A statistical model is a family of distributions, {P θ } θ∈Θ. • X ∼ P θ are the random observatio... more A statistical model is a family of distributions, {P θ } θ∈Θ. • X ∼ P θ are the random observations associated with the model. • L X (θ) = P θ (X) is called the likelihood function. • Parametric model: when Θ is nite dimensional (e.g., R 3). • Non-parametric model: when Θ has innite dimension. • Nuisance parameters: the components of θ which are not of interest (e.g., when θ = µ, σ 2 and one is interested only in µ, then σ 2 is a nuisance parameter). Identiability A model P θ is identiable if for every θ 1 , θ 2 ∈ Θ, θ 1 = θ 2 ⇒ P θ1 = P θ2 • The inverse is trivially true, that is: P θ1 = P θ2 ⇒ θ 1 = θ 2. • Identiability means a one-to-one correspondence between Θ and {P θ } θ∈Θ ; It is a necessary condition for the ability to estimate the parameter θ. Sucient Statistic Assume X ∼ P θ. • A function T (X) is called a statistic if it does not depend on θ. • T (X) is called sucient statistic if for every t in the range of T (·), the distribution of X| T = t does not depend on θ. Regularity Conditions {P θ } θ∈Θ is called regular if either of the following holds: 1. P θ is continuous for every θ ∈ Θ 2. P θ is discrete for every θ ∈ Θ and there exists a countable set X = {ξ i } ∞ i=1 such that ∞ i=1 P θ (X = ξ i) = 1, ∀θ ∈ Θ. Factorization Theorem (Fisher-Neyman) If {P θ } θ∈Θ is regular, then T (X) is a sucient statistic if-and-only-if there exist functions g and h such that
A statistical model is a family of distributions, {P θ } θ∈Θ. • X ∼ P θ are the random observatio... more A statistical model is a family of distributions, {P θ } θ∈Θ. • X ∼ P θ are the random observations associated with the model. • L X (θ) = P θ (X) is called the likelihood function. • Parametric model: when Θ is nite dimensional (e.g., R 3). • Non-parametric model: when Θ has innite dimension. • Nuisance parameters: the components of θ which are not of interest (e.g., when θ = µ, σ 2 and one is interested only in µ, then σ 2 is a nuisance parameter). Identiability A model P θ is identiable if for every θ 1 , θ 2 ∈ Θ, θ 1 = θ 2 ⇒ P θ1 = P θ2 • The inverse is trivially true, that is: P θ1 = P θ2 ⇒ θ 1 = θ 2. • Identiability means a one-to-one correspondence between Θ and {P θ } θ∈Θ ; It is a necessary condition for the ability to estimate the parameter θ. Sucient Statistic Assume X ∼ P θ. • A function T (X) is called a statistic if it does not depend on θ. • T (X) is called sucient statistic if for every t in the range of T (·), the distribution of X| T = t does not depend on θ. Regularity Conditions {P θ } θ∈Θ is called regular if either of the following holds: 1. P θ is continuous for every θ ∈ Θ 2. P θ is discrete for every θ ∈ Θ and there exists a countable set X = {ξ i } ∞ i=1 such that ∞ i=1 P θ (X = ξ i) = 1, ∀θ ∈ Θ. Factorization Theorem (Fisher-Neyman) If {P θ } θ∈Θ is regular, then T (X) is a sucient statistic if-and-only-if there exist functions g and h such that
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