Unit Half Logistic-Geometry distribution

These functions define the density, distribution function, quantile function and random generation for the Unit Half Logistic-Geometry distribution with parameter \(\mu\).

dUHLG(x, mu, log = FALSE)

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

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

rUHLG(n, mu)

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.

Value

dUHLG gives the density, pUHLG gives the distribution function, qUHLG gives the quantile function, rUHLG generates random deviates.

Details

The Unit Half Logistic-Geometry distribution with parameter \(\mu\) has a support in \((0, 1)\) and density given by

\(f(x| \mu) = \frac{2 \mu}{(\mu+(2-\mu)x)^2} \)

for \(0 < x < 1\) and \(\mu > 0\).

References

Ramadan, A. T., Tolba, A. H., & El-Desouky, B. S. (2022). A unit half-logistic geometric distribution and its application in insurance. Axioms, 11(12), 676.

Author

Juan Diego Suarez Hernandez, jsuarezhe@unal.edu.co

Examples

# Example 1
# Plotting the density function for different parameter values
curve(dUHLG(x, mu=0.4), from=0.01, to=0.99,
      ylim=c(0, 5), lwd=2,
      col="black", las=1, ylab="f(x)")

curve(dUHLG(x, mu=1), lwd=2,
      add=TRUE, col="red")

curve(dUHLG(x, mu=2), lwd=2,
      add=TRUE, col="green")

curve(dUHLG(x, mu=7), lwd=2,
      add=TRUE, col="blue")

legend("topright",
       col=c("black", "red", "green", "blue"),
       lty=1, bty="n", lwd=2,
       legend=c("mu=0.4",
                "mu=1",
                "mu=2",
                "mu=7"))


# Example 2
# Checking if the cumulative curves converge to 1
curve(pUHLG(x, mu=0.25), lwd=2,
      from=0.001, to=0.999, col="black", las=1, ylab="F(x)")

curve(pUHLG(x, mu=0.7), lwd=2,
      add=TRUE, col="red")

curve(pUHLG(x, mu=1.8), lwd=2,
      add=TRUE, col="green")

curve(pUHLG(x, mu=2.2), lwd=2,
      add=TRUE, col="blue")

legend("bottomright", col=c("black", "red", "green", "blue"),
       lty=1, bty="n", lwd=2,
       legend=c("mu=0.25",
                "mu=0.7",
                "mu=1.8",
                "mu=2.2"))


# Example 3
# Checking the quantile function
mu <- 2
p <- seq(from=0.01, to=0.99, length.out=100)
plot(x=qUHLG(p, mu=mu), y=p,
     xlab="Quantile", las=1, ylab="Probability")
curve(pUHLG(x, mu=mu), add=TRUE, col="red")


# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rUHLG(n=10000, mu=0.5)
hist(x, freq=FALSE)
curve(dUHLG(x, mu=0.5), lwd=2,
      col="tomato", add=TRUE, from=0.01, to=0.99)