LindleyPowerSeries (version 1.0.1)

plindleylogarithmic: LindleyLogarithmic

Description

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

Usage

plindleylogarithmic(x, lambda, theta, log.p = FALSE)
dlindleylogarithmic(x, lambda, theta)
hlindleylogarithmic(x, lambda, theta)
qlindleylogarithmic(p, lambda, theta)
rlindleylogarithmic(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.

Value

plindleylogarithmic gives the culmulative distribution function

dlindleylogarithmic gives the probability density function

hlindleylogarithmic gives the hazard rate function

qlindleylogarithmic gives the quantile function

rlindleylogarithmic 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

# NOT RUN {
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)
plindleylogarithmic(x, lambda, theta, log.p = FALSE)
dlindleylogarithmic(x, lambda, theta)
hlindleylogarithmic(x, lambda, theta)
qlindleylogarithmic(p, lambda, theta)
rlindleylogarithmic(n, lambda, theta)
# }