statistical 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.

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identifiable parameterization, parameter space

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Definition

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