DiscreteDists (version 1.0.0)

dDLD: The Discrete Lindley distribution

Description

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

Usage

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)

Value

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

generates random deviates.

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.

Author

Yojan Andrés Alcaraz Pérez,

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

bakouch2014newDiscreteDists

See Also

DLD.

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)")