A Description of R Package ‘dhh’
Runlong Tang
2018-07-30
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 functionphh: the cumulative distribution function (CDF)qhh: the quantile functionrhh: 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.
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