Exponentiated Lindley Distribution

Description

Density function, distribution function, quantile function, random number generation and hazard rate function for the exponentiated Lindley distribution with parameters theta and alpha.

Usage

dexplindley(x, theta, alpha, log = FALSE)

pexplindley(q, theta, alpha, lower.tail = TRUE, log.p = FALSE)

qexplindley(p, theta, alpha, lower.tail = TRUE, log.p = FALSE)

rexplindley(n, theta, alpha)

hexplindley(x, theta, alpha, log = FALSE)

Arguments

Details

Probability density function

f(xmid θ,α )=frac{α θ ^{2}}{(1+θ )} (1+x)e^{-θ x}≤ft[ 1-≤ft( 1+frac{θ x}{1+θ } ight) e^{-θ x} ight] ^{α -1}

Cumulative distribution function

F(xmid θ,α )=≤ft[ 1-≤ft( 1+frac{θ x}{1+θ } ight) e^{-θ x} ight] ^{α }

Quantile function

Q(pmid θ ,α )=-1-frac{1}{θ }-{frac{1}{θ }}W_{-1}{≤ft( (p^{frac{1}{α }}-1)≤ft( 1+θ ight) e{^{-≤ft( 1+θ ight) }} ight) }

Hazard rate function

h(xmid θ ,α )={frac{α {θ }^{2}≤ft( 1+x ight) {{e}^{-θ x}}≤ft[ 1-≤ft( 1+{frac{θ x}{1+θ }} ight) e{^{-θ x}} ight] ^{α -1}}{≤ft( 1+θ ight) ≤ft{ 1-≤ft[ 1-≤ft( 1+frac{θ x}{1+θ } ight) e^{-θ x} ight] ^{α} ight} }}

where W_{-1} denotes the negative branch of the Lambert W function.

Particular case: α = 1 the one-parameter Lindley distribution.

Value

dexplindley gives the density, pexplindley gives the distribution function, qexplindley gives the quantile function, rexplindley generates random deviates and hexplindley gives the hazard rate function.

Invalid arguments will return an error message.

Note

Nadarajah et al. (2011) named the exponentiated Lindley distribution as generalized Lindley distribution.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Larissa B. Fernandes lbf.estatistica@gmail.com

Source

[d-h-p-q-r]explindley are calculated directly from the definitions. rexplindley uses the quantile function.

References

Nadarajah, S., Bakouch, H. S., Tahmasbi, R., (2011). A generalized Lindley distribution. Sankhya B, 73, (2), 331-359.

See Also

lambertWm1.

Examples

set.seed(1)
x <- rexplindley(n = 1000, theta = 1.5, alpha = 1.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dexplindley(S, theta = 1.5, alpha = 1.5), xlab = 'x', ylab = 'pdf')
hist(x, prob = TRUE, main = '', add = TRUE)

p <- seq(from = 0.1, to = 0.9, by = 0.1)
q <- quantile(x, prob = p)
pexplindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
pexplindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
qexplindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
qexplindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)

## Relief times data (from Nadarajah et al., 2011)
data(relieftimes)
library(fitdistrplus)
fit <- fitdist(relieftimes, 'explindley', start = list(theta = 1.5, alpha = 1.5))
plot(fit)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(LindleyR)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/LindleyR/EXPLindley.Rd_%03d_medium.png", width=480, height=480)
> ### Name: EXPLindley
> ### Title: Exponentiated Lindley Distribution
> ### Aliases: EXPLindley dexplindley hexplindley pexplindley qexplindley
> ###   relieftimes rexplindley
> 
> ### ** Examples
> 
> set.seed(1)
> x <- rexplindley(n = 1000, theta = 1.5, alpha = 1.5)
> R <- range(x)
> S <- seq(from = R[1], to = R[2], by = 0.1)
> plot(S, dexplindley(S, theta = 1.5, alpha = 1.5), xlab = 'x', ylab = 'pdf')
> hist(x, prob = TRUE, main = '', add = TRUE)
> 
> p <- seq(from = 0.1, to = 0.9, by = 0.1)
> q <- quantile(x, prob = p)
> pexplindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
      10%       20%       30%       40%       50%       60%       70%       80% 
0.1050439 0.2012339 0.3119572 0.3977886 0.4832603 0.5960067 0.7061000 0.7969307 
      90% 
0.9100919 
> pexplindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
       10%        20%        30%        40%        50%        60%        70% 
0.89495611 0.79876614 0.68804279 0.60221138 0.51673970 0.40399330 0.29390004 
       80%        90% 
0.20306929 0.08990809 
> qexplindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
[1] 0.2575551 0.4329961 0.6018380 0.7769918 0.9680684 1.1869294 1.4532823
[8] 1.8092681 2.3862366
> qexplindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
[1] 2.3862366 1.8092681 1.4532823 1.1869294 0.9680684 0.7769918 0.6018380
[8] 0.4329961 0.2575551
> 
> ## Relief times data (from Nadarajah et al., 2011)
> data(relieftimes)
> library(fitdistrplus)
Loading required package: MASS
> fit <- fitdist(relieftimes, 'explindley', start = list(theta = 1.5, alpha = 1.5))
> plot(fit)
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>