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Homestatistical model

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# statistical model

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 X is a set of probability distribution functions of X:

$\mathcal{P}=\{F_{{\textbf{X}}}\}.$ |

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:

$\mathcal{P}=\{f_{{\textbf{X}}}\}.$ |

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 $\textbf{X}=(X_{1},\ldots,X_{n})$ 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:

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

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*

$\Theta\rightarrow\mathcal{P}\mbox{ given by }\theta\mapsto F_{{\theta}}\mbox{ % so that }\mathcal{P}=\{F_{{\theta}}\mid\theta\in\Theta\},$ |

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

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

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.

## Mathematics Subject Classification

62A01*no label found*

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