Description Usage Arguments Details Value Note Author(s) Source References See Also Examples
Density function, distribution function, quantile function, random number generation and hazard rate function for the exponentiated power Lindley distribution with parameters theta, alpha and beta.
1 2 3 4 5 6 7 8 9 | dexpplindley(x, theta, alpha, beta, log = FALSE)
pexpplindley(q, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qexpplindley(p, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
rexpplindley(n, theta, alpha, beta)
hexpplindley(x, theta, alpha, beta, log = FALSE)
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x, q |
vector of positive quantiles. |
theta, alpha, beta |
positive parameters. |
log, log.p |
logical; If TRUE, probabilities p are given as log(p). |
lower.tail |
logical; If TRUE, (default), P(X ≤q x) are returned, otherwise P(X > x). |
p |
vector of probabilities. |
n |
number of observations. If |
Probability density function
f(x\mid θ,α,β )={\frac{β α θ ^{2}}{1 + θ}}(1+x^{α })x^{α -1}e^{-θ x^{α }}≤ft[ 1-≤ft( 1+{\frac{θ x^{α }}{1 + θ}}\right) e^{-θ x^{α }}\right]^{β -1}
Cumulative distribution function
F(x\mid θ,α,β )=≤ft[ 1-≤ft( 1+{\frac{θ x^{α }}{1 + θ}}\right) \ e^{-θ x^{α }}\right] ^{β }
Quantile function
Q(p\mid θ ,α ,β )=≤ft( -1-\frac{1}{θ }-\frac{1}{θ }W_{-1}≤ft( ≤ft( 1+θ \right) ≤ft( p^{\frac{1}{β }}-1\right)e^{-≤ft( 1+θ \right) }\right) \right) ^{\frac{1}{α }}
Hazard rate function
h(x\midθ,α,β )={\frac{β α θ ^{2}(1+x^{α })x^{α -1}e^{-θ x^{α }}≤ft[ 1-≤ft( 1+{\frac{θ x^{α }}{θ+1}}\right) e^{-θ x^{α }}\right] ^{β -1}}{≤ft( θ +1\right) ≤ft\{ 1-≤ft[ 1-≤ft( 1+{\frac{θ x^{α }}{1 + θ}}\right) \ e^{-θ x^{α }}\right] ^{β }\right\} }}
where W_{-1} denotes the negative branch of the Lambert W function.
Particular cases: α = 1 the exponentiated Lindley distribution, β = 1 the power Lindley distribution and (α = 1, β = 1) the one-parameter Lindley distribution. See Warahena-Liyanage and Pararai (2014) for other particular cases.
dexpplindley
gives the density, pexpplindley
gives the distribution function, qexpplindley
gives the quantile function, rexpplindley
generates random deviates and hexpplindley
gives the hazard rate function.
Invalid arguments will return an error message.
Warahena-Liyanage and Pararai (2014) named the exponentiated power Lindley distribution as generalized power Lindley distribution.
Josmar Mazucheli jmazucheli@gmail.com
Larissa B. Fernandes lbf.estatistica@gmail.com
[d-h-p-q-r]expplindley are calculated directly from the definitions. rexpplindley
uses the quantile function.
Ashour, S. K., Eltehiwy, M. A., (2015). Exponentiated power Lindley distribution. Journal of Advanced Research, 6, (6), 895-905.
Warahena-Liyanage, G., Pararai, M., (2014). A generalized power Lindley distribution with applications. Asian Journal of Mathematics and Applications, 2014, 1-23.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | set.seed(1)
x <- rexpplindley(n = 1000, theta = 11.0, alpha = 5.0, beta = 2.0)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.01)
plot(S, dexpplindley(S, theta = 11.0, alpha = 5.0, beta = 2.0), xlab = 'x', ylab = 'pdf')
hist(x, prob = TRUE, main = '', add = TRUE)
p <- seq(from = 0.1, to = 0.9, by = 0.1)
q <- quantile(x, prob = p)
pexpplindley(q, theta = 11.0, alpha = 5.0, beta = 2.0, lower.tail = TRUE)
pexpplindley(q, theta = 11.0, alpha = 5.0, beta = 2.0, lower.tail = FALSE)
qexpplindley(p, theta = 11.0, alpha = 5.0, beta = 2.0, lower.tail = TRUE)
qexpplindley(p, theta = 11.0, alpha = 5.0, beta = 2.0, lower.tail = FALSE)
## bladder cancer data (from Warahena-Liyanage and Pararai, 2014)
data(bladdercancer)
library(fitdistrplus)
fit <- fitdist(bladdercancer, 'expplindley', start = list(theta = 1, alpha = 1, beta = 1))
plot(fit)
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