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1982 | Buch

Theory of Statistical Experiments

verfasst von: H. Heyer

Verlag: Springer New York

Buchreihe : Springer Series in Statistics

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Über dieses Buch

By a statistical experiment we mean the procedure of drawing a sample with the intention of making a decision. The sample values are to be regarded as the values of a random variable defined on some meas­ urable space, and the decisions made are to be functions of this random variable. Although the roots of this notion of statistical experiment extend back nearly two hundred years, the formal treatment, which involves a description of the possible decision procedures and a conscious attempt to control errors, is of much more recent origin. Building upon the work of R. A. Fisher, J. Neyman and E. S. Pearson formalized many deci­ sion problems associated with the testing of hypotheses. Later A. Wald gave the first completely general formulation of the problem of statisti­ cal experimentation and the associated decision theory. These achieve­ ments rested upon the fortunate fact that the foundations of probability had by then been laid bare, for it appears to be necessary that any such quantitative theory of statistics be based upon probability theory. The present state of this theory has benefited greatly from contri­ butions by D. Blackwell and L. LeCam whose fundamental articles expanded the mathematical theory of statistical experiments into the field of com­ parison of experiments. This will be the main motivation for the ap­ proach to the subject taken in this book.

Inhaltsverzeichnis

Frontmatter
Chapter I. Games and Statistical Decisions
Abstract
We start with an introduction to the basic notions and properties of two-person zero sum games and their randomizations. Much emphasis is given to the interpretations of the formal definitions. A few standard examples indicate the route from the theory of games to statistical decision theory.
H. Heyer
Chapter II. Sufficient σ-Algebras and Statistics
Abstract
A notion from the theory of probability which proves to be of basic interest in mathematical statistics, is the conditional probability of a measure with respect to a σ-algebra. In particular we are interested in such σ-algebras, for which there are versions of the corresponding conditional probability that are independent of the individual measures involved. To make this idea precise we present
H. Heyer
Chapter III. Sufficiency under Additional Assumptions
Abstract
In this section we pose the question of how far we can simplify or strengthen certain results concerning sufficiency if the given experiment (Ω,A,P) admits a separable a-algebra A.
H. Heyer
Chapter IV. Testing Experiments
Abstract
The theory of testing statistical hypotheses is based on the notions of testing experiments and tests, which will be introduced purely measure theoretically. Once first results have been established, these notions will gain their concrete statistical meaning. Till now they have been only roughly described, in Examples 3.10 and 3.13.1 of the game theoretical set-up.
H. Heyer
Chapter V. Testing Experiments Admitting an Isotone Likelihood Quotient
Abstract
Testing experiments with an isotone likelihood quotient arise whenever one considers a special class of parametrized experiments (Ω,A, P; χ: P→IR) and investigates those testing experiments (Ω,A, P P0,P1) which are consistent with the given parametrization. Here, parametrizations are understood to be infective mappings χ: P+IR. We shall put θ: = χ (P), and for every θ ∈ Θ we write Pθ: = χ−1(θ). In this context, the mapping βt: Θ → [0,1] defined by
$$ {\beta_t}(\theta ): = \int t \,d{P_{\theta }} = {E_{\theta }}(t) $$
for all θ ∈ Θ appears to be the power function of the test t ∈ℳ(1) (Ω, A).
H. Heyer
Chapter VI. Estimation Experiments
Abstract
In this section we shall deal with parametrized experiments (not necessarily injectively parametrized) whose parameter set is IRk for k ≥ 1. As in Chapter IV we are going to study properties of these parametrized experiments with respect to a given class of functions. While the class of functions considered previously — we chose the set ℳ(1) (Ω,A) of all test functions on (Ω,A) — depends only on the underlying measurable space (Ω,A), we shall now admit more specific classes of functions which are more closely adapted to the given experiment.
H. Heyer
Chapter VII. Information and Sufficiency
Abstract
In this section we introduce the decision theoretic basis to our approach to mathematical statistics. We restrict ourselves to the purely measure theoretic version of decision theory. The section contains the fundamental definitions, some interpretations, and some theorems of an introductory nature designed to give a first insight into the theory of comparison of experiments.
H. Heyer
Chapter VIII. Invariance and the Comparison of Experiments
Abstract
Invariant Markov kernels can be used with success whenever the general theory of comparison of experiments is applied to special classes of experiments like those classical experiments involving location parameters. Our first aim in this section is a strengthening of LeCam’s Markov kernel criterion in the case of invariant experiments.
H. Heyer
Chapter IX. Comparison of Finite Experiments
Abstract
In Section 19 we dealt with general decision problems of the form \( \bar{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{D}}} = (I,D,V) \)= (I,D,V), where I: = (ΩI,AI) and D: = (ΩD,AD) denoted measurable spaces and V a set of separately measurable functions on ΩI × ΩD. From now on, and for the remainder of the chapter, we shall specialize the general framework in two steps: First we shall restrict our attention to decision problems of the form <Inline>2</Inline>k(I): = (I,Dk,V) with Dk: = {1,…,k} (k ≥1) as the decision space and the set V of all bounded, separately measurable functions on ΩI × ΩD as the set of loss functions. Later we shall consider decision problems of the form \( {\bar{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{D}}}_k}\,(I): = (I,{D_k},V) \)k (In) with In = {l, …, n} (n ≥ 1).
H. Heyer
Chapter X. Comparison with Extremely Informative Experiments
Abstract
The topic of this section refers to the comparison of experiments with respect to apriori measures which have been introduced in Section 3. There we formulated the Bayesian principle as one of the basic ideas of modern statistics. Although we did not put much emphasis on the Bayesian approach throughout the exposition we intend at least to touch upon the general scope in handling a few interesting types of examples: We shall study deviations from total information and from total ignorance as measures of information. In other words we shall compute the deficiencies of experiments relative to totally informative and totally uninformative ones respectively. For the corresponding computations apriori distributions are of great value.
H. Heyer
Backmatter
Metadaten
Titel
Theory of Statistical Experiments
verfasst von
H. Heyer
Copyright-Jahr
1982
Verlag
Springer New York
Electronic ISBN
978-1-4613-8218-8
Print ISBN
978-1-4613-8220-1
DOI
https://doi.org/10.1007/978-1-4613-8218-8