dGGEO: The GGEO distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Generalized Geometric distribution with parameters $μ$ and $σ$ .

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)

Value

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

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, vhurtados@unal.edu.co

Details

The GGEO distribution with parameters $μ$ and $σ$ has a support 0, 1, 2, ... and mass function given by

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

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

Note: in this implementation we changed the original parameters $θ$ for $μ$ and $α$ for $σ$ , we did it to implement this distribution within gamlss framework.

References

gomez2010DiscreteDists

Examples

Run this code

# 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)")

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