs over and over.

Examples worked out in the chapter sections also almost never bear any resemblance to the problems students are expected to complete. Although the examples vary in terms of difficulty, a student stuck on an exercise almost definitely will not find any help in the teaching material of the section in completing it simply because the examples never entirely cover the concepts demanded in the exercises.

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it is really good and as the discription. and you know iti s the firat time for me to write the review . i ddid not know this before ! oh my god , i never write this . in a word, it is really a happy experience for me to buy the book .thank you

I took up this book for my course in economics and i found the book clear and examples quite relevant. However, our calculus background was rather weak and we were left to study integrals by oursleves. Because of this, most of us floundered in the text and could not fully appreciate some of the more essential steps of the proofs. The solutions to many examples requires a solid background in calculus to fully appreciate and, at times, even understand since certain steps are ommited. So much so that our eco teacher, with a degree in engineering mind you, admitted the book was a bit too terse and spent most of the time explaining the calculus that we learnt very little of actual statistics.
In short, master your calculus or this book will only give you a rough feel for elementary statistics but will definitely not arm you to take up higher stat courses.