Density function, distribution function, quantile function and random generation for the perturbed uniform distribution having a linear increase of slope \(s\) on an interval \([a,b] \in [0,1]\).
dlin(x, a, b, s)
plin(q, a, b, s)
qlin(p, a, b, s)
rlin(n, a, b, s)dlin gives the values of the density function, plin those of the distribution
function, and qlin those of the quantile function of the PUD at \(x, q,\) and \(p\),
respectively. rlin generates \(n\) random numbers, returned as an ordered vector.
Vector of quantiles.
Vector of probabilities.
Number of observations.
Left interval endpoint, real number in \([0,1)\).
Right interval endpoint, real number in \((0,1]\).
Slope parameter, real number such that \(|s| \le 2/(b-a)\).
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
Guenther Walther, gwalther@stanford.edu,
www-stat.stanford.edu/~gwalther
The what we call perturbed uniform distribution (PUD) with perturbation on an interval \([a,b] \in [0,1]\) with slope parameter \(s\) such that \(|s| \le 2 / (b-a)\) has density function
$$f_{a, b, s}(x) = \Bigl(sx-s\frac{a+b}{2}\Bigr)1\{x \in [a,b)\} + 1\{[0,a) \cup [b,1]\},$$
distribution function
$$F_{a, b, s}(q) = \Bigl(q+\frac{s}{2}(q^2-a^2+(a-x)(a+b)) \Bigr)1\{q \in [a,b)\} + q\{[0,a) \cup [b,1]\},$$
and quantile function
$$F_{a, b, s}^{-1}(p) = \Bigl(-s^{-1}+\frac{a+b}{2}+\frac{s \sqrt{(a-b)^2+\frac{4}{s}(\frac{1}{s}-(a+b)+2p)}}{2|s|} \Bigr) \ 1\{p \in [a,b)\} + p\{[0,a) \cup [b,1]\}.$$
This function was used to carry out the simulations to compute the power curves given in Rufibach and Walther (2010).
Rufibach, K. and Walther, G. (2010). A general criterion for multiscale inference. J. Comput. Graph. Statist., 19, 175--190.