Product model (statistics)

In mathematical statistics, product models are a special class of statistical models . Many common models, such as the normal distribution model, are product models. What all product models have in common is that they arise as the product of smaller statistical models with themselves. In particular, the sample variables in product models are independently and identically distributed . Therefore, product models occur when modeling several, identical tests, the results of which do not influence each other.

definition

A statistical model is given , i.e. a set and a σ-algebra on and a family of probability measures on the measurement space . Where is any index set . ${\displaystyle ({\mathcal {X}},{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle ({\mathcal {X}},{\mathcal {A}})}$ ${\displaystyle \Theta }$

Then means for the statistical model ${\displaystyle n\in \mathbb {N} }$

${\displaystyle ({\mathcal {X}}^{\times n},{\mathcal {A}}^{\otimes n},(P_{\vartheta }^{\otimes n})_{\vartheta \in \Theta })}$

the associated n-fold product model. ${\displaystyle ({\mathcal {X}},{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$

Here referred to

${\displaystyle {\mathcal {X}}^{\times n}=X\times X\times \dots \times X}$ (n times)

the n-fold Cartesian product , denotes the n-fold product σ-algebra of with itself and is the n-fold product measure of with itself. ${\displaystyle {\mathcal {A}}^{\otimes n}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle P_{\vartheta }^{\otimes n}}$ ${\displaystyle P_{\vartheta }}$

Real product models

Gängigster case of a product model is, if the set of real numbers is provided with the canonical Borel σ algebra and any family of probability measures on . Then the n-fold product model is of the form ${\displaystyle {\mathcal {X}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {A}}={\mathcal {B}}(\mathbb {R} )}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),(P_{\vartheta }^{\otimes n})_{\vartheta \in \Theta })}$

there and is. Such product models are also called real product models. ${\displaystyle \mathbb {R} ^{\times n}=\mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {B}}(\mathbb {R} )^{\otimes n}={\mathcal {B}}(\mathbb {R} ^{n})}$

Examples

Normal distribution model

→ Main article : Normal distribution model

The normal distribution model is formulated in several different versions. Only the families and the probability distributions differ, but these are always defined on. The following cases exist: ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

• As a family of probability distributions, all normal distributions with a fixed expected value and any variance are considered. The product model is then of the form ${\displaystyle \mu _{0}}$ ${\displaystyle \sigma ^{2}>0}$

${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),({\mathcal {N}}(\mu _{0},\sigma ^{2})^{\otimes n})_{\sigma ^{2}>0})}$

• As a family of probability distributions, all normal distributions with any expected value and fixed variance are considered. The product model is then of the form ${\displaystyle \mu }$ ${\displaystyle \sigma _{0}^{2}>0}$

${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),({\mathcal {N}}(\mu ,\sigma _{0}^{2})^{\otimes n})_{\mu \in \mathbb {R} })}$

• As a family of probability distributions, all normal distributions with any expected value and any variance are considered. The product model is then of the form ${\displaystyle \mu }$ ${\displaystyle \sigma ^{2}>0}$

${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),({\mathcal {N}}(\mu ,\sigma ^{2})^{\otimes n})_{(\mu ,\sigma ^{2})\in \mathbb {R} \times R_{>0}})}$

Bernoulli model

The so-called Bernoulli model arises as a product model from the basic set , provided with the power set as σ-algebra, so and as probability measures the Bernoulli distributions with . The product model thus takes shape ${\displaystyle {\mathcal {X}}=\{0,1\}}$ ${\displaystyle {\mathcal {A}}={\mathcal {P}}(\{0,1\})}$ ${\displaystyle \operatorname {Ber} _{\vartheta }}$ ${\displaystyle \vartheta \in (0,1)}$

${\displaystyle (\{0,1\}^{n},{\mathcal {P}}(\{0,1\}^{n}),(\operatorname {Ber} _{\vartheta })_{\vartheta \in (0,1)})}$

on. The Bernoulli model occurs when modeling a sequence of similar attempts, in which each attempt can be either a success or a failure. A typical case for this would be tossing a coin. ${\displaystyle n}$

properties

Independence and identical distribution

An important property of product models is that the sample variables are always independently identically distributed with distribution . This simplifies many calculations in product models, since for example applies to all or also to . ${\displaystyle (X_{i})_{i=1,\dots ,n}}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle \operatorname {E} _{\vartheta }(X_{i})=\operatorname {E} _{\vartheta }(X_{j})}$ ${\displaystyle i,j}$ ${\displaystyle \operatorname {E} _{\vartheta }(X_{i}X_{j})=\operatorname {E} _{\vartheta }(X_{i})\cdot \operatorname {E} _{\vartheta }(X_{j})}$ ${\displaystyle i\neq j}$

stability

Product models of parametric models are again parametric models for the same set of parameters . Likewise, product models of standard models are again standard models, because if the probability density function has , then the probability density function has . Likewise, product models of exponential models are again exponential models. ${\displaystyle \Theta }$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle f_{\vartheta }(x)}$ ${\displaystyle P_{\vartheta }^{\otimes n}}$ ${\displaystyle f(x)=\prod _{i=1}^{n}f_{\vartheta }(x_{i})}$

Individual evidence

1. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 197 , doi : 10.1515 / 9783110215274 .
2. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 208 , doi : 10.1515 / 9783110215274 .