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LindleyPowerSeries (version 1.0.1)

plindleybinomial: LindleyBinomial

Description

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

Usage

plindleybinomial(x, lambda, theta, m, log.p = FALSE)

dlindleybinomial(x, lambda, theta, m)

hlindleybinomial(x, lambda, theta, m)

qlindleybinomial(p, lambda, theta, m)

rlindleybinomial(n, lambda, theta, m)

Arguments

x

vector of positive quantiles.

lambda

positive parameter

theta

positive parameter.

m

number of trails.

log.p

logical; If TRUE, probabilities \(p\) are given as \(log(p)\).

p

vector of probabilities.

n

number of observations.

Value

plindleybinomial gives the culmulative distribution function

dlindleybinomial gives the probability density function

hlindleybinomial gives the hazard rate function

qlindleybinomial gives the quantile function

rlindleybinomial gives the random number generatedy by distribution

Invalid arguments will return an error message.

Details

Probability density function $$f(x)=\frac{\theta\lambda^2}{(\lambda+1)A(\theta)}(1+x)exp(-\lambda x)A^{'}(\phi)$$

Cumulative distribution function $$F(x)=\frac{A(\phi)}{A(\theta)}$$

Quantile function $$F^{-1}(p)=-1-\frac{1}{\lambda}-\frac{1}{\lambda}W_{-1}\left\{\frac{\lambda+1}{exp(\lambda+1)}\left[\frac{1}{\theta}A^{-1}\{pA(\theta)\}-1\right]\right\}$$

Hazard rate function $$h(x)=\frac{\theta\lambda^2}{1+\lambda}(1+x)exp(-\lambda x)\frac{A^{'}(\phi)}{A(\theta)-A(\phi)}$$

where \(W_{-1}\) denotes the negative branch of the Lambert W function. \(A(\theta)=\sum_{n=1}^{\infty}a_n\theta^{n}\) is given by specific power series distribution. Note that \(x>0, \lambda>0\) for all members in Lindley Power Series distribution. \(0<\theta<1\) for Lindley-Geometric distribution, Lindley-logarithmic distribution, Lindley-Negative Binomial distribution. \(\theta>0\) for Lindley-Poisson distribution, Lindley-Binomial distribution.

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

Run this code
# NOT RUN {
set.seed(1)
lambda = 1
theta = 0.5
n = 10
m = 10
x <- seq(from = 0.1,to = 6,by = 0.5)
p <- seq(from = 0.1,to = 1,by = 0.1)
plindleybinomial(x, lambda, theta, m, log.p = FALSE)
dlindleybinomial(x, lambda, theta, m)
hlindleybinomial(x, lambda, theta, m)
qlindleybinomial(p, lambda, theta, m)
rlindleybinomial(n, lambda, theta, m)
# }

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