Type: | Package |
Title: | Lindley Power Series Distribution |
Version: | 1.0.1 |
Author: | Saralees Nadarajah & Yuancheng Si, Peihao Wang |
Maintainer: | Yuancheng Si <siyuanchengman@gmail.com> |
Description: | Computes the probability density function, the cumulative distribution function, the hazard rate function, the quantile function and random generation for Lindley Power Series distributions, see Nadarajah and Si (2018) <doi:10.1007/s13171-018-0150-x>. |
License: | GPL-2 | GPL-3 [expanded from: GPL (≥ 2)] |
Encoding: | UTF-8 |
RoxygenNote: | 7.1.1 |
Imports: | stats, lamW(≥ 1.3.0) |
NeedsCompilation: | no |
Packaged: | 2021-07-10 13:16:35 UTC; siyua |
Repository: | CRAN |
Date/Publication: | 2021-07-10 16:50:02 UTC |
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 |
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
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.
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)
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)
LindleyGeometric
Description
distribution function, density function, hazard rate function, quantile function, random number generation
Usage
plindleygeometric(x, lambda, theta, log.p = FALSE)
dlindleygeometric(x, lambda, theta)
hlindleygeometric(x, lambda, theta)
qlindleygeometric(p, lambda, theta)
rlindleygeometric(n, lambda, theta)
Arguments
x |
vector of positive quantiles. |
lambda |
positive parameter |
theta |
positive parameter. |
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
plindleygeometric
gives the culmulative distribution function
dlindleygeometric
gives the probability density function
hlindleygeometric
gives the hazard rate function
qlindleygeometric
gives the quantile function
rlindleygeometric
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
x <- seq(from = 0.1,to = 6,by = 0.5)
p <- seq(from = 0.1,to = 1,by = 0.1)
plindleygeometric(x, lambda, theta, log.p = FALSE)
dlindleygeometric(x, lambda, theta)
hlindleygeometric(x, lambda, theta)
qlindleygeometric(p, lambda, theta)
rlindleygeometric(n, lambda, theta)
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 |
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
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.
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
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)
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)
LindleyPoisson
Description
distribution function, density function, hazard rate function, quantile function, random number generation
Usage
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 |
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
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
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)