DiscreteDists (version 1.1.1)

dDIKUM: The discrete Inverted Kumaraswamy distribution

Description

These functions define the density, distribution function, quantile function and random generation for the discrete Inverted Kumaraswamy, DIKUM(), distribution with parameters \(\mu\) and \(\sigma\).

Usage

dDIKUM(x, mu = 1, sigma = 5, log = FALSE)
pDIKUM(q, mu = 1, sigma = 5, lower.tail = TRUE, log.p = FALSE)
rDIKUM(n, mu = 1, sigma = 5)
qDIKUM(p, mu = 1, sigma = 5, lower.tail = TRUE, log.p = FALSE)

Value

dDIKUM gives the density, pDIKUM gives the distribution function, qDIKUM gives the quantile function, rDIKUM

generates random deviates.

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, \(P[X > x]\).

n

number of random values to return.

p

vector of probabilities.

Author

Daniel Felipe Villa Rengifo,

Details

The discrete Inverted Kumaraswamy distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and density given by

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

with \(\mu > 0\) and \(\sigma > 0\).

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

References

El-Helbawy, A. A., Hegazy, M. A., Al-Dayian, G. R., & Abd EL-Kader, R. E. (2022). A discrete analog of the inverted Kumaraswamy distribution: properties and estimation with application to COVID-19 data. Pakistan Journal of Statistics and Operation Research, 18(1), 297-328.

See Also

DIKUM.

Examples

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

x_max <- 30

probs1 <- dDIKUM(x=0:x_max, mu=1, sigma=5)
probs2 <- dDIKUM(x=0:x_max, mu=1, sigma=20)
probs3 <- dDIKUM(x=0:x_max, mu=1, sigma=50)

# 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 Inverted Kumaraswamy Distribution",
     ylim=c(0, 0.12))
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=1, sigma=5",
                "mu=1, sigma=20",
                "mu=1, sigma=50"))

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

x_max <- 500

cumulative_probs1 <- pDIKUM(q=0:x_max, mu=1, sigma=5)
cumulative_probs2 <- pDIKUM(q=0:x_max, mu=1, sigma=20)
cumulative_probs3 <- pDIKUM(q=0:x_max, mu=1, sigma=50)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for Inverted Kumaraswamy Distribution",
     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")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=1, sigma=5",
                "mu=1, sigma=20",
                "mu=1, sigma=50"))

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

x_max <- 20
probs1 <- dDIKUM(x=0:x_max, mu=3, sigma=20)
names(probs1) <- 0:x_max

x <- rDIKUM(n=1000, mu=3, sigma=20)
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 <- 1
sigma <- 5
p <- seq(from=0.01, to=0.99, by=0.1)
qxx <- qDIKUM(p=p, mu=mu, sigma=sigma, lower.tail=TRUE, log.p=FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
     main="Quantiles of HP(mu = sigma = 3)")