#LyX 1.6.2 created this file. For more info see http://www.lyx.org/
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\author ""
\end_header
\begin_body
\begin_layout Standard
Variance Functions:
\end_layout
\begin_layout Standard
Constant:
\begin_inset Formula $\boldsymbol{1}$
\end_inset
\end_layout
\begin_layout Standard
Power:
\begin_inset Formula $\boldsymbol{X}^{2}$
\end_inset
\end_layout
\begin_layout Standard
Binomial:
\begin_inset Formula $np(1-p)\text{ where }p=\frac{\mu}{n};\,\, V(\mu)=np(1-p)$
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $\frac{\partial\mu}{\partial\eta}$
\end_inset
\end_layout
\begin_layout Standard
Links: initialization of base class returns the actual mean vector
\begin_inset Formula $\boldsymbol{\mu}$
\end_inset
;
\begin_inset Formula $p$
\end_inset
in the logit and subclasses;
\begin_inset Formula $x$
\end_inset
elsewhere.
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Tabular
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Link
\begin_inset Formula $g(p)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Inverse
\begin_inset Formula $g^{-1}(p)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Analytic Derivative
\begin_inset Formula $g^{\prime}(p)$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Logit
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $z=\log\frac{p}{1-p}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p=\frac{e^{z}}{1+e^{z}}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $g^{\prime}(p)=\frac{1}{p(1-p)}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Power
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $z=x^{\text{pow}}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $x=z^{\frac{1}{\text{pow}}}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $g^{\prime}(x)=\text{pow}\cdot x^{\text{power}-1}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
Inverse
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
same as above with
\begin_inset Formula $\text{pow}=-1$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Square Root
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\text{pow}=0.5$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Identity
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\text{pow}=1$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Log
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $z=\log x$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $g^{-1}(z)=e^{z}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $g^{\prime}(x)=\frac{1}{x}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
CDFLink/Probit
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $z=\Phi^{-1}(p)$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p=\Phi(z)$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $g^{\prime}(x)=\frac{1}{\int_{-\infty}^{p}f(t)dt}$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Cauchy
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
same as the above with the Cauchy distribution
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
CLogLog
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $z=\log(-\log p)$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $p=e^{-e^{z}}$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $g^{\prime}(p)=-\frac{1}{p\log p}$
\end_inset
\end_layout
\end_inset
|
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption
\begin_layout Plain Layout
Link Functions
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Initializing the family sets a link property and a variance based on the
link(?)
\end_layout
\begin_layout Standard
\begin_inset Float table
placement H
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Tabular
\begin_inset Text
\begin_layout Plain Layout
Family
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Weights
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Deviance
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
DevResid
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Fitted
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Predict
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Base Class
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\frac{1}{(g^{\prime}(\mu))^{2}\cdot V(\mu)}$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\frac{\sum_{i}\text{DevResid}^{2}}{\text{scale}}$
\end_inset
\end_layout
\end_inset
|
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\left(Y-\mu\right)\cdot\sqrt{\text{weights}}$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\mu=g^{-1}(\eta)$
\end_inset
*
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\eta=g(\mu)$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
Poisson
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\text{sign}\left(Y-\mu\right)\sqrt{2Y\log\frac{Y}{\mu}-2(Y-\mu)}$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
Gaussian
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\frac{\left(Y-\mu\right)}{\text{\sqrt{\text{scale}\cdot V\left(\mu\right)}}}$
\end_inset
\end_layout
\end_inset
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
Gamma
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\begin_inset Text
\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
Bug?
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\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
Binomial
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\text{sign}\left(Y-\mu\right)\sqrt{-2Y\log\frac{\mu}{n}+\left(n-Y\right)\log\left(1-\frac{\mu}{n}\right)}$
\end_inset
\end_layout
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\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
Inverse Gaussian
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
\end_layout
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\begin_inset Text
\begin_layout Plain Layout
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\begin_inset Text
\begin_layout Plain Layout
?
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\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption
\begin_layout Plain Layout
Families
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
*
\begin_inset Formula $\eta$
\end_inset
is the linear predictor ie.,
\begin_inset Formula $X\beta$
\end_inset
in the generalized linear model
\end_layout
\end_inset
\end_layout
\end_body
\end_document