# DWLindley: Discrete Weighted Lindley Distribution In LindleyR: The Lindley Distribution and Its Modifications

## Description

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

## Usage

 1 2 3 4 5 6 7 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 [email protected]

Ricardo P. de Oliveira [email protected]

## 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).

WLindley.
  1 2 3 4 5 6 7 8 9 10 11 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)