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29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

In statistics, a parametric model or parametric family or finite-dimensional model is a family of distributions that can be described using a finite number of parameters. These parameters are usually collected together to form a single k-dimensional parameter vector θ = (θ₁, θ₂, …, θ_k).

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:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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.^[1] 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.^[2] This difficulty can be avoided by considering only “smooth” parametric models.

Definition

Template:Inline A parametric model is a collection of probability distributions such that each member of this collection, P_θ, is described by a finite-dimensional parameter θ. The set of all allowable values for the parameter is denoted Θ ⊆ R^k, and the model itself is written as

𝒫 = {P_{θ} | θ \in Θ} .

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

𝒫 = {f_{θ} | θ \in Θ} .

The parametric model is called identifiable if the mapping θ ↦ P_θ is invertible, that is there are no two different parameter values θ₁ and θ₂ such that P_θ₁ = P_θ₂.

Examples

The Poisson family of distributions is parametrized by a single number λ > 0:
$𝒫 = {p_{λ} (j) = \frac{λ^{j}}{j!} e^{- λ}, j = 0, 1, 2, 3, \dots | λ > 0},$
where p_λ is the probability mass function. This family is an exponential family.
The normal family is parametrized by θ = (μ,σ), where μ ∈ R is a location parameter, and σ > 0 is a scale parameter. This parametrized family is both an exponential family and a location-scale family:
$𝒫 = {f_{θ} (x) = \frac{1}{\sqrt{2 π} σ} e^{- \frac{1}{2 σ^{2}} (x - μ)^{2}} | μ \in ℝ, σ > 0} .$
The Weibull translation model has three parameters θ = (λ, β, μ):
$𝒫 = {f_{θ} (x) = \frac{β}{λ} {(\frac{x - μ}{λ})}^{β - 1} \exp (- (\frac{x - μ}{λ})^{β}) 1_{{x > μ}} | λ > 0, β > 0, μ \in ℝ} .$
This model is not regular (see definition below) unless we restrict β to lie in the interval (2, +∞).

Regular parametric model

Let μ be a fixed σ-finite measure on a probability space (Ω, ℱ), and $ℳ_{μ}$ the collection of all probability measures dominated by μ. Then we will call $𝒫 = {P_{θ} | θ \in Θ} \subseteq ℳ_{μ}$ a regular parametric model if the following requirements are met:^[3]

Θ is an open subset of R^k.
The map
$θ \mapsto s (θ) = \sqrt{d P_{θ} / d μ}$
from Θ to L²(μ) is Fréchet differentiable: there exists a vector $\dot{s} (θ) = ({\dot{s}}_{1} (θ), \dots, {\dot{s}}_{k} (θ))$ such that
$‖ s (θ + h) - s (θ) - \dot{s} (θ)^{'} h ‖ = o (| h |) as h \to 0,$
where ′ denotes matrix transpose.
The map $θ \mapsto \dot{s} (θ)$ (defined above) is continuous on Θ.
The k×k Fisher information matrix
$I (θ) = 4 \int \dot{s} (θ) \dot{s} (θ)^{'} d μ$
is non-singular.

Properties

Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒ_θ are following:^[4]
1. The density function ƒ_θ(x) is continuously differentiable in θ for μ-almost all x, with gradient ∇ƒ_θ.
2. The score function
  $z_{θ} = \frac{\nabla f_{θ}}{f_{θ}} \cdot 1_{{f_{θ} > 0}}$
  belongs to the space L²(P_θ) of square-integrable functions with respect to the measure P_θ.
3. The Fisher information matrix I(θ), defined as
  $I_{θ} = \int z_{θ} {z_{θ}}^{'} d P_{θ}$
  is nonsingular and continuous in θ.
If conditions (i)−(iii) hold then the parametric model is regular.
Local asymptotic normality.
If the regular parametric model is identifiable then there exists a uniformly $\sqrt{n}$ -consistent and efficient estimator of its parameter θ.^[5]

Notes

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References

Template:Refbegin

20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Template:MR

Template:Refend

↑ Template:Harvnb, ch.7.4
↑ Template:Harvnb
↑ Template:Harvnb
↑ Template:Harvnb, p.13, prop.2.1.1
↑ Template:Harvnb, Theorems 2.5.1, 2.5.2

[1] Template:Harvnb, ch.7.4

[2] Template:Harvnb

[3] Template:Harvnb

[4] Template:Harvnb, p.13, prop.2.1.1

[5] Template:Harvnb, Theorems 2.5.1, 2.5.2

[1]

[2]

[3]

[4]

[5]

@@ Line 1: / Line 1: @@
-Hi there, I am Andrew Berryhill. My wife and I live in Kentucky. Office supervising is exactly where her main income arrives from but she's already applied for another 1. The preferred hobby for him and his children is to play lacross and he'll be beginning some thing else along with it.<br><br>Feel free to surf to my page: [http://Modenpeople.Co.kr/modn/qna/292291 real psychic]
+{{about|statistics|mathematical and computer representation of objects|Solid modeling}}
+In [[statistics]], a '''parametric model''' or '''parametric family''' or '''finite-dimensional model''' is a family of [[probability distribution|distribution]]s that can be described using a finite number of [[parameter]]s. These parameters are usually collected together to form a single ''k''-dimensional ''parameter vector'' ''θ'' = (''θ''<sub>1</sub>, ''θ''<sub>2</sub>, …, ''θ''<sub>''k''</sub>).
+Parametric models are contrasted with the [[semiparametric model|semi-parametric]], [[semi-nonparametric model|semi-nonparametric]], and [[non-parametric model]]s, 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 parameter]]s;
+* 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.<ref>{{harvnb|LeCam|2000}}, ch.7.4</ref> It can also be noted that the set of all probability measures has [[cardinality]] of [[Continuum (set theory)|continuum]], and therefore it is possible to parametrize any model at all by a single number in (0,1) interval.<ref>{{harvnb|Bickel|1998|page=2}}</ref> This difficulty can be avoided by considering only “smooth” parametric models.
+==Definition==
+{{inline|section|date=May 2012}}
+A '''parametric model''' is a collection of [[probability distribution]]s such that each member of this collection, ''P<sub>θ</sub>'', is described by a finite-dimensional parameter ''θ''. The set of all allowable values for the parameter is denoted Θ ⊆ '''R'''<sup>''k''</sup>, and the model itself is written as
+: <math>
+    \mathcal{P} = \big\{ P_\theta\ \big|\ \theta\in\Theta \big\}.
+  </math>
+When the model consists of absolutely continuous distributions, it is often specified in terms of corresponding [[probability density function]]s:
+: <math>
+    \mathcal{P} = \big\{ f_\theta\ \big|\ \theta\in\Theta \big\}.
+  </math>
+The parametric model is called [[identifiable]] if the mapping ''θ'' ↦ ''P<sub>θ</sub>'' is invertible, that is there are no two different parameter values ''θ''<sub>1</sub> and ''θ''<sub>2</sub> such that ''P''<sub>''θ''<sub>1</sub></sub> = ''P''<sub>''θ''<sub>2</sub></sub>.
+===Examples===
+<ul>
+<li> The [[Poisson distribution|Poisson family]] of distributions is parametrized by a single number ''λ'' > 0:
+: <math>
+    \mathcal{P} = \Big\{\ p_\lambda(j) = \tfrac{\lambda^j}{j!}e^{-\lambda},\ j=0,1,2,3,\dots \ \Big|\ \lambda>0 \ \Big\},
+  </math>
+where ''p<sub>λ</sub>'' is the [[probability mass function]]. This family is an [[exponential family]].
+<li> The [[Normal distribution|normal family]] is parametrized by ''θ'' = (''μ'',''σ''), where ''μ'' ∈ '''R''' is a location parameter, and ''σ'' > 0 is a scale parameter. This parametrized family is both an [[exponential family]] and a [[location-scale family]]:
+: <math>
+    \mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} e^{ -\frac{1}{2\sigma^2}(x-\mu)^2 }\ \Big|\ \mu\in\mathbb{R}, \sigma>0 \ \Big\}.
+  </math>
+<li> The [[Weibull distribution|Weibull translation model]] has three parameters ''θ'' = (''λ'', ''β'', ''μ''):
+: <math>
+    \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\}.
+  </math>
+This model is '''not''' regular (see definition below) unless we restrict ''β'' to lie in the interval (2, +∞).
+</ul>
+==Regular parametric model==
+Let ''μ'' be a fixed [[σ-finite measure]] on a [[probability space]] (Ω, ℱ), and <math>\scriptstyle \mathcal{M}_\mu</math> the collection of all probability measures [[measure domination|dominated]] by ''μ''. Then we will call  <math style="position:relative;top:-.2em">\mathcal{P}\!=\!\{ P_\theta|\, \theta\in\Theta \} \subseteq \mathcal{M}_\mu</math>  a '''regular parametric model''' if the following requirements are met:<ref>{{harvnb|Bickel|1998|page=12}}</ref>
+<ol>
+<li> Θ is an [[open set|open subset]] of '''R'''<sup>''k''</sup>.
+<li> The map
+: <math>\theta\mapsto s(\theta)=\sqrt{dP_\theta/d\mu}</math>
+from Θ to [[L2_space#lp_spaces|''L''<sup>2</sup>(''μ'')]] is [[Fréchet derivative|Fréchet differentiable]]: there exists a vector <math style="position:relative;top:-.3em">\dot{s}(\theta) = (\dot{s}_1(\theta),\,\ldots,\,\dot{s}_k(\theta))</math> such that
+: <math>
+    \lVert s(\theta+h) - s(\theta) - \dot{s}(\theta)'h \rVert = o(|h|)\ \ \text{as }h \to 0,
+  </math>
+where ′ denotes matrix [[transpose]].
+<li> The map <math style="position:relative;top:-.2em">\theta\mapsto\dot{s}(\theta)</math> (defined above) is [[continuous function|continuous]] on Θ.
+<li> The ''k×k'' [[Fisher information matrix]]
+: <math>I(\theta) = 4\int \dot{s}(\theta)\dot{s}(\theta)'d\mu</math>
+is [[invertible matrix|non-singular]].
+</ol>
+===Properties===
+<ul>
+<li> Sufficient conditions for regularity of a parametric model in terms of ordinary differentiability of the density function ƒ<sub>''θ''</sub> are following:<ref>{{harvnb|Bickel|1998}}, p.13, prop.2.1.1</ref>
+  <ol type="i">
+  <li>The density function ƒ<sub>''θ''</sub>(''x'') is continuously differentiable in ''θ'' for ''μ''-almost all ''x'', with gradient ∇ƒ<sub>''θ''</sub>.
+  <li> The score function
+: <math>
+    z_\theta = \frac{\nabla f_\theta}{f_\theta} \cdot \mathbf{1}_{\{f_\theta>0\}}
+  </math>
+belongs to the space [[L2_space#lp_spaces|''L''²(''P<sub>θ</sub>'')]] of square-integrable functions with respect to the measure ''P<sub>θ</sub>''.
+  <li> The Fisher information matrix ''I''(''θ''), defined as
+: <math>
+    I_\theta = \int \!z_\theta z_\theta' \,dP_\theta
+  </math>
+is nonsingular and continuous in ''θ''.
+  </ol>
+If conditions (i)−(iii) hold then the parametric model is regular.
+<li>[[Local asymptotic normality]].
+<li>If the regular parametric model is identifiable then there exists a uniformly <math>\scriptstyle \sqrt{n}</math>-consistent and efficient estimator of its parameter ''θ''.<ref>{{harvnb|Bickel|1998}}, Theorems 2.5.1, 2.5.2</ref>
+</ul>
+== See also ==
+* [[Statistical model]]
+* [[Parametric family]]
+* [[Parametrization]] (i.e., [[coordinate system]])
+* [[Parsimony]] (with regards to the trade-off of many or few parameters in data fitting)
+==Notes==
+{{Reflist}}
+==References==
+{{refbegin}}
+* {{cite book
+  | author = Bickel, Peter J. and Doksum, Kjell A.
+  | title = Mathematical Statistics: Basic and Selected Topics, Volume 1.
+  | volume =
+  | edition = Second (updated printing 2007)
+  | year = 2001
+  | publisher = Pearson Prentice-Hall
+  }}
+* {{cite book
+  | authors = Bickel, Peter J.; Klaassen, Chris A.J.; Ritov, Ya’acov; Wellner Jon A.
+  | title = Efficient and adaptive estimation for semiparametric models
+  | publisher = Springer: New York
+  | year = 1998
+  | isbn = 0-387-98473-9
+  | ref = CITEREFBickel1998
+  }}
+* {{cite book
+  | last = Davidson
+  | first = A.C.
+  | title = Statistical Models
+  | publisher = Cambridge University Press
+  | year = 2003
+  }}
+* {{cite book
+  | author = [[David Freedman (statistician)|Freedman, David A.]]
+  | title = Statistical Models: Theory and Practice
+  | publisher = [http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521671057 Cambridge University Press]
+  | isbn = 978-0-521-67105-7
+  | year = 2009
+  | edition = Second
+  }}
+* {{cite book
+  | author = [[Lucien Le Cam|Le Cam, Lucien]]
+  | coauthors = Lo Yang, Grace
+  | title =  Asymptotics in statistics: some basic concepts
+  | year = 2000
+  | publisher = Springer
+  | isbn = 0-387-95036-2
+  | ref = CITEREFLeCam2000
+  }}
+* {{cite book
+  | author = [[Erich Leo Lehmann|Lehmann, Erich]]
+  | title = Theory of Point Estimation
+  | year = 1983
+  }}
+* {{cite book
+  | author = [[Erich Leo Lehmann|Lehmann, Erich]]
+  | title = Testing Statistical Hypotheses
+  | year = 1959
+  }}
+* {{cite book
+  | author = Liese, Friedrich and Miescke, Klaus-J.
+  | title = Statistical Decision Theory: Estimation, Testing, and Selection
+  | year = 2008
+  | publisher = Springer
+  }}
+* {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann
+|authorlink= <!-- Johann Pfanzagl -->
+|last2=with the assistance of R. Hamböker
+|year=1994|publisher=Walter de Gruyter
+|isbn=3-11-013863-8
+}} {{MR|1291393}}
+{{refend}}
+{{DEFAULTSORT:Parametric Model}}
+[[Category:Statistical theory]]
+[[Category:Statistical models]]

Clock hypothesis: Difference between revisions

Latest revision as of 16:34, 22 December 2012

Contents

Definition

Examples

Regular parametric model

Properties

See also

Notes

References

Navigation menu

Clock hypothesis: Difference between revisions

Latest revision as of 16:34, 22 December 2012

Definition

Examples

Regular parametric model

Properties

See also

Notes

References

Navigation menu

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