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lin: 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 $[0,1)$.
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.