(Ebook) Modes of Parametric Statistical Inference by Seymour Geisser, Wesley M. Johnson ISBN 9780471667261, 0471667269

(Ebook) Modes of Parametric Statistical Inference by Seymour Geisser, Wesley M. Johnson ISBN 9780471667261, 0471667269

This book provides a graduate level discussion of four basic modes of statistical

inference: (i) frequentist, (ii) likelihood, (iii) Bayesian and (iv) Fisher’s fiducial

method. Emphasis is given throughout on the foundational underpinnings of these

four modes of inference in addition to providing a moderate amount of technical

detail in developing and critically analyzing them. The modes are illustrated with

numerous examples and counter examples to highlight both positive and potentially

negative features. The work is heavily influenced by the work three individuals:

George Barnard, Jerome Cornfield and Sir Ronald Fisher, because of the author’s

appreciation of and admiration for their work in the field. The clear intent of the

book is to augment a previously acquired knowledge of mathematical statistics by

presenting an overarching overview of what has already been studied, perhaps

from a more technical viewpoint, in order to highlight features that might have

remained salient without taking a further, more critical, look. Moreover, the

author has presented several historical illustrations of the application of various

modes and has attempted to give corresponding historical and philosophical perspectives

on their development.

The basic prerequisite for the course is a master’s level introduction to probability

and mathematical statistics. For example, it is assumed that students will have

already seen developments of maximum likelihood, unbiased estimation and

Neyman-Pearson testing, including proofs of related results. The mathematical

level of the course is entirely at the same level, and requires only basic calculus,

though developments are sometimes quite sophisticated. There book is suitable

for a one quarter, one semester, or two quarter course. The book is based on a

two quarter course in statistical inference that was taught by the author at the University

of Minnesota for many years. Shorter versions would of course involve

selecting particular material to cover.

Chapter 1 presents an example of the application of statistical reasoning by the

12th century theologian, physician and philosopher, Maimonides, followed by a discussion

of the basic principles guiding frequentism in Chapter 2. The law of likelihood

is then introduced in Chapter 3, followed by an illustration involving the

assessment of genetic susceptibility, and then by the various forms of the likelihood

principle. Significance testing is introduced and comparisons made between likelihood

and frequentist based inferences where they are shown to disagree. Principles

of conditionality are introduced.

Chapter 4, entitled “Testing Hypotheses” covers the traditional gamut of material

on the Neyman-Pearson (NP) theory of hypothesis testing including most powerful

(MP) testing for simple versus simple and uniformly most powerful testing (UMP)

for one and two sided hypotheses. A careful proof of the NP fundamental lemma is

given. The relationship between likelihood based tests and NP tests is explored

through examples and decision theory is introduced and briefly discussed as it relates

to testing. An illustration is given to show that, for a particular scenario without the

monotone likelihood ratio property, a UMP test exists for a two sided alternative.

The chapter ends by showing that a necessary condition for a UMP test to exist in

the two sided testing problem is that the derivative of the log likelihood is a nonzero

constant.

Chapter 5 discusses unbiased and invariant tests. This proceeds with the usual

discussion of similarity and Neyman structure, illustrated with several examples.

The sojourn into invariant testing gives illustrations of the potential pitfalls of this

approach. Locally best tests are developed followed by the construction of likelihood

ratio tests (LRT). An example of a worse-than-useless LRT is given. It is

suggested that pre-trial test evaluation may be inappropriate for post-trial evaluation.

Criticisms of the NP theory of testing are given and illustrated and the chapter

ends with a discussion of the sequential probability ratio test.

Chapter 6 introduces Bayesianism and shows that Bayesian testing for a simple

versus simple hypotheses is consistent. Problems with point null and composite

alternatives are discussed through illustrations. Issues related to prior selection in

binomial problems are discussed followed by a presentation of de Finetti’s theorem

for binary variates. This is followed by de Finetti’s proof of coherence of the

Bayesian method in betting on horse races, which is presented as a metaphor for

making statistical inferences. The chapter concludes with a discussion of Bayesian

model selection.

Chapter 7 gives an in-depth discussion of various theories of estimation. Definitions

of consistency, including Fisher’s, are introduced and illustrated by example.

Lower bounds on the variance of estimators, including those of Cramer-Rao and

Bhattacharya, are derived and discussed. The concepts of efficiency and Fisher

information are developed and thoroughly discussed followed by the presentation

of the Blackwell-Rao result and Bayesian sufficiency. Then a thorough development

of the theory of maximum likelihood estimation is presented, and the chapter

concludes with a discussion of the implications regarding relationships among the

various statistical principles.

The last chapter, Chapter 8, develops set and interval estimation. A quite general

method of obtaining a frequentist confidence set is presented and illustrated, followed

by discussion of criteria for developing intervals including the concept of

conditioning on relevant subsets, which was originally introduced by Fisher. The

use of conditioning is illustrated by Fisher’s famous “Problem of the Nile.” Bayesian

interval estimation is then developed and illustrated, followed by development of

Fisher’s fiducial inference and a rather thorough comparison between it and

Bayesian inference. The chapter and the book conclude with two complex but relevant

illustrations, first the Fisher-Behrens problem, which considered inferences

for the difference in means for the two sample normal problem with unequal variances,

and the second, the Fieller-Creasy problem in the same setting but making

inferences about the ratio of two means.