Jay L. Devore - Modern Mathematical Statistics with Applications (Springer Texts in Statistics)

A number of currently available mathematical statistics texts are heavily oriented toward a rigorous mathematical development of probability and statistics, with much emphasis on theorems, proofs, and derivations. The focus is more on mathematics than on statistical practice. Even when applied material is included, the scenarios are often contrived (many examples and exercises involving dice, coins, cards, widgets, or a comparison of treatment A to treatment B).

Our exposition is an attempt to provide a reasonable balance between mathematical rigor and statistical practice. We believe that showing students the applicability of statistics to real-world problems is extremely effective in inspiring them to pursue further coursework and even career opportunities in statistics. Opportunities for exposure to mathematical foundations will follow in due course. In our view, it is more important for students coming out of this course to be able to carry out and interpret the results of a two-sample t test or simple regression analysis, and appreciate how these are based on underlying theory, than to manipulate joint moment generating functions or discourse on various modes of convergence.

The book certainly does include core material in probability (Chap. ) is definitely more oriented to dealing with real data than with theoretical properties of models. We also show many examples of output from commonly used statistical software packages, something noticeably absent in most other books pitched at this audience and level.

The challenge for students at this level should lie with mastery of statistical concepts as well as with mathematical wizardry. Consequently, the mathematical prerequisites and demands are reasonably modest. Mathematical sophistication and quantitative reasoning ability are certainly important, especially as they facilitate dealing with symbolic notation and manipulation. Students with a solid grounding in univariate calculus and some exposure to multivariate calculus should feel comfortable with what we are asking of them. The few sections where matrix algebra appears (transformations in Chap. ) can easily be deemphasized or skipped entirely. Proofs and derivations are included where appropriate, but we think it likely that obtaining a conceptual understanding of the statistical enterprise will be the major challenge for readers.

There should be more than enough material in our book for a year-long course. Those wanting to emphasize some of the more theoretical aspects of the subject (e.g., moment generating functions, conditional expectation, transformations, order statistics, sufficiency) should plan to spend correspondingly less time on inferential methodology in the latter part of the book. We have opted not to mark certain sections as optional, preferring instead to rely on the experience and tastes of individual instructors in deciding what should be presented. We would also like to think that students could be asked to read an occasional subsection or even section on their own and then work exercises to demonstrate understanding, so that not everything would need to be presented in class. Remember that there is never enough time in a course of any duration to teach students all that wed like them to know!