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 one-parameter Lindley distribution with parameter theta.
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x, q |
vector of positive quantiles. |
theta |
positive parameter. |
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 )=\frac{θ ^{2}}{(1+θ )}(1+x)e^{-θ x}
Cumulative distribution function
F(x\mid θ ) =1 - ≤ft(1+ \frac{θ x}{1+θ }\right)e^{-θ x}
Quantile function
Q(p\mid θ )=-1-\frac{1}{θ }-\frac{1}{θ }W_{-1}≤ft((1+θ)( p-1)e^{-(1+θ) }\right)
Hazard rate function
h(x\mid θ )=\frac{θ ^{2}}{1+θ +θ x}(1+x)
where W_{-1} denotes the negative branch of the Lambert W function.
dlindley
gives the density, plindley
gives the distribution function, qlindley
gives the quantile function, rlindley
generates random deviates and hlindley
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]lindley are calculated directly from the definitions. rlindley
uses either a two-component mixture of the gamma distributions or the quantile function.
Ghitany, M. E., Atieh, B., Nadarajah, S., (2008). Lindley distribution and its application. Mathematics and Computers in Simulation, 78, (4), 49-506.
Jodra, P., (2010). Computer generation of random variables with Lindley or Poisson-Lindley distribution via the Lambert W function. Mathematics and Computers in Simulation, 81, (4), 851-859.
Lindley, D. V., (1958). Fiducial distributions and Bayes' theorem. Journal of the Royal Statistical Society. Series B. Methodological, 20, 102-107.
Lindley, D. V., (1965). Introduction to Probability and Statistics from a Bayesian View-point, Part II: Inference. Cambridge University Press, New York.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | set.seed(1)
x <- rlindley(n = 1000, theta = 1.5, mixture = TRUE)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dlindley(S, theta = 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)
plindley(q, theta = 1.5, lower.tail = TRUE)
plindley(q, theta = 1.5, lower.tail = FALSE)
qlindley(p, theta = 1.5, lower.tail = TRUE)
qlindley(p, theta = 1.5, lower.tail = FALSE)
## waiting times data (from Ghitany et al., 2008)
data(waitingtimes)
library(fitdistrplus)
fit <- fitdist(waitingtimes, 'lindley', start = list(theta = 0.1))
plot(fit)
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