dapprox_unif: Compute Density Function of Approximated (Differentiably) Uniform Distribution.
Description
Compute Density Function of Approximated (Differentiably) Uniform Distribution.Usage
dapprox_unif(x, scale, tildec = 13.86294)
Details
The density of the uniform distribution for $\tau$ is approximated by
$$p(\tau)=(1/(1+exp(\tau\tilde{c}/\theta-\tilde{c})))/(\theta(1+log(1+exp(-\tilde{c}))))$$.
This results in $$p(\tau^2)=0.5*(\tau^2)^(-1/2)(1/(1+exp((\tau^2)^(1/2)\tilde{c}/\theta-\tilde{c})))/(\theta(1+log(1+exp(-\tilde{c}))))$$ for $tau^2$.
$\tilde{c}$ is chosen such that $P(\tau<=\theta)>=0.95$.References
Nadja Klein and Thomas Kneib (2015). Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression.
Working Paper.