Keywords: Probability Theory, Statistics.
Notes
Quotes
Table of Contents
- Mathematics and Chance p.1
- Principles of Modelling Chance p.7
- Stochastic Standard Models p.26
- Conditional Probabilities and Independence p.50
- Expectation and Variance p.90
- The Law of Large Numbers and the Central Limit Theorem p.117
- Markov Chains p.149
- Estimation p.187
- Around the Normal Distributions p.241
- Hypothesis Testing p.255
- Asymptotic Tests and Rank Tests p.283
- Regression Models and Analysis of Variance p.318
Overview
- This book is a translation of the third edition of the highly successful German textbook "Stochastik." This new English edition presents the fundamental ideas and results of both probability theory and statistics, and comprises the material of a one-year course. The stochastic concepts, models and methods are motivated by examples and problems and then developed and analysed systematically.
8. Estimation
p.190
- Definition. A statistical model is a triple (X F, Pv : v ∈ ϴ) consisting of a sample space X, a σ-algebra F on X, and a class {Pv : v ∈ ϴ} of (at least two) probability measures on (X, F), ,which are indexed by an Index Set ϴ.
- Since we have to deal with many (or at least two) different probability measures, we must indicate the respective probability measure when taking expectations. Therefore we write
Ev for the expectation and Vv for the variance with respect to Pv. - Although self-evident, we would like to emphasize that the first basic task of a statistician is to choose the right statistical model! For, it is evident that a statistical procedure can only make sense if the underlying class of probability measures is appropiate for (or, at least, an acceptable approximation to) the application at hand.
- The statistical models that will be considered later on will typically have one of the basic following properties.
- Definition. (a) A statistical model M = (X F, Pv : v ∈ ϴ) is called a parametric model if ϴ ⊂
Rd for some d ∈ N. For d=1, M is called a one-parameter model.