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statistical model

(Definition)

Let $\textbf{X}=(X_1,\ldots,X_n)$ be a random vector with a given realization $\textbf{X}(\omega)=(x_1,\ldots,x_n)$ , where $\omega$ is the outcome (of an observation or an experiment) in the sample space $\Omega$ . A statistical model $\mathcal{P}$ based on $\textbf{X}$ is a set of probability distribution functions of $\textbf{X}$ :

$\displaystyle \mathcal{P}=\lbrace F_{\textbf{X}} \rbrace.$

If it is known in advance that this family of distributions comes from a set of continuous distributions, the statistical model $\mathcal{P}$ can be equivalently defined as a set of probability density functions:

$\displaystyle \mathcal{P}=\lbrace f_{\textbf{X}} \rbrace.$

As an example, a coin is tossed times and the results are observed. The probability of landing a head during one toss is . Assume that each toss is independent of one another. If $\textbf{X}=(X_1,\ldots,X_n)$ is defined to be the vector of the ordered outcomes, then a statistical model based on $\textbf{X}$ can be a family of Bernoulli distributions

$\displaystyle \mathcal{P}=\lbrace \prod_{i=1}^n p^{x_i}(1-p)^{1-x_i} \rbrace,$

where $X_i(\omega)=x_i$ and

if $\omega$ is the outcome that the

th toss lands a head and

if $\omega$ is the outcome that the

th toss lands a tail.

Next, suppose is the number of tosses where a head is observed, then a statistical model based on can be a family binomial distributions:

$\displaystyle \mathcal{P}=\lbrace {n\choose x}p^x(1-p)^{n-x} \rbrace,$

where $X(\omega)=x$ , where $\omega$ is the outcome that

heads (out of

tosses) are observed.

A statistical model is usually parameterized by a function, called a parameterization

$\displaystyle \Theta\rightarrow\mathcal{P}$ given by $\displaystyle \theta\mapsto F_{\theta}$ so that $\displaystyle \mathcal{P}=\lbrace F_{\theta} \mid \theta\in\Theta \rbrace,$

where $\Theta$ is called a parameter space. $\Theta$ is usually a subset of $\mathbb{R}^n$ . However, it can also be a function space.

In the first part of the above example, the statistical model is parameterized by

$\displaystyle p\mapsto\prod_{i=1}^n p^{x_i}(1-p)^{1-x_i}.$

If the parameterization is a one-to-one function, it is called an identifiable parameterization and $\theta$ is called a parameter. The in the above example is a parameter.

"statistical model" is owned by CWoo. [ full author list (2) ]

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Also defines:

identifiable parameterization, parameter space

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Cross-references: parameter, one-to-one, function space, subset, function, binomial distributions, number, Bernoulli distributions, vector, independent, continuous, distributions, probability distribution functions, observation, outcome, random vector
There are 14 references to this entry.

This is version 7 of statistical model, born on 2004-08-24, modified 2006-09-11.
Object id is 6107, canonical name is StatisticalModel.
Accessed 11429 times total.

Classification:

AMS MSC:

62A01 (Statistics :: Foundational and philosophical topics)

Pending Errata and Addenda

None.

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