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 extended power Lindley distribution with parameters theta, alpha and beta.
1 2 3 4 5 6 7 8 9 | dextplindley(x, theta, alpha, beta, log = FALSE)
pextplindley(q, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qextplindley(p, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
rextplindley(n, theta, alpha, beta, mixture = TRUE)
hextplindley(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. |
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{α θ ^{2}}{θ +β }}(1+β x^{α })\ x^{α -1}\ e^{-θ x^{α }}
Cumulative distribution function
F(x\mid θ,α,β )=1-≤ft( 1+{\frac{β θ x^{α }}{θ +β }}\right) \ e^{-θ x^{α }}
Quantile function
Q(p\mid θ ,α ,β )={≤ft[ -\frac{1}{θ }-\frac{1}{β }-{\frac{1}{θ }}W_{-1}{≤ft( \frac{1}{β }≤ft( p-1\right) ≤ft( β +θ \right) e{{^{-≤ft( {\frac{β +θ }{β }}\right) }}}\right) }\right] }^{\frac{1}{α }}
Hazard rate function
h(x\mid θ ,α ,β )={\frac{α {θ }^{2}≤ft( 1+β {x}^{α }\right) {x}^{α -1}}{≤ft( β +θ \right) {≤ft(1+{\frac{β θ {x}^{α }}{β +θ }}\right) }}}
where W_{-1} denotes the negative branch of the Lambert W function.
Particular cases: β = 1 the power Lindley distribution, α = 1 the two-parameter Lindley distribution and (α = 1, β = 1) the one-parameter Lindley distribution.
dextplindley
gives the density, pextplindley
gives the distribution function, qextplindley
gives the quantile function, rextplindley
generates random deviates and hextplindley
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]extplindley are calculated directly from the definitions. rextplindley
uses either a two-component mixture of gamma distributions or the quantile function.
Alkarni, S. H., (2015). Extended power Lindley distribution: A new statistical model for non-monotone survival data. European Journal of Statistics and Probability, 3, (3), 19-34.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | set.seed(1)
x <- rextplindley(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, dextplindley(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)
pextplindley(q, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = TRUE)
pextplindley(q, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = FALSE)
qextplindley(p, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = TRUE)
qextplindley(p, theta = 1.5, alpha = 1.5, beta = 1.5, lower.tail = FALSE)
library(fitdistrplus)
fit <- fitdist(x, 'extplindley', start = list(theta = 1.5, alpha = 1.5, beta = 1.5))
plot(fit)
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