ashw | R Documentation |
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)
.
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)
x, q |
vector of positive quantiles. |
mu |
location parameter indicating the |
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 |
vector of probabilities. |
n |
number of observations. If |
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]
.
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.
Josmar Mazucheli
André F. B. Menezes
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.
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)
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