DiscreteDists (version 1.0.0)

dPOISXL: The Discrete Poisson XLindley

Description

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

Usage

dPOISXL(x, mu = 0.3, log = FALSE)
pPOISXL(q, mu = 0.3, lower.tail = TRUE, log.p = FALSE)
qPOISXL(p, mu = 0.3, lower.tail = TRUE, log.p = FALSE)
rPOISXL(n, mu = 0.3)

Value

dPOISXL gives the density, pPOISXL gives the distribution function, qPOISXL gives the quantile function, rPOISXL

generates random deviates.

Arguments

x, q

vector of (non-negative integer) quantiles.

mu

vector of the mu 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

Mariana Blandon Mejia,

Details

The Discrete Poisson XLindley distribution with parameters \(\mu\) has a support 0, 1, 2, ... and mass function given by

\(f(x | \mu) = \frac{\mu^2(x+\mu^2+3(1+\mu))}{(1+\mu)^{4+x}}\); with \(\mu>0\).

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

References

ahsan2022DiscreteDists

See Also

POISXL.

Examples

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

x_max <- 20
probs1 <- dPOISXL(x=0:x_max, mu=0.2)
probs2 <- dPOISXL(x=0:x_max, mu=0.5)
probs3 <- dPOISXL(x=0:x_max, mu=1.0)

# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="dodgerblue", las=1,
     ylab="P(X=x)", xlab="X", main="Probability for Poisson XLindley",
     ylim=c(0, 0.50))
points(x=0:x_max, y=probs2, type="o", lwd=2, col="tomato")
points(x=0:x_max, y=probs3, type="o", lwd=2, col="green4")
legend("topright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=0.2", "mu=0.5", "mu=1.0"))

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

x_max <- 20

plot_discrete_cdf(x=0:x_max,
                  fx=dPOISXL(x=0:x_max, mu=0.2), col="dodgerblue",
                  main="CDF for Poisson XLindley with mu=0.2")

plot_discrete_cdf(x=0:x_max,
                  fx=dPOISXL(x=0:x_max, mu=0.5), col="tomato",
                  main="CDF for Poisson XLindley with mu=0.5")

plot_discrete_cdf(x=0:x_max,
                  fx=dPOISXL(x=0:x_max, mu=1.0), col="green4",
                  main="CDF for Poisson XLindley with mu=1.0")

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

x_max <- 15
probs1 <- dPOISXL(x=0:x_max, mu=0.3)
names(probs1) <- 0:x_max

x <- rPOISXL(n=3000, mu=0.3)
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 <- 0.3
p <- seq(from=0, to=1, by = 0.01)
qxx <- qPOISXL(p, mu, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
     main="Quantiles for Poisson XLindley mu=0.3")