Unit-Power Half-Normal distribution

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

dUPHN(x, mu, sigma, log = FALSE)

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

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

rUPHN(n, mu, sigma)

Arguments

x, q

vector of (non-negative integer) quantiles.

mu

vector of the mu parameter.

sigma

vector of the sigma 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 Unit-Power Half-Normal distribution with parameters \(\mu\) and \(\sigma\) has a support in \((0, 1)\) and density given by

\(f(x| \mu, \sigma) = \frac{2\mu}{\sigma x^2} \phi(\frac{1-x}{\sigma x}) (2 \Phi(\frac{1-x}{\sigma x})-1)^{\mu-1}\)

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

References

Santoro, K. I., Gómez, Y. M., Soto, D., & Barranco-Chamorro, I. (2024). Unit-Power Half-Normal Distribution Including Quantile Regression with Applications to Medical Data. Axioms, 13(9), 599.

See also

dUPHN.

Author

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

Examples

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

curve(dUPHN(x, mu=2, sigma=1),
      add=TRUE, col= "red")

curve(dUPHN(x, mu=3, sigma=1),
      add=TRUE, col="seagreen")

curve(dUPHN(x, mu=4, sigma=1),
      add=TRUE, col="royalblue2")

legend("topright", col=c("black", "red", "seagreen", "royalblue2"),
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=2, sigma=1",
                "mu=3, sigma=1",
                "mu=4, sigma=1"))


# Example 2
# Checking if the cumulative curves converge to 1
curve(pUPHN(x, mu=1, sigma=1),
       from=0.01, to=0.99, col="black", las=1, ylab="F(x)")

curve(pUPHN(x, mu=2, sigma=1),
       add=TRUE, col= "red")

curve(pUPHN(x, mu=3, sigma=1),
       add=TRUE, col="seagreen")

curve(pUPHN(x, mu=4, sigma=1),
       add=TRUE, col="royalblue2")

legend("topleft", col=c("black", "red", "seagreen", "royalblue2"),
       lty=1, bty="n",
       legend=c("mu=1, sigma=1",
                "mu=2, sigma=1",
                "mu=3, sigma=1",
                "mu=4, sigma=1"))


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


# Example 4
# Comparing the random generator output with
# the theoretical density
x <- rUPHN(n= 10000, mu=4, sigma=1)
hist(x, freq=FALSE)
curve(dUPHN(x, mu=4, sigma=1),
      col="tomato", add=TRUE, from=0.01, to=0.99)