dDLD: The Discrete Lindley distribution

In DiscreteDists: Discrete Statistical Distributions

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)

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.

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.

Value

dDLD gives the density, pDLD gives the distribution function, qDLD gives the quantile function, rDLD generates random deviates.

Author(s)

Yojan Andrés Alcaraz Pérez, [e-mail]

References

\insertRef

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)")

DiscreteDists documentation built on Sept. 14, 2024, 1:07 a.m.