lin {modehunt} | R Documentation |
Perturbed Uniform Distribution
Description
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 |
b |
Right interval endpoint, real number in |
s |
Slope parameter, real number such that |
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).
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.
Author(s)
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
Guenther Walther, gwalther@stanford.edu,
www-stat.stanford.edu/~gwalther
References
Rufibach, K. and Walther, G. (2010). A general criterion for multiscale inference. J. Comput. Graph. Statist., 19, 175–190.