Discrete Weighted Lindley Distribution

Description

Probability mass function, distribution function, quantile function and random number generation for the discrete weighted Lindley distribution with parameters theta and alpha.

Usage

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ddwlindley(x, theta, alpha, log = FALSE)

pdwlindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE)

qdwlindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE)

rdwlindley(n, theta, alpha)

Arguments

x, q

vector of integer positive quantiles.

theta, alpha

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 length(n) > 1, the length is taken to be the number required.

Details

Probability mass function

P(X=x\mid θ ,α )=\frac{1}{≤ft( θ +α \right) Γ ≤ft( α \right) }∑\limits_{i=0}^{1}≤ft( -1\right) ^{i}≤ft\{ ≤ft( θ +α \right) Γ ≤ft[ α ,θ ≤ft( x+i\right) \right] +≤ft[θ ≤ft( x+i\right) \right] ^{α }e^{-θ ≤ft( x+i\right)}\right\}

where Γ ≤ft(α,θ x\right) = \int_{θ x}^{∞}x^{α -1}e^{-x}dx is the upper incomplete gamma function.

Particular case: α = 1 the one-parameter discrete Lindley distribution.

Value

ddwlindley gives the probability mass function, pdwlindley gives the distribution function, qdwlindley gives the quantile function and rdwlindley generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Ricardo P. de Oliveira rpuziol.oliveira@gmail.com

Source

[d-p-q-r]dwlindley are calculated directly from the definitions. rdwlindley uses the discretize values.

References

Al-Mutairi, D. K., Ghitany, M. E., Kundu, D., (2015). Inferences on stress-strength reliability from weighted Lindley distributions. Communications in Statistics - Theory and Methods, 44, (19), 4096-4113.

Bashir, S., Rasul, M., (2015). Some properties of the weighted Lindley distribution. EPRA Internation Journal of Economic and Business Review, 3, (8), 11-17.

Ghitany, M. E., Alqallaf, F., Al-Mutairi, D. K. and Husain, H. A., (2011). A two-parameter weighted Lindley distribution and its applications to survival data. Mathematics and Computers in Simulation, 81, (6), 1190-1201.

Mazucheli, J., Louzada, F., Ghitany, M. E., (2013). Comparison of estimation methods for the parameters of the weighted Lindley distribution. Applied Mathematics and Computation, 220, 463-471.

Mazucheli, J., Coelho-Barros, E. A. and Achcar, J. (2016). An alternative reparametrization on the weighted Lindley distribution. Pesquisa Operacional, (to appear).

See Also

WLindley.

Examples

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set.seed(1)
x <- rdwlindley(n = 1000, theta = 1.5, alpha = 1.5)
plot(table(x) / sum(table(x)))
points(unique(x),ddwlindley(unique(x), theta = 1.5, alpha = 1.5))

## fires in Greece data (from Bakouch et al., 2014)
data(fires)
library(fitdistrplus)
fit <- fitdist(fires, 'dwlindley', start = list(theta = 0.30, alpha = 1.0), discrete = TRUE)
gofstat(fit, discrete = TRUE)
plot(fit)