Introduction to Mathematical Statistics

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9780023557224: Introduction to Mathematical Statistics

An exceptionally clear and impeccably accurate presentation of statistical applications and more advanced theory. Included is a chapter on the distribution of functions of random variables as well as an excellent chapter on sufficient statistics. More modern technology is used in considering limiting distributions, making the presentations more clear and uniform.

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Exceptionally clear and impeccably accurate, the fifth edition of this trusted text offers a careful presentation of the probability needed for mathematical statistics and the mathematics of statistical inference. Offering a strong background for those who wish to go on to study statistical applications or more advanced theory, this text presents the most thorough treatment of the mathematics of statistics of any competing text.

Excerpt. Reprinted by permission. All rights reserved.:

Since Allen T. Craig's death in 1978, Bob Hogg has revised the later editions of this text. However, when Prentice Hall asked him to consider a sixth edition, he thought of his good friend, Joe McKean, and asked him to help. That was a great choice for Joe made many excellent suggestions on which we both could agree and these changes are outlined later in this preface.

In addition to Joe's ideas, our colleague Jon Cryer gave us his marked up copy of the fifth edition from which we changed a number of items. Moreover, George Woodworth and Kate Cowles made a number of suggestions concerning the new Bayesian chapter; in particular, Woodworth taught us about a "Dutch book" used in many Bayesian proofs. Of course, in addition to these three, we must thank others, both faculty and students, who have made worthwhile suggestions. However, our greatest debts of gratitude are for our special friend, Tom Hettmansperger of Penn State University, who used our revised notes in his mathematical statistics course during the 2002-2004 academic years and Suzanne Dubnicka of Kansas State University who used our notes in her mathematical statistics course during Fall of 2003. From these experiences, Tom and Suzanne and several of their students provided us with new ideas and corrections.

While in earlier editions, Hogg and Craig had resisted adding any "real" problems, Joe did insert a few among his more important changes. While the level of the book is aimed for beginning graduate students in Statistics, it is also suitable for senior undergraduate mathematics, statistics and actuarial science majors.

The major differences between this edition and the fifth edition are:

  • It is easier to find various items because more definitions, equations, and theorems are given by chapter, section, and display numbers. Moreover, many theorems, definitions, and examples are given names in bold faced type for easier reference.
  • Many of the distribution finding techniques, such as transformations and moment generating methods, are in the first three chapters. The concepts of expectation and conditional expectation are treated more thoroughly in the first two chapters.
  • Chapter 3 on special distributions now includes contaminated normal distributions, the multivariate normal distribution, the t- and F-distributions, and a section on mixture distributions.
  • Chapter 4 presents large sample theory on convergence in probability and 1 distribution and ends with the Central Limit Theorem. In the first semester, if the instructor is pressed for time he or she can omit this chapter and proceed to Chapter 5.
  • To enable the instructor to include some statistical inference in the first semester, Chapter 5 introduces sampling, confidence intervals and testing. These include many of the normal theory procedures for one and two sample location problems and the corresponding large sample procedures. The chapter concludes with an introduction to Monte Carlo techniques and bootstrap procedures for confidence intervals and testing. These procedures are used throughout the later chapters of the book.
  • Maximum likelihood methods, Chapter 6, have been expanded. For illustration, the regularity conditions have been listed which allows us to provide better proofs of a number of associated theorems, such as the limiting distributions of the maximum likelihood procedures. This forms a more complete inference for these important methods. The EM algorithm is discussed and is applied to several maximum likelihood situations.
  • Chapters 7-9 contain material on sufficient statistics, optimal tests of hypotheses, and inferences about normal models.
  • Chapters 10-12 contain new material. Chapter 10 presents nonparametric procedures for the location models and simple linear regression. It presents estimation and confidence intervals as well as testing. Sections on optimal scores and adaptive methods are presented. Chapter 11 offers an introduction to Bayesian methods. This includes traditional Bayesian procedures as well as Markov Chain Monte Carlo procedures, including the Gibbs sampler, for hierarchical and empirical Bayes procedures. Chapter 12 offers a comparison of robust and traditional least squares methods for linear models. It introduces the concepts of influence functions and breakdown points for estimators. Not every instructor will include these new chapters in a two-semester course, but those interested in one of these areas will find their inclusion very worthwhile. These last three chapters are independent of one another.
  • We have occasionally made use of the statistical softwares R, (Ihaka and Gentleman, 1996), and S-PLUS, (S-PLUS, 2000), in this edition; see Venables and Ripley (2002). Students do not need resource to these packages to use the text but the use of one (or that of another package) does add a computational flavor. The package R is freeware which can be downloaded for free at the site: http://lib.stat.cmu.edu/R/CRAN/
  • There are versions of R for unix, pc and mac platforms. We have written some R functions for several procedures in the text. These we have listed in Appendix B but they can also be downloaded at the site: http://www.stat.wmich.edu/mckean/HMC/Rcode - These functions will run in S-PLUS also.
  • The reference list has been expanded so that instructors and students can find the original sources better.
  • The order of presentation has been greatly improved and more exercises have been added. As a matter of fact, there are now over one thousand exercises and, further, many new examples have been added.

Most instructors will find selections from the first nine chapters sufficient for a two-semester course. However, we hope that many will want to insert one of the three topic chapters into their course. As a matter of fact, there is really enough material for a three semester sequence, which at one time we taught at the University of Iowa. A few optional sections have been marked with an asterisk.

We would like to thank the following reviewers who read through earlier versions of the manuscript: Walter Freiberger, Brown University; John Leahy, University of Oregon; Bradford Crain, Portland State University; Joseph S. Verducci, Ohio State University. and Hosam M. Mahmoud, George Washington University. Their suggestions were helpful in editing the final version.

Finally, we would like to thank George Lobell and Prentice Hall who provided funds to have the fifth edition converted to LATEX 2e and Kimberly Crimin who carried out this work. It certainly helped us in writing the sixth edition in LATF,X 2E. Also, a special thanks to Ash Abebe for technical assistance. Last, but not least, we must thank our wives, Ann and Marge, who provided great support for our efforts. Let's hope the readers approve of the results.

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