DiscreteDists (version 1.1.1)

dDGEII: Discrete generalized exponential distribution - a second type

Description

These functions define the density, distribution function, quantile function and random generation for the Discrete generalized exponential distribution a second type with parameters \(\mu\) and \(\sigma\).

Usage

dDGEII(x, mu = 0.5, sigma = 1.5, log = FALSE)
pDGEII(q, mu = 0.5, sigma = 1.5, lower.tail = TRUE, log.p = FALSE)
rDGEII(n, mu = 0.5, sigma = 1.5)
qDGEII(p, mu = 0.5, sigma = 1.5, lower.tail = TRUE, log.p = FALSE)

Value

dDGEII gives the density, pDGEII gives the distribution function, qDGEII gives the quantile function, rDGEII

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

Valentina Hurtado Sepulveda,

Details

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

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

with \(0 < \mu < 1\) and \(\sigma > 0\). If \(\sigma=1\), the DGEII distribution reduces to the geometric distribution with success probability \(1-\mu\).

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

References

Nekoukhou, V., Alamatsaz, M. H., & Bidram, H. (2013). Discrete generalized exponential distribution of a second type. Statistics, 47(4), 876-887.

See Also

DGEII.

Examples

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

x_max <- 40
probs1 <- dDGEII(x=0:x_max, mu=0.1, sigma=5)
probs2 <- dDGEII(x=0:x_max, mu=0.5, sigma=5)
probs3 <- dDGEII(x=0:x_max, mu=0.9, sigma=5)

# 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 DGEII",
     ylim=c(0, 0.60))
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=0.1, sigma=5",
                "mu=0.5, sigma=5",
                "mu=0.9, sigma=5"))

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

#plot1
x_max <- 10
plot_discrete_cdf(x=0:x_max,
                  fx=dDGEII(x=0:x_max, mu=0.3, sigma=15),
                  col="dodgerblue",
                  main="CDF for DGEII",
                  lwd=3)
legend("bottomright", legend="mu=0.3, sigma=15",
       col="dodgerblue", lty=1, lwd=2, cex=0.8)


#plot2
plot_discrete_cdf(x=0:x_max,
                  fx=dDGEII(x=0:x_max, mu=0.5, sigma=30),
                  col="tomato",
                  main="CDF for DGEII",
                  lwd=3)
legend("bottomright", legend="mu=0.5, sigma=30",
       col="tomato", lty=1, lwd=2, cex=0.8)


#plot3
plot_discrete_cdf(x=0:x_max,
                  fx=dDGEII(x=0:x_max, mu=0.5, sigma=50),
                  col="green4",
                  main="CDF for DGEII",
                  lwd=3)
legend("bottomright", legend="mu=0.5, sigma=50",
       col="green4", lty=1, lwd=2, cex=0.8)


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

x_max <- 15
probs1 <- dDGEII(x=0:x_max, mu=0.5, sigma=5)
names(probs1) <- 0:x_max

x <- rDGEII(n=1000, mu=0.5, sigma=5)
probs2 <- prop.table(table(x))

cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
              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 <- 0.5
sigma <- 5
p <- seq(from=0, to=1, by=0.01)
qxx <- qDGEII(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 DDGEII(mu=0.5, sigma=5)")