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 extended Lindley distribution with parameters theta, alpha and beta.
1 2 3 4 5 6 7 8 9 10 | dextlindley(x, theta, alpha, beta, log = FALSE)
pextlindley(q, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qextlindley(p, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE,
L = 1e-04, U = 50)
rextlindley(n, theta, alpha, beta, L = 1e-04, U = 50)
hextlindley(x, theta, alpha, beta, log = TRUE)
|
x, q |
vector of positive quantiles. |
theta |
positive parameter. |
alpha |
\rm I\!R^{-}\cup (0,1). |
beta |
greater than or equal to zero. |
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. |
L, U |
interval which |
n |
number of observations. If |
Probability density function
f(x\mid θ ,α ,β )=\frac{θ }{≤ft( 1+θ \right) }≤ft( 1+\frac{θ x}{1+θ }\right) ^{α -1}≤ft[ β ≤ft( 1+θ +θ x\right) ≤ft( θ x\right) ^{β -1}-α \right]e^{-≤ft( θ x\right) ^{β }}
Cumulative distribution function
F(x\mid θ ,α ,β )=1-≤ft( 1+\frac{θ x}{1+θ }\right)^{α }e^{-≤ft( θ x\right) ^{β }}
Quantile function
\code{does not have a closed mathematical expression}
Hazard rate function
h(x\mid θ ,α ,β )=\frac{β ≤ft( 1+θ +θ x\right)θ ^{β }x^{β -1}-α θ }{≤ft( 1+θ +θ x\right) }
Particular cases: (α = 1, β = 1) the one-parameter Lindley distribution, (α = 0, β = 1) the exponential distribution and for α = 0 the Weibull distribution. See Bakouch et al. (2012) for other particular cases.
dextlindley
gives the density, pextlindley
gives the distribution function, qextlindley
gives the quantile function, rextlindley
generates random deviates and hextlindley
gives the hazard rate function.
Invalid arguments will return an error message.
The uniroot
function with default arguments is used to find out the quantiles.
Josmar Mazucheli jmazucheli@gmail.com
Larissa B. Fernandes lbf.estatistica@gmail.com
[d-h-p-q-r]extlindley are calculated directly from the definitions. rextlindley
uses the quantile function.
Bakouch, H. S., Al-Zahrani, B. M., Al-Shomrani, A. A., Marchi, V. A. A., Louzada, F., (2012). An extended Lindley distribution. Journal of the Korean Statistical Society, 41, (1), 75-85.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | set.seed(1)
x <- rextlindley(n = 1000, theta = 5.0, alpha = -1.0, beta = 5.0)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.01)
plot(S, dextlindley(S, theta = 5.0, alpha = -1.0, beta = 5.0), 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)
pextlindley(q, theta = 5.0, alpha = -1.0, beta = 5.0, lower.tail = TRUE)
pextlindley(q, theta = 5.0, alpha = -1.0, beta = 5.0, lower.tail = FALSE)
qextlindley(p, theta = 5.0, alpha = -1.0, beta = 5.0, lower.tail = TRUE)
qextlindley(p, theta = 5.0, alpha = -1.0, beta = 5.0, lower.tail = FALSE)
library(fitdistrplus)
fit <- fitdist(x, 'extlindley', start = list(theta = 5.0, alpha = -1.0, beta = 5.0))
plot(fit)
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