Gamma

Parameters

shape (\(a\)) and scale (\(b\)), alternatively, degrees of freedom (\(\nu \)) and mean (\(\mu \)). The RATS functions use the first of these. The relationship between them is \(a=\nu /2\) and

\(b=\dfrac{2\mu {\nu }}\). The chi-squared distribution with \(\nu \) degrees of freedom is a special case with \(\mu =\nu \)

Kernel

\(x^{a-1}\exp \left( -\dfrac{x}{b}\right) \) or \(x^{\left(v/2\right) -1}\exp \left( -\dfrac{x\nu }{2\mu }\right) \)

Support

\(\left[ 0,\infty \right) \)

Mean

\(ba\) or \(\mu \)

Variance

\(b^{2}a\) or \(\dfrac{2\mu ^{2}}{\nu }\)

Main Uses

Prior, exact and approximate posterior for the precision (reciprocal of variance) of residuals or other shocks in a model.

Density Function

%LOGGAMMADENSITY(x,a,b). For the \(\left\{\nu,\mu \right\}\) parameterization, use %LOGGAMMADENSITY(x,.5*nu,2.0*mu/nu)

Random Draws

%RANGAMMA(a) draws one or more (depending upon the target) independent Gammas with unit scale factor. Use b*%RANGAMMA(a) to get a draw from \(Gamma(a,b)\). If you are using the \(\left\{\nu ,\mu \right\}\) parameterization, use 2.0*mu*%RANGAMMA(.5*nu)/nu. You can also use mu*%RANCHISQR(nu)/nu.

Moment Matching

%GammaParms(mean,sd) (external function) returns the 2-vector of parameters ((a,b) parameterization) for a gamma with the given mean and standard deviation.