# Statistical model

A statistical model , sometimes also called statistical space , is a term from mathematical statistics , the sub-area of statistics that uses the methods of stochastics and probability theory . A statistical model clearly summarizes all the initial information: which values ​​can the data assume, which sets of values ​​should a probability be assigned and which probability measures are possible or should be taken into account?

## definition

A statistical model is a triple consisting of ${\ displaystyle {\ mathcal {E}}}$${\ displaystyle {\ mathcal {E}} = ({\ mathcal {X}}, {\ mathcal {A}}, {\ mathcal {P}})}$

• a basic set that contains all possible results of the experiment,${\ displaystyle {\ mathcal {X}}}$
• a σ-algebra on the basic set and${\ displaystyle {\ mathcal {A}}}$${\ displaystyle {\ mathcal {X}}}$
• an amount of probability measures on${\ displaystyle {\ mathcal {P}}}$${\ displaystyle ({\ mathcal {X}}, {\ mathcal {A}})}$

It is often more handy to note the set of probability measures as a family (with any index set) in order to be able to more easily access the identified elements. The set of probability measures is then also noted. This does not necessarily mean that it is a parametric model. ${\ displaystyle (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta}}$

## Alternative definitions

There are several alternative definitions of a statistical model that differ in their level of detail.

On the one hand, the description of a statistical model is found as a random variable that assumes values ​​in the measurement space according to the distributions . The underlying probability space of the random variable is not specified in more detail because it is not relevant for the distributions. In contrast to the description above, this description makes it clearer that the samples, i.e. the elements , are to be seen as a realization of a random variable with an unknown distribution. The set is then also called the sampling area . ${\ displaystyle X}$ ${\ displaystyle ({\ mathcal {X}}, {\ mathcal {A}})}$${\ displaystyle {\ mathcal {P}}}$${\ displaystyle {\ mathcal {X}}}$${\ displaystyle {\ mathcal {X}}}$

On the other hand, the description of a statistical model can only be found as a family or set of probability measures . The corresponding basic space then results implicitly from the defined probability measures, the σ-algebra used is accordingly the canonical choice (power set in the discrete case, Borelian σ-algebra otherwise). ${\ displaystyle {\ mathcal {P}}}$

## Classification of statistical models

### Parametric and non-parametric models

If the set of probability measures can be described using a set of parameters, also called a parameter space, is

${\ displaystyle {\ mathcal {P}} = \ {P _ {\ vartheta} \, | \, \ vartheta \ in \ Theta \}}$

for a set of parameters , one speaks of a parametric model , otherwise of a nonparametric model . Is , then one speaks of a one-parameter model . ${\ displaystyle \ Theta \ subset \ mathbb {R} ^ {n}}$${\ displaystyle \ Theta \ subset \ mathbb {R}}$

### Discrete models

If finite or countably infinite and if the power set is , then one speaks of a discrete model . The probability measures can then be described by probability functions. ${\ displaystyle {\ mathcal {X}}}$${\ displaystyle {\ mathcal {A}} = 2 ^ {\ mathcal {X}}}$

### Continuous models

Is a Borel set of and is restricting the Borel σ-algebra on this amount, that is , and, each of the probability measures in a probability density , it is called a continuous model . ${\ displaystyle {\ mathcal {X}}}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle {\ mathcal {A}} = {\ mathcal {B}} (\ mathbb {R} ^ {n}) | _ {\ mathcal {X}}}$${\ displaystyle {\ mathcal {P}}}$

### Standard models

If it is a continuous model or a discrete model, one speaks of a standard model . In the case of standard models, there is in particular a probability density or a probability function. Some authors also call these models regular models.

### Regular models

Regular statistical models are single-parameter standard models in which requirements are still placed on the existence of derivatives of the density function. They are needed to formulate the Cramér-Rao inequality .

### Location and scale models

Statistical models whose distribution class is a location class, i.e. which are created by shifting a single probability distribution, are called location models, and statistical models with scale families are also called scale models .

### Product models

Product models arise when you form the multiple product of a statistical model with yourself. They formalize the idea that an experiment is carried out several times in a row and that the results of the individual experiments do not influence each other. Many of the popular models, such as the normal distribution model, are product models.

## Examples

An example of a statistical model is the basic space provided with the σ-algebra and the set as the set of probability measures ${\ displaystyle {\ mathcal {X}} = \ {0.1, \ dots, 100 \}}$${\ displaystyle {\ mathcal {A}} = 2 ^ {\ mathcal {X}}}$

${\ displaystyle {\ mathcal {P}} = \ {\ operatorname {Bin} _ {100, \ vartheta} \, | \, \ vartheta \ in [0,1] \}}$

of all binomial distributions with parameters 100 and . This statistical model could be chosen, for example, if you toss a coin 100 times and count the number of successes. This is binomially distributed, but for an unknown parameter, since it is not clear whether the coin is forged or not. This model is a one-parameter model, there is. It is also a discrete model, since the basic set is finite and the σ-algebra is defined by the power set. It is therefore automatically a standard model. The set of probability measures ${\ displaystyle \ vartheta}$${\ displaystyle \ vartheta \ in \ Theta = [0,1] \ subset \ mathbb {R}}$

${\ displaystyle {\ mathcal {P}} = \ {P \, | \, P {\ text {is W dimension on}} ({\ mathcal {X}}, {\ mathcal {A}}) \} }$,

however, results in a non-parametric model.

## Individual evidence

1. ^ Rüschendorf: Mathematical Statistics. 2014, p. 18.
2. Czado, Schmidt: Mathematical Statistics. 2011, p. 39.
3. ^ Georgii: Stochastics. 2009, p. 197.
4. Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 41 , doi : 10.1007 / 978-3-642-17261-8 .