Page 13 - Probability and Statistical Inference

vi Preface

scale and location-scale families of distributions are also addressed.
The method of moment estimator, maximum likelihood estimator (MLE),
Rao-Blackwell Theorem, Rao-Blackwellization, Cramér-Rao inequality,
uniformly minimum variance unbiased estimator (UMVUE) and Lehmann-
Scheffé Theorems are developed in Chapter 7. Chapter 8 provides the
Neyman-Pearson theory of the most powerful (MP) and uniformly most
powerful (UMP) tests of hypotheses as well as the monotone likelihood
ratio (MLR) property. The concept of a UMP unbiased (UMPU) test is
briefly addressed in Section 8.5.3. The confidence interval and confidence
region methods are elaborated in Chapter 9. Chapter 10 is devoted entirely
to the Bayesian methods for developing the concepts of the highest posterior
density (HPD) credible intervals, the Bayes point estimators and tests of
hypotheses.
Two-sided alternative hypotheses, likelihood ratio (LR) and other tests
are developed in Chapter 11. Chapter 12 presents the basic ideas of large-
sample confidence intervals and test procedures, including variance stabilizing
transformations and properties of MLE. In Section 12.4, I explain how one
p
–1
arrives at the customary sin ( ), , and tanh (ρ) transformations in the
–1
case of Binomial (p), Poisson (λ), and the correlation coefficient ρ,
respectively.
Chapter 13 introduces two-stage sampling methodologies for determining
the required sample size needed to solve two simple problems in statistical
inference for which, unfortunately, no fixed-sample-size solution exists. This
material is included to emphasize that there is much more to explore beyond
what is customarily covered in a standard one-year statistics course based
on Chapters 1 -12.
Chapter 14 (Appendix) presents (i) a list of notation and abbreviations,
(ii) short biographies of selected luminaries, and (iii) some of the standard
statistical tables computed with the help of MAPLE. One can also find
some noteworthy remarks and examples in the section on statistical tables.
An extensive list of references is then given, followed by a detailed index.
In a two-semester sequence, probability theory is covered in the first
part, followed by statistical inference in the second. In the first semester,
the core material may consist of Chapters 1-4 and some parts of Chapter 5.
In the second semester, the core material may consist of the remainder of
Chapter 5 and Chapters 6-10 plus some selected parts of Chapters 11-13.
The book covers more than enough ground to allow some flexibility in the
selection of topics beyond the core. In a three-quarter system, the topics
will be divided somewhat differently, but a year’s worth of material taught
in either a two-semester or three-quarter sequence will be similar.