Parametric model

Type of statistical model In statistics, a parametric model or parametric family or finite-dimensional model is a particular class of statistical models. Specifically, a parametric model is a family of probability distributions that has a finite number of parameters. Definition A statistical model is a collection of probability distributions on some sample space. We assume that the collection, P {\displaystyle {\mathcal {P}}} , is indexed by some set Θ. The set Θ is called the parameter set or, more commonly, the parameter space. For each θ ∈ Θ, let Fθ denote the corresponding member of the collection; so Fθ is a cumulative distribution function. Then a statistical model can be written as P = { F θ | θ ∈ Θ } . {\displaystyle {\mathcal {P}}={\big \{}F_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.} The model is a parametric model if Θ ⊆ ℝk for some positive integer k. When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions: P = { f θ | θ ∈ Θ } . {\displaystyle {\mathcal {P}}={\big \{}f_{\theta }\ {\big |}\ \theta \in \Theta {\big \}}.} Examples The Poisson family of distri