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 inverse Lindley distribution with parameters theta, alpha and beta.
1 2 3 4 5 6 7 8 9 | dextilindley(x, theta, alpha, beta, log = FALSE)
pextilindley(q, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
qextilindley(p, theta, alpha, beta, lower.tail = TRUE, log.p = FALSE)
rextilindley(n, theta, alpha, beta, mixture = TRUE)
hextilindley(x, theta, alpha, beta, log = TRUE)
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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 inverse-gamma distributions, otherwise from the quantile function. #' |
Probability density function
f(x\mid θ ,α ,β )=\frac{β θ ^{2}}{θ +α }≤ft( \frac{α +x^{β }}{x^{2β +1}}\right) e^{-\frac{θ }{ x^{β }}}
Cumulative distribution function
F(x\mid θ ,α ,β )=≤ft( 1+\frac{θ α }{≤ft( θ +α \right) }\frac{1}{{x}^{β }}\right) e{{^{-{\frac{θ }{ x^{β }}}}}}
Quantile function
Q(p\mid θ ,α ,β) =≤ft[ -\frac{1}{θ }-\frac{1}{α }-\frac{1}{θ }W_{-1}{≤ft( -\frac{p}{α }≤ft( θ+α \right) {e{^{-≤ft( {\frac{θ +α }{α }}\right) }}}\right) }\right] ^{-\frac{1}{β }}
Hazard rate function
h(x\mid θ ,α ,β )=\frac{β θ ^{2}≤ft( α+x^{β }\right) e^{-\frac{θ }{x^{β }}}}{≤ft( θ +α\right) x^{2β +1}≤ft[ 1-≤ft( 1+\frac{θ α }{≤ft( θ+α \right) }\frac{1}{{x}^{β }}\right) e{{^{-{\frac{θ }{x^{β }}}}}}\right] }
where W_{-1} denotes the negative branch of the Lambert W function.
Particular cases: α = 1, β = 1 the inverse Lindley distribution, α = 1 the generalized inverse Lindley distribution and for α = 0 the inverse Weibull distribution.
dextilindley
gives the density, pextilindley
gives the distribution function, qextilindley
gives the quantile function, rextilindley
generates random deviates and hextilindley
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]extilindley are calculated directly from the definitions. rextilindley
uses either a two-component mixture of generalized inverse gamma distributions or the quantile function.
Alkarni, S. H., (2015). Extended inverse Lindley distribution: properties and application. SpringerPlus, 4, (1), 690-703.
Mead, M. E., (2015). Generalized inverse gamma distribution and its application in reliability. Communication in Statistics - Theory and Methods, 44, 1426-1435.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | set.seed(1)
x <- rextilindley(n = 10000, theta = 5, alpha = 20, beta = 10)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.01)
plot(S, dextilindley(S, theta = 5, alpha = 20, beta = 20), 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)
pextilindley(q, theta = 5, alpha = 20, beta = 10, lower.tail = TRUE)
pextilindley(q, theta = 5, alpha = 20, beta = 10, lower.tail = FALSE)
qextilindley(p, theta = 5, alpha = 20, beta = 10, lower.tail = TRUE)
qextilindley(p, theta = 5, alpha = 20, beta = 10, lower.tail = FALSE)
library(fitdistrplus)
fit <- fitdist(x, 'extilindley', start = list(theta = 5, alpha = 20, beta = 10))
plot(fit)
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