Two-Parameter Lindley Distribution

Description

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

Usage

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

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

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

rslindley(n, theta, alpha, mixture = TRUE)

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

Arguments

Details

Probability density function

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

Cumulative distribution function

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

Quantile function

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

Hazard rate function

h(xmid θ )=frac{θ ^{2}}{≤ft( θ + α +αθ x ight) }(1+α x)

where θ > 0, α > -θ and W_{-1} denotes the negative branch of the Lambert W function.

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

Value

dslindley gives the density, pslindley gives the distribution function, qslindley gives the quantile function, rslindley generates random deviates and hslindley gives the hazard rate function.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

Larissa B. Fernandes lbf.estatistica@gmail.com

Source

[d-h-p-q-r]slindley are calculated directly from the definitions. rslindley uses either a two-component mixture of the gamma distributions or the quantile function.

References

Shanker, R., Sharma, S. and Shanker, R. (2013). A two-parameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics, 4, (2), 363-368.

See Also

lambertWm1.

Examples

set.seed(1)
x <- rslindley(n = 1000, theta = 1.5, alpha = 1.5, mixture = TRUE)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.1)
plot(S, dslindley(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)
pslindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
pslindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
qslindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
qslindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)

library(fitdistrplus)
fit <- fitdist(x, 'slindley', 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/SLindley.Rd_%03d_medium.png", width=480, height=480)
> ### Name: SLindley
> ### Title: Two-Parameter Lindley Distribution
> ### Aliases: SLindley dslindley hslindley pslindley qslindley rslindley
> 
> ### ** Examples
> 
> set.seed(1)
> x <- rslindley(n = 1000, theta = 1.5, alpha = 1.5, mixture = TRUE)
> R <- range(x)
> S <- seq(from = R[1], to = R[2], by = 0.1)
> plot(S, dslindley(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)
> pslindley(q, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
       10%        20%        30%        40%        50%        60%        70% 
0.09043296 0.19088344 0.30270821 0.42976697 0.53676494 0.63408868 0.72294231 
       80%        90% 
0.81556379 0.90557371 
> pslindley(q, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
       10%        20%        30%        40%        50%        60%        70% 
0.90956704 0.80911656 0.69729179 0.57023303 0.46323506 0.36591132 0.27705769 
       80%        90% 
0.18443621 0.09442629 
> qslindley(p, theta = 1.5, alpha = 1.5, lower.tail = TRUE)
[1] 0.1341529 0.2729038 0.4205503 0.5820360 0.7641288 0.9774690 1.2413431
[8] 1.5981840 2.1812080
> qslindley(p, theta = 1.5, alpha = 1.5, lower.tail = FALSE)
[1] 2.1812080 1.5981840 1.2413431 0.9774690 0.7641288 0.5820360 0.4205503
[8] 0.2729038 0.1341529
> 
> library(fitdistrplus)
Loading required package: MASS
> fit <- fitdist(x, 'slindley', start = list(theta = 1.5, alpha = 1.5))
> plot(fit)
> 
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>