dGGEO: The GGEO distribution

In DiscreteDists: Discrete Statistical Distributions

The GGEO distribution

Description

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

Usage

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.

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.

Value

dGGEO gives the density, pGGEO gives the distribution function, qGGEO gives the quantile function, rGGEO generates random deviates.

Author(s)

Valentina Hurtado Sepulveda, [e-mail]

References

\insertRef

gomez2010DiscreteDists

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)")

DiscreteDists documentation built on Sept. 14, 2024, 1:07 a.m.