paretoconv

An R package to calculate the n-fold convolution of Pareto distributions. f(x)=a x^1-a, for x>0, a>1 using the techniques devised by Colin Ramsay in

1. ‘The Distribution of Sums of Certain I.I.D. Pareto Variates’ (Communications in Statistics - Theory and Methods 35:395-405, 2006); and

2. ‘The Distribution of Sums of I.I.D. Pareto Random Variables with Arbitrary Shape Parameter’ (Communications in Statistics - Theory and Methods 37:2177-2184, 2008).

The package contains only one function:

paretoconv (x, a, n, cdf=FALSE)

where n specifies the number of convolutions. Both this and a must be single-valued, while x can be a vector. cdf generates the cumulative distribution function, otherwise the probability density function is returned.

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Installation

paretoconv is not (yet) on CRAN, and can be installed with any of the following options:

remotes::install_git("https://git.sr.ht/~mpadge/paretoconv")
remotes::install_bitbucket("mpadge/paretoconv")
remotes::install_gitlab("mpadge/paretoconv")
remotes::install_github("mpadge/paretoconv")

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Example

Solid lines in the figure below are a reproduction of Ramsay’s (2006) Figure 2 of probability density functions for the first 5 convolutions of the Pareto pdf with shape parameter of a=5. Dashed lines are analogous values for the non-integer value of a=4.5.

x <- 1:50 / 10
n <- 1:5
yint <- lapply (n, function (i) paretoconv (x=x, a=5, n=i))
ynon <- lapply (n, function (i) paretoconv (x=x, a=4.5, n=i))
cols <- rainbow (length (n))
plot (NULL, NULL, xlim=range (x), ylim=range (yint, na.rm=TRUE), xlab='x', ylab='p')
for (i in n) {
    lines (x, yint [[i]], col=cols [i])
    lines (x, ynon [[i]], col=cols [i], lty=2)
}
legend ('topright', lwd=1, col=cols, bty='n', 
        legend=sapply (seq (n), function (i) paste0 ('n=', i)))