Essentials of Statistical Inference (Cambridge Series in Statistical and Probabilistic Mathematics, Series Number 16) by G. A. Young (PDF)

Description

This textbook presents the concepts and results underlying the Bayesian, frequentist, and Fisherian approaches to statistical inference, with particular emphasis on the contrasts between them. Aimed at advanced undergraduates and graduate students in mathematics and related disciplines, it covers basic mathematical theory as well as more advanced material, including such contemporary topics as Bayesian computation, higher-order likelihood theory, predictive inference, bootstrap methods, and conditional inference.

User’s Reviews

Editorial Reviews: Review “This is a delightful book! It gives a well-written exposure to inference issues in statistics, very suitable for a first-year graduate course…The authors present the material in a very good pedagogical manner. The examples are excellent, and the exercises are very instructive…very much up to date and includes recent developments in the field.” MAA Reviews”This is a solid book, ideal for advanced classes in the mathematical justification for statistical inference.” Journal of Recreational Mathematics”I wish that I had had such a textbook during my student days…this new book presents the core ideas of statistical inference in the unifying framework of decision theory and includes a fruitful discussion of the different foundational standpoints (Bayesian, Fisherian and frequentist)…[it is] sufficiently precise to satisfy a mathematician and yet omitting too much technical detail that could hide the core of the ideas. Carefully selected examples from a rainbow of application areas such as baseball, coal-mining disasters or gene expression data make it even more enjoyable to read…this book is a very nice graduate level textbook.” Journal of the Royal Statistical Society”[T]his book gives a clear and comprehensive account of the basic elements of statistical theory. It should make a good text for an advanced course on statistical inference…Students will find it informative and challenging.” ISI Short Book Reviews”Essentials of Statistical Inference is a book worth having.” Jane L. Harvill, Baylor University for the Journal of The American Statistician”The book is comprehensively written without dwelling in unnecessary details.” Iris Pigeot, Biometrics”This book is very unique in that the authors present the foundations of all three schools of inference and produce the essential theoretical results in each approach. This text also contains a great bibliography that is partially annotated. Readers should pay attention to the annotations as they are very enlightening. This book could easily be used for a modern first graduate level course in mathematical statistics.” Michael R. Chernick, Significance Book Description This textbook presents the concepts and results underlying the Bayesian, frequentist, and Fisherian approaches to statistical inference. About the Author G. A. Young is Professor of Statistics at Imperial College London.R. L. Smith is Mark L. Reed Distinguished Professor of Statistics at the University of North Carolina, Chapel Hill. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This short book covers all the major topics in statistical theory. I feel like the authors strike the perfect balance between building intuition and mathematical detail. Most standard results are accompanied by proofs. However instead of full mathematical generality, the proofs are often presented with simplified assumptions, that still bring out the essence of the main ideas. Measure theory is kept to the minimum, whenever it’s needed the authors use discrete setting instead.The book could be used as a refresher, reference, or a study guide to accompany another statistics book (such as Casella & Berger). It is definitely geared towards readers who have studied statistical inference before. The minimum mathematical prerequisites are: calculus at the level of elementary analysis, linear algebra, discrete and continuous probability, and familiarity with convergence of random variables. Basic group theory is used in one place to define transformation families.It’s incredible how much the authors cover in this short book. It starts out with decision theory in chapter 2, which is unifies frequentist and bayesian approaches to statistical inference under finding decisions to minimize risk. Chapter 3 is fully devoted to Bayesian analysis and covers conjugate priors, Bayesian confidence intervals, empirical bayes, hierarchical models, prediction, and computational techniques. There’s also a nice description of the James-Stein estimator, a surprising result of using seemingly unrelated data to improve an estimator. Chapters 4-8 then cover the standard statistical inference topics from frequentist perspective: hypothesis testing and interval estimation, sufficient and ancillary statistics, conditionality principle, exponential models, maximum likelihood and fisher information. I especially enjoyed chapter 8 on maximum likelihood, the authors do a superb job intuitively justifying the asymptotic normality of maximum likelihood estimators.Next three chapters deal with topics not usually encountered in first courses on statistics. Chapter 9 is a dense chapter on asymptotic density approximations. It deals with approximations such as CLT, but with more terms to improve the convergence speed. There is not much motivation or intuition here, most results are simply stated without proofs. It could serve as a good first exposure to this material, but to get any deeper understanding will require consulting other books devoted to this subject.Chapter 10 deals with predicting random variables and their distributions, rather than inference about unknown parameters. This is probably the closest chapter to what is currently known as “machine learning”. Here it is shown that better predictions can be made directly, instead of by first estimating the parameters and then plugging them into the distribution to make predictions. Usually the topic of formulating predictive distributions is covered only from the Bayesian viewpoint, but this chapter shows how it can also be done using the frequentist appoach. It covers several such methods: pivots, predictive likelihood, and bootstrap. This chapter is also short on intuition and proofs, and only scratches the surface of this subject. It could serve as a first exposure to build your interest, but you’ll have to explore the references to make it useful in applied work. The section on predictive likelihood looks very similar to the two papers “Predictive inference a review” by Bjornstad and “Predictive inference” by Hinkley. I suggest reading those papers instead, since the book doesn’t do such a good job motivating these ideas.Last Chapter 11 deals with bootstrap. This has now become a very popular method of building empirical distributions for estimators, without relying on analytic derivations or asymptotic approximations. Notation is very important when presenting bootstrap methods, since it needs to be kept clear which is the modeling distribution, which are the distributions for drawing bootstrap samples, and what are the various parameters and their estimates. I feel like the authors do a good job at presenting this material without ambiguity.Each chapter is relatively short and usually focuses on a few key concepts. Authors do a very good job using examples to motivate and elucidate the material. I would actually like to see even more examples, especially of various sufficient and ancillary statistics. The exercises following each chapter are not incredibly hard, and focus on applying the theory in various special cases, providing more examples to the reader.In summary, this is a very readable and concise summary of the main topics in statistical inference, with just enough mathematical detail to be interesting and elucidating, while not becoming too dry. I would definitely recommend it to anyone who have studied statistical theory before and would like another source of this material for reference.

⭐Not as clear and concise as I expected. I bought the book because of the three five-star reviews and I was disappointed.EDIT: I’ve returned recently to give this book another shot. I STRONGLY recommend against buying this as either a study tool or reference. First, the organization and general structure is insanely poor. It begins with decision theory, then Bayesian inference / shrinkage, then moves to Neyman Pearson lemma for UMP, then exponential families, then Fisher sufficiency, but then back to Neyman Pearson hypothesis testing mixed with Fisher’s conditional likelihood, but then back AGAIN to Fisherian likelihood theory, then a random collection of ideas grouped into “higher-order theory”, “predictive inference”, which are both more theoretical miscellany that the authors decided to toss in to pad out some pages, and a very bad review of the bootstrap (no bagging or connection to Bayesian methods). There is no attempt to sequence the material either thematically or historically. To really appreciate these ideas, a reader must understand the problem at-hand that motivated such solutions, and also understand the historical development of one concept to the other. Reading cover-to-cover is incredibly disjointed and without context, despite the single subsection 3.6 that gives superficial history of the Bayes-non-Bayes disagreement. The more complex decision theory problems should not precede MLE or hypothesis testing, since the whole theory was developed from multi-stage testing and experiment design. Second, the writing style is mostly inscrutable in the way that most bad math writing is: here’s a proof, we review this, here’s a definition, lemma, etc. without any attempt to situate the abstract definitions. Never mind that in an age of enhanced computational techniques, this book is hopelessly outdated in helping a practitioner in a non-parametric, non-experimental setting, since it devotes so much space to asymptotic optimality theory without unifying the various historical constructions. I would recommend directly reading Fisher, Neyman-Pearson, Wald, and Efron, and not this poor excuse for synthesis.

⭐There are many excellent books for a first graduate level course in mathematical statistics. It is quite common to see texts such as Lehmann’s book on hypothesis testing and his second one on estimation based on the frequentist approach. There are also at this time, many books written from the Bayesian approach with DeGroot’s being one of the earliest. At this time it is rare to see a text written from the Fisherian approach partly because the concept of fiducial inference has largely been discredited. Of course Fisher’s texts were written from his perspective although not always so clearly illucidated.This book is very unique in that the authors present the foundations of all three schools of inference and produce the essential theoretical results in each approach. The book is concise but provides the key theorems and results of the methods developed in the 20th century. It includes many of the modern advances of the latter portion of the twentieth century from 1980 – 2000. This includes resampling methods especially the bootstrap which has its own chapter (Chapter 11).The book also contains a great bibliography that is partially annotated. Readers should pay attention to the annotations as they are very enlightening. Another computer-intensive method, Markov Chain Monte Carlo, is covered under the Bayesian paradigm where it has very important applications. This book could easily be used for a modern first graduate level course in mathematical statistics.

⭐A modern account of the theory of statistical inference comparing the Fisherian, Bayesian and frequentists paradigms. It is clear, concise and direct. Not so easy to follow if you, like me, do not have a string background in maths. However, with some googling every now and then following this book is not impossible for non mathematicians. i really recommend it for anybody who needs to understand the fundamentals and more.

⭐This book can be described by its title. Essentials, nothing more. Very useful, but for PhD students level and above.

⭐This review is being written even before having used the book to study.As soon as I got this book I was so disappointed…Yes, it’s a scientific work, but to make justice to its content some care must be taken.One does not simply publish a manual like this, with this quality… The paper quality is so lousy that reminds me that of the lousiest pocket books/paperback that I’ve ever bought. Another thing really bad is the font size. I might be getting old, but the font size is similar to some very very cheap pocket size book edition with impossible-to-read font size.What did go through the publisher’s head to publish this very good, content-wise, book, with this material quality???Furthermore, how did the authors allowed it?The material is so bad, that one is tempted just to find a online version, and do a home printing, with the text fitted to some decent A4 pages.Cambridge, Cambridge…where art thou?

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