统计推断原理详解：方法与理论的交融

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《统计推断原则》是一本由D.R. Cox撰写的重要著作，它系统地探讨了统计推断理论的核心概念，重点关注了自18世纪以来在统计学基础问题上存在的主要观点和争议。作为统计思想领域的长期贡献者，Cox教授以其深厚的专业素养提供了当前统计领域所需的全面且平衡的阐述。书中特别关注了频率主义和贝叶斯方法两种统计推断的主要途径，通过深入比较，读者能够自行评价这两种方法的优势和不足。两个附录分别提供了统计历史的简短回顾和作者对不同理念价值的个人见解，使得读者可以从更广阔的历史视角来理解和评估统计学的发展。尽管这本书并未专门处理具体应用案例，但其内容深受科学和相关技术领域应用的驱动，强调理论与实践的紧密联系。数学基础被尽可能地保持简洁，但需要读者具备一定的统计学基础知识。本书适合所有严肃的统计用户或学生，尤其是那些希望深入了解统计推断原理以及各种方法如何应用于实际问题的人士。通过阅读《统计推断原则》，读者不仅能掌握理论框架，还能提升批判性思维和方法选择的能力，从而更好地应对日益复杂的数据分析挑战。

Preface xv

The book is in broadly three parts. The ﬁrst three chapters are largely intro-

ductory, 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 com-

plications and developments that are needed from time to time in applications.

Chapter 9 deals with something almost completely different, the possibil-

ity 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 com-

fortable 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.

I appreciate greatly the care devoted to this book by Diana Gillooly, Com-

missioning Editor, and Emma Pearce, Production Editor, Cambridge University

Press.

2 Preliminaries

where θ ⊂ 

is unknown. The distribution may depend also on design fea-

tures of the study that generated the data. We typically simplify the notation to

(y; θ), although the explanatory variables z are frequently essential in speciﬁc

applications.

To choose the model appropriately is crucial to fruitful application.

We follow the very convenient, although deplorable, practice of using the term

density both for continuous random variables and for the probability function

of discrete random variables. The deplorability comes from the functions being

dimensionally different, probabilities per unit of measurement in continuous

problems and pure numbers in discrete problems. In line with this convention

in what follows integrals are to be interpreted as sums where necessary. Thus

we write

E(Y ) = E(Y ; θ) =



(y; θ)dy (1.2)

for the expectation of Y, showing the dependence on θ only when relevant. The

integral is interpreted as a sum over the points of support in a purely discrete case.

Next, for each aspect of the research question we partition θ as (ψ , λ), where ψ

is called the parameter of interest and λ is included to complete the speciﬁcation

and commonly called a nuisance parameter. Usually, but not necessarily, ψ and

λ are variation independent in that 

is the Cartesian product 

×

. That

is, any value of ψ may occur in connection with any value of λ. The choice of

ψ is a subject-matter question. In many applications it is best to arrange that ψ

is a scalar parameter, i.e., to break the research question of interest into simple

components corresponding to strongly focused and incisive research questions,

but this is not necessary for the theoretical discussion.

It is often helpful to distinguish between the primary features of a model

and the secondary features. If the former are changed the research questions of

interest have either been changed or at least formulated in an importantly differ-

ent way, whereas if the secondary features are changed the research questions

are essentially unaltered. This does not mean that the secondary features are

unimportant but rather that their inﬂuence is typically on the method of estima-

tion to be used and on the assessment of precision, whereas misformulation of

the primary features leads to the wrong question being addressed.

We concentrate on problems where 

is a subset of R

, i.e., d-dimensional

real space. These are so-called fully parametric problems. Other possibilities

are to have semiparametric problems or fully nonparametric problems. These

typically involve fewer assumptions of structure and distributional form but

usually contain strong assumptions about independencies. To an appreciable

1.3 Some simple models 3

extent the formal theory of semiparametric models aims to parallel that of

parametric models.

The probability model and the choice of ψ serve to translate a subject-matter

question into a mathematical and statistical one and clearly the faithfulness of

the translation is crucial. To check on the appropriateness of a new type of model

to represent a data-generating process it is sometimes helpful to consider how

the model could be used to generate synthetic data. This is especially the case

for stochastic process models. Understanding of new or unfamiliar models can

be obtained both by mathematical analysis and by simulation, exploiting the

power of modern computational techniques to assess the kind of data generated

by a speciﬁc kind of model.

1.2 Role of formal theory of inference

The formal theory of inference initially takes the family of models as given and

the objective as being to answer questions about the model in the light of the

data. Choice of the family of models is, as already remarked, obviously crucial

but outside the scope of the present discussion. More than one choice may be

needed to answer different questions.

A second and complementary phase of the theory concerns what is sometimes

called model criticism, addressing whether the data suggest minor or major

modiﬁcation of the model or in extreme cases whether the whole focus of

the analysis should be changed. While model criticism is often done rather

informally in practice, it is important for any formal theory of inference that it

embraces the issues involved in such checking.

1.3 Some simple models

General notation is often not best suited to special cases and so we use more

conventional notation where appropriate.

Example 1.1. The normal mean. Whenever it is required to illustrate some

point in simplest form it is almost inevitable to return to the most hackneyed

of examples, which is therefore given ﬁrst. Suppose that Y

, ..., Y

are inde-

pendently normally distributed with unknown mean µ and known variance σ

Here µ plays the role of the unknown parameter θ in the general formulation.

In one of many possible generalizations, the variance σ

also is unknown. The

parameter vector is then (µ, σ

). The component of interest ψ would often be µ

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