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 generalized Lindley distribution with parameters theta, alpha and beta.
1 2 3 4 5 6 7 8 9 10 | dgenlindley(x, theta, alpha, beta, log = FALSE)
pgenlindley(q, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qgenlindley(p, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE,
L = 1e-04, U = 50)
rgenlindley(n, theta, alpha, beta, mixture = TRUE, L = 1e-04, U = 50)
hgenlindley(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. |
L, U |
interval which |
n |
number of observations. If |
mixture |
logical; If TRUE, (default), random deviates are generated from a two-component mixture of gamma distributions, otherwise from the quantile function. |
Probability density function
f( x\mid θ,α,β) =\frac{θ ^{α +1}}{≤ft( θ +β \right) Γ ≤ft( α +1\right) }x^{α -1}≤ft( α +β x\right) e^{-θ x}
Cumulative distribution function
F(x\mid θ,α,β )=∑_{j=0}^{1}≤ft\vert j-\frac{θ }{≤ft( θ +β \right) }\right\vert \frac{Γ ≤ft( α -j,θ x\right) }{Γ ≤ft( α -j\right) }
Quantile function
\code{does not have a closed mathematical expression}
Hazard rate function
h(x\mid θ,α,β )=\frac{θ ^{α +1}x^{α -1}≤ft(α +β x\right) e^{-θ x}}{≤ft( θ +β \right) Γ ≤ft( α +1\right) ∑\limits_{j=0}^{1}≤ft\vert j-\frac{θ }{ ≤ft( θ +β \right) }\right\vert \frac{Γ ≤ft( α -j,θ x\right) }{Γ ≤ft( α -j\right) }}
where Γ ≤ft( a,b\right) is the lower incomplete gamma function.
Particular cases: (α=1, β = 1) the one-parameter Lindley distribution, α=1 the two-parameter Lindley distribution, (α=1,β=0) the exponential distribution, β = 0 the gamma distribution and for β=α the weighted Lindley distribution.
dgenlindley
gives the density, pgenlindley
gives the distribution function, qgenlindley
gives the quantile function, rgenlindley
generates random deviates and hgenlindley
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]genlindley are calculated directly from the definitions. rgenlindley
uses either a two-component mixture of the gamma distributions or the quantile function.
Zakerzadeh, H., Dolati, A., (2009). Generalized Lindley distribution. Journal of Mathematical Extension, 3, (2), 13–25.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | set.seed(1)
x <- rgenlindley(n = 1000, theta = 1.5, alpha = 1.5, beta = 1.5, mixture = TRUE)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dgenlindley(S, theta = 1.5, alpha = 1.5, beta = 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)
pgenlindley(q, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = TRUE)
pgenlindley(q, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = FALSE)
qgenlindley(p, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = TRUE)
qgenlindley(p, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = FALSE)
library(fitdistrplus)
fit <- fitdist(x, 'genlindley', start = list(theta = 1.5, alpha = 1.5, beta = 1.5))
plot(fit)
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