A Description of R Package ‘dhh’

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 \[f(x) = \frac{\alpha (b-a)^{-\alpha}}{(x-a)^{1-\alpha}}I\{a<x<b\},\] with parameters \(a<b\) and \(\alpha>0\). 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)
#> [1] 0.1866066
dhh(0.5, 0, 1, 0.1)
#> [1] 0.1866066

The argument x can be a vector.

dhh(c(0.5, 0.7))
#> [1] 0.1866066 0.1378516

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)
#> [1] 0.933033
phh(0.5, 0, 1, 0.1)
#> [1] 0.933033

The argument x can be a vector.

phh(c(0.5, 0.7))
#> [1] 0.9330330 0.9649611

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)
#> [1] 0.3486784
qhh(0.9, a=0, b=1, alpha=0.1)
#> [1] 0.3486784

The argument p can be a vector.

phh(qhh((1:9)/10))
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

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)
#> [1] 1.252758e-04 4.515955e-03 1.327694e-02 1.447834e-13 1.917654e-02
rhh(5, a = 0, b = 1, alpha = 0.1)
#> [1] 8.962788e-01 1.679130e-01 1.665611e-06 1.211562e-04 3.247341e-01

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))
#> [1]    0.7943282    6.3095734   50.1187234  398.1071706 3162.2776602

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