# The Argus distribution Density, distribution function, quantile function and random generation for the Argus distribution with parameter `chi`. `rargus()` uses very efficient random variate generation methods. Inversion or the Ratio of Uniforms method can be selected. UTF-8 ```r rargus(n=length(chi) , chi, method = c("inversion","RoU") ) dargus(x, chi, log = FALSE) pargus(x, chi, lower = TRUE, log.p = FALSE) qargus(p, chi, lower = TRUE, log.p = FALSE) ``` ## Arguments - `n`: Number of observations - `chi`: parameter vector `chi>0` - `method`: random variate generation method to be used (details are below) - `x`: x-value for the density or CDF - `log`: If `TRUE` the logarithm of the density is returned. - `lower`: If `FALSE` 1-CDF or its inverse is returned - `log.p`: `log(p)` the logarithm of the probability is used or returned - `p`: probability of the quantile ## Details The Argus distribution has the density (pdf) which is proportional to $$ f(x) = x \sqrt{1-x^2}\exp(-0.5\chi^2(1-x^2)), \mbox{for} 0 \leq x \leq 1 \mbox{and} \chi > 0%f(x) = x \sqrt(1-x^2)\exp(-0.5\chi^2(1-x^2)), for 0 <= x <= 1 and \chi > 0 $$ The generators used for `rargus()` are very efficient also for the varying parameter case, i.e. when the vector `chi` has length `n`. `method = "inversion"`: transforms the uniform variate into an argus variate using a close to exact approximation of the inverse CDF. It is also well suited for the varying parameter case as the theorem that an argus variate can be represented as a transformed truncated Gamma(1.5) variate allows to express the inverse CDF of the argus distribution as a simple transform of the inverse CDF of the Gamma(1.5) distribution. The use of the `pinv.new()` function of the `Runuran` package makes that evaluation very fast. When loading the argus-package `pinv.new()` calculates and stores the required tables. `method = "RoU"`: uses the Ratio of Uniforms method which requires 2 or more uniform variates to generate one argus variate. It is also a bit slower than the inversion method and is added here mainly for the case that the `Runuran`-package is not available. ## Returns `rargus` creates a random sample of size `n`. `dargus` gives the density, `dargus` give the CDF and `qargus` gives the quantile function. The length of the result for these 3 functions is the maximum of the lengths of the numerical arguments that are recycled to the length of the result. Only the first elements of the logical arguments are used. ## Author(s) Wolfgang and Christoph Baumgarten ## References Christoph Baumgarten: Random Variate Generation by Fast Numerical Inversion in the Varying Parameter Case. ## Examples ```r library(argus) # pdf, cdf and quantile function dargus((0:10)/10,.6) qargus(0.9,2) pargus(qargus(0.9,2),2) y<-rargus(n=1.e5,chi=0.3,method="inversion") hist(y,breaks=100,main="Argus pdf with chi=0.3",freq=FALSE) lines(xv<-seq(0,1,0.001),dargus(xv,chi=0.3),lwd=0.2,col=2) n=1.e5 chiv <- 1. +runif(n)*5 # chi-values in (1,5) system.time({set.seed(123);y<-rargus(chi=chiv)}) set.seed(123);u<- runif(n) mean(abs(pargus(y,chiv)-u)) # random variate generation with the inversion method y<-rargus(n=1.e5,chi=0.3,method="inversion") hist(y,breaks=100,main="Argus pdf with chi=0.3",freq=FALSE) lines(xv<-seq(0,1,0.001),dargus(xv,chi=0.3),lwd=0.2,col=2) # using Ratio of Uniforms y<-rargus(n=1.e5,chi=3,method="RoU") hist(y,breaks=100,main="Argus pdf with chi=3",freq=FALSE) lines(xv<-seq(0,1,0.001),dargus(xv,chi=3),lwd=2,col=2) # generating for different chi values y<-rargus(n=100,chi=runif(100)*2,method="RoU") ```

rargus function