dGGEO function

The GGEO distribution

These functions define the density, distribution function, quantile function and random generation for the Generalized Geometric distribution with parameters $\mu$ and $\sigma$.

dGGEO(x, mu = 0.5, sigma = 1, log = FALSE)

pGGEO(q, mu = 0.5, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rGGEO(n, mu = 0.5, sigma = 1)

qGGEO(p, mu = 0.5, sigma = 1, 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

dGGEO gives the density, pGGEO gives the distribution function, qGGEO gives the quantile function, rGGEO

generates random deviates.

Details

The GGEO distribution with parameters $\mu$ and $\sigma$

has a support 0, 1, 2, ... and mass function given by

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

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

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

Examples

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

x_max <- 80
probs1 <- dGGEO(x=0:x_max, mu=0.5, sigma=10)
probs2 <- dGGEO(x=0:x_max, mu=0.7, sigma=30)
probs3 <- dGGEO(x=0:x_max, mu=0.9, 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 GGEO",
     ylim=c(0, 0.20))
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=0.5, sigma=10",
                "mu=0.7, sigma=30",
                "mu=0.9, sigma=50"))

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

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

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

plot_discrete_cdf(x=0:x_max,
                  fx=dGGEO(x=0:x_max, mu=0.5, sigma=50),
                  col="green4",
                  main="CDF for GGEO",
                  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 <- dGGEO(x=0:x_max, mu=0.5, sigma=5)
names(probs1) <- 0:x_max

x <- rGGEO(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 <- qGGEO(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 GGEO(mu=0.5, sigma=0.5)")

References

\insertRef gomez2010DiscreteDists

See Also

GGEO .

Author(s)

Valentina Hurtado Sepulveda, vhurtados@unal.edu.co