dDGEII function

Discrete generalized exponential distribution - a second type

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 $\sigma$.

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)

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

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

generates random deviates.

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.

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])
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 <- 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)")

References

\insertRef nekoukhou2013DiscreteDists

See Also

DGEII .

Author(s)

Valentina Hurtado Sepulveda, vhurtados@unal.edu.co