The Discrete Lindley distribution

These functions define the density, distribution function, quantile function and random generation for the Discrete Lindley distribution with parameter \(\mu\).

dDLD(x, mu, log = FALSE)

pDLD(q, mu, lower.tail = TRUE, log.p = FALSE)

qDLD(p, mu, lower.tail = TRUE, log.p = FALSE)

rDLD(n, mu = 0.5)

Arguments

x, q: vector of (non-negative integer) quantiles.
mu: vector of positive values of this parameter.
log, log.p: logical; if TRUE, probabilities p are given as log(p).
lower.tail: logical; if TRUE (default), probabilities are \(P[X <= x]\), otherwise, \(P[X > x]\).
p: vector of probabilities.
n: number of random values to return.

Value

dDLD gives the density, pDLD gives the distribution function, qDLD gives the quantile function, rDLD

generates random deviates.

Details

The Discrete Lindley distribution with parameters \(\mu\) has a support 0, 1, 2, ... and density given by

\(f(x | \mu) = \frac{e^{-\mu x}}{1 + \mu} \left[ \mu(1 - 2e^{-\mu}) + (1- e^{-\mu})(1+\mu x)\right]\)

Note: in this implementation we changed the original parameters \(\theta\) for \(\mu\), we did it to implement this distribution within gamlss framework.

References

Bakouch HS, Jazi MA, Nadarajah S (2014). “A new discrete distribution.” Statistics, 48(1), 200--240.

Author

Yojan Andrés Alcaraz Pérez, yalcaraz@unal.edu.co

Examples

# Example 1
# Plotting the mass function for different parameter values

plot(x=0:25, y=dDLD(x=0:25, mu=0.2),
     type="h", lwd=2, col="dodgerblue", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 0.1),
     main="Probability mu=0.2")


plot(x=0:15, y=dDLD(x=0:15, mu=0.5),
     type="h", lwd=2, col="tomato", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 0.25),
     main="Probability mu=0.5")


plot(x=0:8, y=dDLD(x=0:8, mu=1),
     type="h", lwd=2, col="green4", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 0.5),
     main="Probability mu=1")


plot(x=0:5, y=dDLD(x=0:5, mu=2),
     type="h", lwd=2, col="red", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 1),
     main="Probability mu=2")


# Example 2
# Checking if the cumulative curves converge to 1

x_max <- 10
cumulative_probs1 <- pDLD(q=0:x_max, mu=0.2)
cumulative_probs2 <- pDLD(q=0:x_max, mu=0.5)
cumulative_probs3 <- pDLD(q=0:x_max, mu=1)
cumulative_probs4 <- pDLD(q=0:x_max, mu=2)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for Lindley",
     xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
points(x=0:x_max, y=cumulative_probs4, type="o", col="magenta")
legend("bottomright",
       col=c("dodgerblue", "tomato", "green4", "magenta"), lwd=3,
       legend=c("mu=0.2",
                "mu=0.5",
                "mu=1",
                "mu=2"))


# Example 3
# Comparing the random generator output with
# the theoretical probabilities

mu <- 0.6
x <- rDLD(n = 1000, mu = mu)
x_max <- max(x)
probs1 <- dDLD(x = 0:x_max, mu = mu)
names(probs1) <- 0:x_max

probs2 <- prop.table(table(x))

cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
nombres <- cn

mp <- barplot(height, beside = TRUE, names.arg = nombres,
              col=c('dodgerblue3','firebrick3'), las=1,
              xlab='X', ylab='Proportion')
legend('topright',
       legend=c('Theoretical', 'Simulated'),
       bty='n', lwd=3,
       col=c('dodgerblue3','firebrick3'), lty=1)


# Example 4
# Checking the quantile function

mu <- 0.9
p <- seq(from=0, to=1, by=0.01)
qxx <- qDLD(p, mu, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type="S", lwd=2, col="green3", ylab="quantiles",
     main="Quantiles of DL(mu=0.9)")