dDIKUM function

The discrete Inverted Kumaraswamy distribution

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

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)

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.

Returns

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

generates random deviates.

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.

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)")

References

\insertRef EL_Helbawy2022DiscreteDists

See Also

DIKUM .

Author(s)

Daniel Felipe Villa Rengifo, dvilla@unal.edu.co