Description Usage Arguments Details Value Author(s) Source References See Also Examples
Density function, distribution function, quantile function, random number generation and hazard rate function for the Marshall-Olkin extended Lindley distribution with parameters theta and alpha.
1 2 3 4 5 6 7 8 9 | dmolindley(x, theta, alpha, log = FALSE)
pmolindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE)
qmolindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE)
rmolindley(n, theta, alpha)
hmolindley(x, theta, alpha, log = FALSE)
|
x, q |
vector of positive quantiles. |
theta, alpha |
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) =\frac{α θ^{2}(1+x)e^{-θ x}}{(1+θ )≤ft[ 1-\overline{α }≤ft( 1+\frac{θ x}{1+θ }\right) e^{-θ x}\right] ^{2}}
Cumulative distribution function
F(x\mid θ,α) =1-\frac{α ≤ft( 1+\frac{θ x}{1+θ }\right) e^{-θ x}}{1-\overline{α }≤ft( 1+\frac{θ x}{1+θ }\right) e^{-θ x}}
Quantile function
Q(p\mid θ,α )=-1-\frac{1}{θ }-\frac{1}{θ }W_{-1}≤ft( \frac{(θ +1)}{e^{1 + θ}}\frac{(p-1)}{≤ft( 1-\overline{α }p\right) }\right)
Hazard rate function
h(x\mid θ,α) =\frac{θ ^{2}≤ft( 1+x\right) }{≤ft( 1+θ +θ x\right) ≤ft[ 1-\overline{α }≤ft( 1+\frac{θ x}{1+θ }\right) e^{-θ x}\right] }
where \overline{α}=(1 - α) and W_{-1} denotes the negative branch of the Lambert W function.
Particular case: α=1 the one-parameter Lindley distribution.
dmolindley
gives the density, pmolindley
gives the distribution function, qmolindley
gives the quantile function, rmolindley
generates random deviates and hmolindley
gives the hazard rate function.
Invalid arguments will return an error message.
Josmar Mazucheli jmazucheli@gmail.com
Larissa B. Fernandes lbf.estatistica@gmail.com
[d-h-p-q-r]molindley are calculated directly from the definitions. rmolindley
uses the quantile function.
do Espirito Santo, A. P. J., Mazucheli, J., (2015). Comparison of estimation methods for the Marshall-Olkin extended Lindley distribution. Journal of Statistical Computation and Simulation, 85, (17), 3437-3450.
Ghitany, M. E., Al-Mutairi, D. K., Al-Awadhi, F. A. and Al-Burais, M. M., (2012). Marshall-Olkin extended Lindley distribution and its application. International Journal of Applied Mathematics, 25, (5), 709-721.
Marshall, A. W., Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, (3), 641.652.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | set.seed(1)
x <- rmolindley(n = 1000, theta = 5, alpha = 5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dmolindley(S, theta = 5, alpha = 5), 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)
pmolindley(q, theta = 5, alpha = 5, lower.tail = TRUE)
pmolindley(q, theta = 5, alpha = 5, lower.tail = FALSE)
qmolindley(p, theta = 5, alpha = 5, lower.tail = TRUE)
qmolindley(p, theta = 5, alpha = 5, lower.tail = FALSE)
## bladder cancer data (from Warahena-Liyanage and Pararai, 2014)
data(bladdercancer)
library(fitdistrplus)
fit <- fitdist(bladdercancer, 'molindley', start = list(theta = 0.1, alpha = 1.0))
plot(fit)
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