plindleynb {LindleyPowerSeries} | R Documentation |
LindleyNegativeBinomial
Description
distribution function, density function, hazard rate function, quantile function, random number generation
Usage
plindleynb(x, lambda, theta, m, log.p = FALSE)
dlindleynb(x, lambda, theta, m)
qlindleynb(p, lambda, theta, m)
rlindleynb(n, lambda, theta, m)
Arguments
x |
vector of positive quantiles. |
lambda |
positive parameter |
theta |
positive parameter. |
m |
target for number of successful trials. Must be strictly positive, need not be integer. |
log.p |
logical; If |
p |
vector of probabilities. |
n |
number of observations. |
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.
Value
plindleynb
gives the culmulative distribution function
dlindleynb
gives the probability density function
hlindleynb
gives the hazard rate function
qlindleynb
gives the quantile function
rlindleynb
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
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)
plindleynb(x, lambda, theta, m, log.p = FALSE)
dlindleynb(x, lambda, theta, m)
hlindleynb(x, lambda, theta, m)
qlindleynb(p, lambda, theta, m)
rlindleynb(n, lambda, theta, m)