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 Gamma Lindley distribution with parameters theta and alpha.
1 2 3 4 5 6 7 8 9 | dgamlindley(x, theta, alpha, log = FALSE)
pgamlindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE)
qgamlindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE)
rgamlindley(n, theta, alpha, mixture = TRUE)
hgamlindley(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 |
mixture |
logical; If TRUE, (default), random deviates are generated from a mixture of gamma and one-parameter Lindley distributions, otherwise from the quantile function. |
Probability density function
f(x\mid θ ,α )=\frac{{θ }^{2}}{α ≤ft( 1+θ \right) }≤ft[ ≤ft( α +α θ -θ \right) x+1\right] e{^{-θ x}}
Cumulative distribution function
F(x\mid θ ,α )=\frac{1}{α ≤ft( 1+θ \right) }≤ft[≤ft( α +α θ -θ \right) ≤ft( 1+θ x\right) +θ \right] e{^{-θ x}}
Quantile function
Q(p\mid θ ,α )=-\frac{α ≤ft( 1+θ \right) }{θ ≤ft[ ≤ft( α +α θ -θ \right) \right] }-\frac{1}{θ }W_{-1}≤ft( {\frac{≤ft( 1+θ \right) α ≤ft(p-1\right) }{α +α θ -θ }}e{{^{-{\frac{≤ft( 1+θ \right) α }{α θ +α -θ }}}}}\right)
Hazard rate function
h(x\mid θ ,α )=\frac{{θ }^{2}≤ft[ ≤ft( α +α θ -θ \right) x+1\right] }{{θ }≤ft( α +α θ-θ \right) x+α ≤ft( 1+θ \right) }
where W_{-1} denotes the negative branch of the Lambert W function.
Particular case: α = 1 the one-parameter Lindley distribution.
dgamlindley
gives the density, pgamlindley
gives the distribution function, qgamlindley
gives the quantile function, rgamlindley
generates random deviates and hgamlindley
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]gamlindley are calculated directly from the definitions. rgamlindley
uses either a mixture of gamma and one-parameter Lindley distributions or the quantile function.
Nedjar, S. and Zeghdoudi (2016). On gamma Lindley distribution: Properties and simulations. Journal of Computational and Applied Mathematics, 298, 167-174.
Zeghdoudi, H, and Nedjar, S. (2015) Gamma Lindley distribution and its application. Journal of Applied Probability and Statistics, 11, (1), 1-11.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | set.seed(1)
x <- rgamlindley(n = 1000, theta = 1.5, alpha = 1.5, mixture = TRUE)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dgamlindley(S, theta = 1.5, alpha = 1.5), 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)
pgamlindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
pgamlindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
qgamlindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
qgamlindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
library(fitdistrplus)
fit <- fitdist(x, 'gamlindley', start = list(theta = 1.5, alpha = 1.5))
plot(fit)
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