Principles of statistical inference
Cox, David RPreface
Most statistical work is concerned directly with the provision and implementation
of methods for study design and for the analysis and interpretation of data.
The theory of statistics deals in principle with the general concepts underlying
all aspects of suchwork and from this perspective the formal theory of statistical
inference is but a part of that full theory. Indeed, from the viewpoint of individual
applications, it may seem rather a small part. Concern is likely to be more
concentrated on whether models have been reasonably formulated to address
the most fruitful questions, on whether the data are subject to unappreciated
errors or contamination and, especially, on the subject-matter interpretation of
the analysis and its relation with other knowledge of the field.
Yet the formal theory is important for a number of reasons. Without some
systematic structure statistical methods for the analysis of data become a collection
of tricks that are hard to assimilate and interrelate to one another, or
for that matter to teach. The development of new methods appropriate for new
problems would become entirely a matter of ad hoc ingenuity. Of course such
ingenuity is not to be undervalued and indeed one role of theory is to assimilate,
generalize and perhaps modify and improve the fruits of such ingenuity.
Much of the theory is concerned with indicating the uncertainty involved in
the conclusions of statistical analyses, and with assessing the relative merits of
different methods of analysis, and it is important even at a very applied level to
have some understanding of the strengths and limitations of such discussions.
This is connected with somewhat more philosophical issues connected with
the nature of probability. A final reason, and a very good one, for study of the
theory is that it is interesting.
The object of the present book is to set out as compactly as possible the
key ideas of the subject, in particular aiming to describe and compare the main
ideas and controversies over more foundational issues that have rumbled on at
varying levels of intensity for more than 200 years. I have tried to describe the
various approaches in a dispassionate way but have added an appendix with a
more personal assessment of the merits of different ideas.
Some previous knowledge of statistics is assumed and preferably some
understanding of the role of statistical methods in applications; the latter
understanding is important because many of the considerations involved are
essentially conceptual rather than mathematical and relevant experience is
necessary to appreciate what is involved.
The mathematical level has been kept as elementary as is feasible and is
mostly that, for example, of a university undergraduate education in mathematics
or, for example, physics or engineering or one of the more quantitative
biological sciences. Further, as I think is appropriate for an introductory discussion
of an essentially applied field, the mathematical style used here eschews
specification of regularity conditions and theorem–proof style developments.
Readers primarily interested in the qualitative concepts rather than their development
should not spend too long on the more mathematical parts of the
book.
The discussion is implicitly strongly motivated by the demands of applications,
and indeed it can be claimed that virtually everything in the book has
fruitful application somewhere across the many fields of study to which statistical
ideas are applied. Nevertheless I have not included specific illustrations.
This is partly to keep the book reasonably short, but, more importantly, to focus
the discussion on general concepts without the distracting detail of specific
applications, details which, however, are likely to be crucial for any kind of
realism.
The subject has an enormous literature and to avoid overburdening the reader
I have given, by notes at the end of each chapter, only a limited number of key
references based on an admittedly selective judgement. Some of the references
are intended to give an introduction to recentwork whereas others point towards
the history of a theme; sometimes early papers remain a useful introduction to
a topic, especially to those that have become suffocated with detail. A brief
historical perspective is given as an appendix.
The book is a much expanded version of lectures given to doctoral students of
the Institute of Mathematics, Chalmers/Gothenburg University, and I am very
grateful to Peter Jagers and NannyWermuth for their invitation and encouragement.
It is a pleasure to thank Ruth Keogh, Nancy Reid and Rolf Sundberg for
their very thoughtful detailed and constructive comments and advice on a preliminary
version. It is a pleasure to thank also Anthony Edwards and Deborah
Mayo for advice on more specific points. I am solely responsible for errors of
fact and judgement that remain.
introductory,
setting out the formulation of problems, outlining in a simple case
the nature of frequentist and Bayesian analyses, and describing some special
models of theoretical and practical importance. The discussion continues with
the key ideas of likelihood, sufficiency and exponential families.
Chapter 4 develops some slightly more complicated applications. The long
Chapter 5 is more conceptual, dealing, in particular, with the various meanings
of probability as it is used in discussions of statistical inference. Most of the key
concepts are in these chapters; the remaining chapters, especially Chapters 7
and 8, are more specialized.
Especially in the frequentist approach, many problems of realistic complexity
require approximate methods based on asymptotic theory for their resolution
and Chapter 6 sets out the main ideas. Chapters 7 and 8 discuss various complications
and developments that are needed from time to time in applications.
Chapter 9 deals with something almost completely different, the possibility
of inference based not on a probability model for the data but rather on
randomization used in the design of the experiment or sampling procedure.
I have written and talked about these issues for more years than it is comfortable
to recall and am grateful to all with whom I have discussed the topics,
especially, perhaps, to those with whom I disagree. I am grateful particularly
to David Hinkley with whom I wrote an account of the subject 30 years ago.
The emphasis in the present book is less on detail and more on concepts but the
eclectic position of the earlier book has been kept.