unitquantreg (version 0.0.6)

ashw: The arcsecant hyperbolic Weibull distribution

Description

Density function, distribution function, quantile function and random number generation function for the arcsecant hyperbolic Weibull distribution reparametrized in terms of the \(\tau\)-th quantile, \(\tau \in (0, 1)\).

Usage

dashw(x, mu, theta, tau = 0.5, log = FALSE)
pashw(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
qashw(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)
rashw(n, mu, theta, tau = 0.5)

Value

dashw gives the density, pashw gives the distribution function, qashw gives the quantile function and rashw generates random deviates.

Invalid arguments will return an error message.

Arguments

x, q

vector of positive quantiles.

mu

location parameter indicating the \(\tau\)-th quantile, \(\tau \in (0, 1)\).

theta

shape parameter.

tau

the parameter to specify which quantile use in the parametrization.

log, log.p

logical; If TRUE, probabilities p are given as log(p).

lower.tail

logical; If TRUE, (default), \(P(X \leq x)\) are returned, otherwise \(P(X > x)\).

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

Author

Josmar Mazucheli

André F. B. Menezes

Details

Probability density function $$f(y;\alpha, \theta)=\frac{\alpha \theta}{y\sqrt{1-y^2}} \mathrm{arcsech}(y)^{\theta-1}\exp\left [ -\alpha \mathrm{arcsech}(y)^\theta \right ]$$

Cumulative distribution function $$F(y;\alpha, \theta)=\exp\left [ -\alpha \mathrm{arcsech}(y)^\theta \right ]$$

Quantile function $$Q(\tau;\alpha, \theta)= \mathrm{sech}\left \{ \left [ -\alpha^{-1} \log(\tau)\right ]^{\frac{1}{\theta}} \right \}$$

Reparameterization $$\alpha = g^{-1}(\mu) = -\frac{\log(\tau)}{\mathrm{arcsech}(\mu)^\theta}$$

where \(\theta >0\) is the shape parameter and \(\mathrm{arcsech}(y)= \log\left[\left( 1+\sqrt{1-y^2} \right)/y \right]\).

References

Korkmaz, M. C., Chesneau, C. and Korkmaz, Z. S., (2021). A new alternative quantile regression model for the bounded response with educational measurements applications of OECD countries. Journal of Applied Statistics, 1--25.

Examples

set.seed(6969)
x <- rashw(n = 1000, mu = 0.5, theta = 2.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1L], to = R[2L], by = 0.01)
hist(x, prob = TRUE, main = 'arcsecant hyperbolic Weibull')
lines(S, dashw(x = S, mu = 0.5, theta = 2.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pashw(q = S, mu = 0.5, theta = 2.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qashw(p = S, mu = 0.5, theta = 2.5, tau = 0.5), col = 2)