# LindleyPoisson: LindleyPoisson In LindleyPowerSeries: Lindley Power Series Distribution

## Description

distribution function, density function, hazard rate function, quantile function, random number generation

## Usage

 1 2 3 4 5 6 7 8 9 plindleypoisson(x, lambda, theta, log.p = FALSE) dlindleypoisson(x, lambda, theta) hlindleypoisson(x, lambda, theta) qlindleypoisson(p, lambda, theta) rlindleypoisson(n, lambda, theta) 

## Arguments

 x vector of positive quantiles. lambda positive parameter theta positive parameter. log.p logical; If TRUE, probabilities p are given as log(p). p vector of probabilities. n number of observations.

## Details

Probability density function

f(x)=\frac{θλ^2}{(λ+1)A(θ)}(1+x)exp(-λ x)A^{'}(φ)

Cumulative distribution function

F(x)=\frac{A(φ)}{A(θ)}

Quantile function

F^{-1}(p)=-1-\frac{1}{λ}-\frac{1}{λ}W_{-1}≤ft\{\frac{λ+1}{exp(λ+1)}≤ft[\frac{1}{θ}A^{-1}\{pA(θ)\}-1\right]\right\}

Hazard rate function

h(x)=\frac{θλ^2}{1+λ}(1+x)exp(-λ x)\frac{A^{'}(φ)}{A(θ)-A(φ)}

where W_{-1} denotes the negative branch of the Lambert W function. A(θ)=∑_{n=1}^{∞}a_nθ^{n} is given by specific power series distribution. Note that x>0, λ>0 for all members in Lindley Power Series distribution. 0<θ<1 for Lindley-Geometric distribution, Lindley-logarithmic distribution, Lindley-Negative Binomial distribution. θ>0 for Lindley-Poisson distribution, Lindley-Binomial distribution.

## Value

plindleypoisson gives the culmulative distribution function

dlindleypoisson gives the probability density function

hlindleypoisson gives the hazard rate function

qlindleypoisson gives the quantile function

rlindleypoisson gives the random number generatedy by distribution

Invalid arguments will return an error message.

## Author(s)

Saralees Nadarajah & Yuancheng Si siyuanchengman@gmail.com

Peihao Wang

## References

Si, Y. & Nadarajah, S., (2018). Lindley Power Series Distributions. Sankhya A, 9, pp1-15.

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.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 set.seed(1) lambda = 1 theta = 0.5 n = 10 x <- seq(from = 0.1,to = 6,by = 0.5) p <- seq(from = 0.1,to = 1,by = 0.1) plindleypoisson(x, lambda, theta, log.p = FALSE) dlindleypoisson(x, lambda, theta) hlindleypoisson(x, lambda, theta) qlindleypoisson(p, lambda, theta) rlindleypoisson(n, lambda, theta) 

LindleyPowerSeries documentation built on July 10, 2021, 5:07 p.m.