Density function, distribution function, quantile function and
random generation for the perturbed uniform distribution having a linear increase of slope $s$
on an interval $[a,b] \in [0,1]$.
Usage
dlin(x, a, b, s)
plin(q, a, b, s)
qlin(p, a, b, s)
rlin(n, a, b, s)
Arguments
x, q
Vector of quantiles.
p
Vector of probabilities.
n
Number of observations.
a
Left interval endpoint, real number in $[0,1)$.
b
Right interval endpoint, real number in $(0,1]$.
s
Slope parameter, real number such that $|s| \le 2/(b-a)$.
Value
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.
Details
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).
References
Rufibach, K. and Walther, G. (2010).
A general criterion for multiscale inference.
J. Comput. Graph. Statist., 19, 175--190.