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.
Value
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.
References
Gómez-Déniz E (2010). “Another generalization of the geometric distribution.” Test, 19, 399-415.
See also
GGEO.
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)")
