Compute Density Function of Approximated (Differentiably) Uniform Distribution.
dapprox_unif(x, scale, tildec = 13.86294)denotes the argument of the density function.
the scale parameter originally defining the upper bound of the uniform distribution.
denotes the ratio between scale parameter \(\theta\) and \(s\). The latter is responsible for the closeness of the approximation to the uniform distribution. See also below for further details and the default value.
the density.
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 \(P(\tau<=\theta)>=0.95\).
Nadja Klein and Thomas Kneib (2015). Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression. Working Paper.