Definition
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Univariate case
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There are two versions to formulate an exponential dispersion model.
Additive exponential dispersion model
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In the univariate case, a real-valued random variable X {\displaystyle X} belongs to the additive exponential dispersion model with canonical parameter θ {\displaystyle \theta } and index parameter λ {\displaystyle \lambda } , X ∼ E D ∗ ( θ , λ ) {\displaystyle X\sim \mathrm {ED} ^{*}(\theta ,\lambda )} , if its probability density function can be written as
f X ( x ∣ θ , λ ) = h ∗ ( λ , x ) exp ( θ x − λ A ( θ ) ) . {\displaystyle f_{X}(x\mid \theta ,\lambda )=h^{*}(\lambda ,x)\exp \left(\theta x-\lambda A(\theta )\right)\,\!.} Reproductive exponential dispersion model
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The distribution of the transformed random variable Y = X λ {\displaystyle Y={\frac {X}{\lambda }}} is called reproductive exponential dispersion model , Y ∼ E D ( μ , σ 2 ) {\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})} , and is given by
f Y ( y ∣ μ , σ 2 ) = h ( σ 2 , y ) exp ( θ y − A ( θ ) σ 2 ) , {\displaystyle f_{Y}(y\mid \mu ,\sigma ^{2})=h(\sigma ^{2},y)\exp \left({\frac {\theta y-A(\theta )}{\sigma ^{2}}}\right)\,\!,} with σ 2 = 1 λ {\displaystyle \sigma ^{2}={\frac {1}{\lambda }}} and μ = A ′ ( θ ) {\displaystyle \mu =A'(\theta )} , implying θ = ( A ′ ) − 1 ( μ ) {\displaystyle \theta =(A')^{-1}(\mu )} .
The terminology dispersion model stems from interpreting σ 2 {\displaystyle \sigma ^{2}} as dispersion parameter . For fixed parameter σ 2 {\displaystyle \sigma ^{2}} , the E D ( μ , σ 2 ) {\displaystyle \mathrm {ED} (\mu ,\sigma ^{2})} is a natural exponential family .
Multivariate case
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In the multivariate case, the n -dimensional random variable X {\displaystyle \mathbf {X} } has a probability density function of the following form[1]
f X ( x | θ , λ ) = h ( λ , x ) exp ( λ ( θ ⊤ x − A ( θ ) ) ) , {\displaystyle f_{\mathbf {X} }(\mathbf {x} |{\boldsymbol {\theta }},\lambda )=h(\lambda ,\mathbf {x} )\exp \left(\lambda ({\boldsymbol {\theta }}^{\top }\mathbf {x} -A({\boldsymbol {\theta }}))\right)\,\!,} where the parameter θ {\displaystyle {\boldsymbol {\theta }}} has the same dimension as X {\displaystyle \mathbf {X} } .
Properties
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Cumulant-generating function
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The cumulant-generating function of Y ∼ E D ( μ , σ 2 ) {\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})} is given by
K ( t ; μ , σ 2 ) = log E [ e t Y ] = A ( θ + σ 2 t ) − A ( θ ) σ 2 , {\displaystyle K(t;\mu ,\sigma ^{2})=\log \operatorname {E} [e^{tY}]={\frac {A(\theta +\sigma ^{2}t)-A(\theta )}{\sigma ^{2}}}\,\!,} with θ = ( A ′ ) − 1 ( μ ) {\displaystyle \theta =(A')^{-1}(\mu )}
Mean and variance
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Mean and variance of Y ∼ E D ( μ , σ 2 ) {\displaystyle Y\sim \mathrm {ED} (\mu ,\sigma ^{2})} are given by
E [ Y ] = μ = A ′ ( θ ) , Var [ Y ] = σ 2 A ″ ( θ ) = σ 2 V ( μ ) , {\displaystyle \operatorname {E} [Y]=\mu =A'(\theta )\,,\quad \operatorname {Var} [Y]=\sigma ^{2}A''(\theta )=\sigma ^{2}V(\mu )\,\!,} with unit variance function V ( μ ) = A ″ ( ( A ′ ) − 1 ( μ ) ) {\displaystyle V(\mu )=A''((A')^{-1}(\mu ))} .
Reproductive
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If Y 1 , … , Y n {\displaystyle Y_{1},\ldots ,Y_{n}} are i.i.d. with Y i ∼ E D ( μ , σ 2 w i ) {\displaystyle Y_{i}\sim \mathrm {ED} \left(\mu ,{\frac {\sigma ^{2}}{w_{i}}}\right)} , i.e. same mean μ {\displaystyle \mu } and different weights w i {\displaystyle w_{i}} , the weighted mean is again an E D {\displaystyle \mathrm {ED} } with
∑ i = 1 n w i Y i w ∙ ∼ E D ( μ , σ 2 w ∙ ) , {\displaystyle \sum _{i=1}^{n}{\frac {w_{i}Y_{i}}{w_{\bullet }}}\sim \mathrm {ED} \left(\mu ,{\frac {\sigma ^{2}}{w_{\bullet }}}\right)\,\!,} with w ∙ = ∑ i = 1 n w i {\displaystyle w_{\bullet }=\sum _{i=1}^{n}w_{i}} . Therefore Y i {\displaystyle Y_{i}} are called reproductive .
Unit deviance
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The probability density function of an E D ( μ , σ 2 ) {\displaystyle \mathrm {ED} (\mu ,\sigma ^{2})} can also be expressed in terms of the unit deviance d ( y , μ ) {\displaystyle d(y,\mu )} as
f Y ( y ∣ μ , σ 2 ) = h ~ ( σ 2 , y ) exp ( − d ( y , μ ) 2 σ 2 ) , {\displaystyle f_{Y}(y\mid \mu ,\sigma ^{2})={\tilde {h}}(\sigma ^{2},y)\exp \left(-{\frac {d(y,\mu )}{2\sigma ^{2}}}\right)\,\!,} where the unit deviance takes the special form d ( y , μ ) = y f ( μ ) + g ( μ ) + h ( y ) {\displaystyle d(y,\mu )=yf(\mu )+g(\mu )+h(y)} or in terms of the unit variance function as d ( y , μ ) = 2 ∫ μ y y − t V ( t ) d t {\displaystyle d(y,\mu )=2\int _{\mu }^{y}\!{\frac {y-t}{V(t)}}\,dt} .
Examples
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References
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^ a b Jørgensen, B. (1987). Exponential dispersion models (with discussion). Journal of the Royal Statistical Society , Series B, 49 (2), 127–162.
^ Jørgensen, B. (1992). The theory of exponential dispersion models and analysis of deviance. Monografias de matemática, no. 51.
^ Marriott, P. (2005) "Local Mixtures and Exponential Dispersion
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