A Description of R Package 'dhh'"
In dhh: A Heavy-Headed Distribution

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Introduction

The name of this R package "dhh" can be understood as "Distributions with Heavy Heads". The probability density function of the heavy-headed distribution is $$f(x) = \frac{\alpha (b-a)^{-\alpha}}{(x-a)^{1-\alpha}}I{a0$. For the motivation and derivation of this distribution, please see the references.

This package contains four functions on a heavy-headed distribution:

dhh: the probability density function
phh: the cumulative distribution function (CDF)
qhh: the quantile function
rhh: the random variables

Next, I will introduce these functions with examples.

To use this library, we can first install it and then use the following command.

library(dhh)

On `dhh`

This function dhh(x, a, b, alpha) gives the values of the density of the heavy-headed distribution.

The default values of a, b and alpha are 0, 1 and 0.1, respectively. Then, the following two commands give the same result, the value of the density at 0.5.

dhh(0.5)
dhh(0.5, 0, 1, 0.1)

The argument x can be a vector.

dhh(c(0.5, 0.7))

We can take a look at the plots of some density functions.

curve(dhh(x, a=0, b=1, alpha=0.1), -1, 2)
curve(dhh(x, a=0, b=10, alpha=0.1), -1, 11)

On `phh`

This function phh(x, a, b, alpha) gives the values of the CDF of the heavy-headed distribution.

The default values of a, b and alpha are 0, 1 and 0.1, respectively. Then, the following two commands give the same result, the value of the density at 0.5.

phh(0.5)
phh(0.5, 0, 1, 0.1)

The argument x can be a vector.

phh(c(0.5, 0.7))

We can take a look at the plots of some CDF's.

curve(phh(x, a=0, b=1, alpha=0.1), -1, 2)
curve(phh(x, a=0, b=10, alpha=0.1), -1, 11)

On `qhh`

This function qhh(p, a, b, alpha) gives the quantiles of the heavy-headed distribution.

The default values of a, b and alpha are 0, 1 and 0.1, respectively. Then, the following two commands give the same result, the value of the density at 0.5.

qhh(0.9)
qhh(0.9, a=0, b=1, alpha=0.1)

The argument p can be a vector.

phh(qhh((1:9)/10))

We can take a look at the plots of some quantile functions.

curve(qhh(x, 0, 1, 0.1), from = 0.1, to = 0.9)
curve(qhh(x, a=10, b=100, alpha = 0.1), from = 0.1, to = 0.9)

On `rhh`

This function rhh(n, a, b, alpha1) generate $n$ i.i.d. random variables following the heavy-headed distribution.

The default values of a, b and alpha are 0, 1 and 0.1, respectively. Then, the following two commands are the same.

rhh(5)
rhh(5, a = 0, b = 1, alpha = 0.1)

We can generate $10000$ i.i.d. random variables and plot the histogram. Then we can add the density plot.

hist(rhh(10000), freq=FALSE)
curve(dhh, add = TRUE, col = 2)

We can also check the fact that the density goes to infinity as a goes to a.

dhh(c(0.1, 0.01, 0.001, 0.0001, 0.00001))

References

Runlong Tang (2018) A Note On Finite Moments, Rediscovery Of The Pareto Distribution and Distributions With Heavy Tails and Heads (v1) https://sites.google.com/site/tangrunlong/notes-on-finance