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 quasi Lindley distribution with parameters theta and alpha.
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x, q |
vector of positive quantiles. |
theta |
positive parameter. |
alpha |
greater than -1. |
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 two-component mixture of gamma distributions, otherwise from the quantile function. |
Probability density function
f(x\mid θ ,α )={\frac{θ ≤ft( α +θ x\right) {{e}^{-θ x}}}{1+α }}
Cumulative distribution function
F(x\mid θ ,α )=1-{\frac{≤ft( 1+α +θ x\right) }{1+α }{e}^{-θ x}}
Quantile function
Q(p\mid θ ,α )=-\frac{1}{{θ }}-{\frac{α }{θ }}-\frac{1}{{θ }}{W}_{-1}≤ft( ≤ft( p-1\right) ≤ft( 1+α \right) {{e}^{-1-α }}\right)
Hazard rate function
h(x\mid θ ,α )=\frac{θ ≤ft( α +θ x\right) }{≤ft( 1+α +θ x\right) }
where W_{-1} denotes the negative branch of the Lambert W function.
Particular cases: α = θ the one-parameter Lindley distribution and for α=0 the gamma distribution with shape = 2 and scale = θ.
dqlindley
gives the density, pqlindley
gives the distribution function, qqlindley
gives the quantile function, rqlindley
generates random deviates and hqlindley
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]qlindley are calculated directly from the definitions. rqlindley
uses either a two-component mixture of gamma distributions or the quantile function.
Shanker, R. and Mishra, A. (2013). A quasi Lindley distribution. African Journal of Mathematics and Computer Science Research, 6, (4), 64-71.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | set.seed(1)
x <- rqlindley(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, dqlindley(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)
pqlindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
pqlindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
qqlindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
qqlindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
library(fitdistrplus)
fit <- fitdist(x, 'qlindley', start = list(theta = 1.5, alpha = 1.5))
plot(fit)
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