parametric model

{{no footnotes|section|date=May 2012}}

A statistical model is a collection of probability distributions on some sample space. We assume that the collection, {{math|𝒫}}, is indexed by some set {{math|Θ}}. The set {{math|Θ}} is called the parameter set or, more commonly, the parameter space. For each {{math|θ ∈ Θ}}, let {{math|F_θ}} denote the corresponding member of the collection; so {{math|F_θ}} is a cumulative distribution function. Then a statistical model can be written as

: $$

\mathcal{P} = \big\{ F_\theta\ \big|\ \theta\in\Theta \big\}.

The model is a parametric model if {{math|Θ ⊆ ℝ^k}} for some positive integer {{math|k}}.

When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding probability density functions:

: $$

\mathcal{P} = \big\{ f_\theta\ \big|\ \theta\in\Theta \big\}.

• The Poisson family of distributions is parametrized by a single number {{math|λ > 0}}:

: $$

\mathcal{P} = \Big\{\ p_\lambda(j) = \tfrac{\lambda^j}{j!}e^{-\lambda},\ j=0,1,2,3,\dots \ \Big|\;\; \lambda>0 \ \Big\},

where {{math|p_λ}} is the probability mass function. This family is an exponential family.

• The normal family is parametrized by {{math|θ {{=}} (μ, σ)}}, where {{math|μ ∈ ℝ}} is a location parameter and {{math|σ > 0}} is a scale parameter:

: $$

\mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} \exp\left(-\tfrac{(x-\mu)^2}{2\sigma^2}\right)\ \Big|\;\; \mu\in\mathbb{R}, \sigma>0 \ \Big\}.

This parametrized family is both an exponential family and a location-scale family.

• The Weibull translation model has a three-dimensional parameter {{math|θ {{=}} (λ, β, μ)}}:

: $$

\mathcal{P} = \Big\{\

f_\theta(x) = \tfrac{\beta}{\lambda}

\left(\tfrac{x-\mu}{\lambda}\right)^{\beta-1}\!

\exp\!\big(\!-\!\big(\tfrac{x-\mu}{\lambda}\big)^\beta \big)\,

\mathbf{1}_{\{x>\mu\}}

\ \Big|\;\;

\lambda>0,\, \beta>0,\, \mu\in\mathbb{R}

\ \Big\}.

• The binomial model is parametrized by {{math|θ {{=}} (n, p)}}, where {{math|n}} is a non-negative integer and {{math|p}} is a probability (i.e. {{math|p ≥ 0}} and {{math|p ≤ 1}}):

: $$

\mathcal{P} = \Big\{\ p_\theta(k) = \tfrac{n!}{k!(n-k)!}\, p^k (1-p)^{n-k},\ k=0,1,2,\dots, n \ \Big|\;\; n\in\mathbb{Z}_{\ge 0},\, p \ge 0 \land p \le 1\Big\}.

This example illustrates the definition for a model with some discrete parameters.

A parametric model is called identifiable if the mapping {{math|θ ↦ P_θ}} is invertible, i.e. there are no two different parameter values {{math|θ₁}} and {{math|θ₂}} such that {{math|P_θ1 {{=}} P_θ2}}.

Parametric models are contrasted with the semi-parametric, semi-nonparametric, and non-parametric models, all of which consist of an infinite set of "parameters" for description. The distinction between these four classes is as follows:{{Citation needed|date=October 2010}}

• in a "parametric" model all the parameters are in finite-dimensional parameter spaces;
• a model is "non-parametric" if all the parameters are in infinite-dimensional parameter spaces;
• a "semi-parametric" model contains finite-dimensional parameters of interest and infinite-dimensional nuisance parameters;
• a "semi-nonparametric" model has both finite-dimensional and infinite-dimensional unknown parameters of interest.

Some statisticians believe that the concepts "parametric", "non-parametric", and "semi-parametric" are ambiguous.{{harvnb|Le Cam| Yang|2000}}, §7.4 It can also be noted that the set of all probability measures has cardinality of continuum, and therefore it is possible to parametrize any model at all by a single number in (0,1) interval.{{harvnb|Bickel|Klaassen| Ritov| Wellner| 1998|page=2}} This difficulty can be avoided by considering only "smooth" parametric models.

• Parametric family
• Parametric statistics
• Statistical model
• Statistical model specification

{{Reflist}}

{{refbegin}}

• {{citation

|author1-last = Bickel | author1-first = Peter J. | author1-link = Peter J. Bickel

|author2-last = Doksum | author2-first = Kjell A.

| title = Mathematical Statistics: Basic and selected topics

| volume = 1

| edition = Second (updated printing 2007)

| year = 2001

| publisher = Prentice-Hall

}}

• {{citation

| author1-last = Bickel | author1-first = Peter J. | author1-link = Peter J. Bickel

| author2-last = Klaassen | author2-first = Chris A. J.

| author3-last = Ritov | author3-first = Ya’acov

| author4-first = Jon A. | author4-last = Wellner

| year = 1998

| title= Efficient and Adaptive Estimation for Semiparametric Models

| publisher = Springer

}}

• {{citation

| last = Davison

| first = A. C.

| title = Statistical Models

| publisher = Cambridge University Press

| year = 2003

}}

• {{citation

| author-last = Le Cam | author-first = Lucien

| author-link = Lucien Le Cam

| author2-last = Yang | author2-first = Grace Lo | author2-link = Grace Yang

| title = Asymptotics in Statistics: Some basic concepts

| edition = 2nd

| year = 2000

| publisher = Springer

}}

• {{citation

| author1-last = Lehmann | author1-first = Erich L.

| author1-link = Erich Leo Lehmann

| author2-last = Casella | author2-first = George

| author2-link = George Casella

| title = Theory of Point Estimation

| edition = 2nd

| year = 1998

| publisher = Springer

}}

• {{citation

| author1-last = Liese| author1-first = Friedrich

| author2-last = Miescke| author2-first = Klaus-J.

| title = Statistical Decision Theory: Estimation, testing, and selection

| year = 2008

| publisher = Springer

}}

• {{citation

| title = Parametric Statistical Theory

| last1 = Pfanzagl

| first1 = Johann

| last2 = with the assistance of R. Hamböker

| year = 1994

| publisher = Walter de Gruyter

|mr=1291393}}

{{refend}}

{{DEFAULTSORT:Parametric Model}}

Category:Parametric statistics

Category:Statistical models