Compute Cumulative Distribution Function of Approximated (Differentiably) Uniform Distribution.
papprox_unif(x, scale, tildec = 13.86294)denotes the argument of cumulative distribution 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 cumulative distribution function.
The cumulative distribution function of dapprox_unif is given by
$$(1/(log(1+exp(-\tilde{c}))+\tilde{c}))*(\tilde{c}*(\tau^2)^(1/2)/\theta-log(exp((\tau^2)^(1/2)*\tilde{c}/\theta)+exp(\tilde{c})))$$
\(\tilde{c}\) is chosen such that \(P(\tau^2<=\theta)>=0.95\).
Nadja Klein and Thomas Kneib (2015). Scale-Dependent Priors for Variance Parameters in Structured Additive Distributional Regression. Working Paper.