# statistical model

Let $\text{\mathbf{X}}=({X}_{1},\mathrm{\dots},{X}_{n})$ be a random vector with a given realization
$\text{\mathbf{X}}(\omega )=({x}_{1},\mathrm{\dots},{x}_{n})$, where $\omega $ is the outcome (of an observation or an experiment) in the sample space $\mathrm{\Omega}$. A *statistical model* $\mathcal{P}$ based on X is a set of probability distribution functions of X:

$$\mathcal{P}=\{{F}_{\text{\mathbf{X}}}\}.$$ |

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:

$$\mathcal{P}=\{{f}_{\text{\mathbf{X}}}\}.$$ |

As an example, a coin is tossed $n$ times and the results are observed. The probability of landing a head during one toss is $p$. Assume that each toss is independent of one another. If $\text{\mathbf{X}}=({X}_{1},\mathrm{\dots},{X}_{n})$ is defined to be the vector of the $n$ ordered outcomes, then a statistical model based on X can be a family of Bernoulli distributions^{}

$$\mathcal{P}=\{\prod _{i=1}^{n}{p}^{{x}_{i}}{(1-p)}^{1-{x}_{i}}\},$$ |

where ${X}_{i}(\omega )={x}_{i}$ and ${x}_{i}=1$ if $\omega $ is the outcome that the $i$th toss lands a head and ${x}_{i}=0$ if $\omega $ is the outcome that the $i$th toss lands a tail.

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

$$\mathcal{P}=\{\left(\genfrac{}{}{0pt}{}{n}{x}\right){p}^{x}{(1-p)}^{n-x}\},$$ |

where $X(\omega )=x$, where $\omega $ is the outcome that $x$ heads (out of $n$ tosses) are observed.

A statistical model is usually *parameterized* by a function^{}, called a *parameterization*

$$\mathrm{\Theta}\to \mathcal{P}\text{given by}\theta \mapsto {F}_{\theta}\text{so that}\mathcal{P}=\{{F}_{\theta}\mid \theta \in \mathrm{\Theta}\},$$ |

where $\mathrm{\Theta}$ is called a *parameter space*. $\mathrm{\Theta}$ 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

$$p\mapsto \prod _{i=1}^{n}{p}^{{x}_{i}}{(1-p)}^{1-{x}_{i}}.$$ |

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

Title | statistical model |
---|---|

Canonical name | StatisticalModel |

Date of creation | 2013-03-22 14:33:18 |

Last modified on | 2013-03-22 14:33:18 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 10 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 62A01 |

Defines | identifiable parameterization |

Defines | parameter space |