DiscreteDists (version 1.1.1)

dBerG: Bernoulli-geometric distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Bernoulli-geometric distribution with parameters \(\mu\) and \(\sigma\).

Usage

dBerG(x, mu, sigma, log = FALSE)
pBerG(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rBerG(n, mu, sigma)
qBerG(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)

Value

dBerG gives the density, pBerG gives the distribution function, qBerG gives the quantile function, rBerG

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

Hermes Marques,

Details

The BerG distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and mass function given by

\(f(x | \mu, \sigma) = \frac{(1-\mu+\sigma)}{(1+\mu+\sigma)}\) if \(x=0\),

\(f(x | \mu, \sigma) = 4 \mu \frac{(\mu+\sigma-1)^{x-1}}{(\mu+\sigma+1)^{x+1}}\) if \(x=1, 2, ...\),

with \(\mu > 0\), \(\sigma > 0\) and \(\sigma>|\mu-1|\).

References

Bourguignon, M., & de Medeiros, R. M. (2022). A simple and useful regression model for fitting count data. Test, 31(3), 790-827.

See Also

BerG.

Examples

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

x_max <- 20
probs1 <- dBerG(x=0:x_max, mu=0.7, sigma=0.5)
probs2 <- dBerG(x=0:x_max, mu=0.3, sigma=1)
probs3 <- dBerG(x=0:x_max, mu=2, sigma=3)

# 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 BerG",
     ylim=c(0, 0.80))
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=0.7, sigma=0.5",
                "mu=0.3, sigma=1",
                "mu=2, sigma=3"))

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

#plot1
x_max <- 10
plot_discrete_cdf(x=0:x_max,
                  fx=dBerG(x=0:x_max, mu=1, sigma=2),
                  col="dodgerblue",
                  main="CDF for BerG",
                  lwd=3)
legend("bottomright", legend="mu=1, sigma=2",
       col="dodgerblue", lty=1, lwd=2, cex=0.8)


#plot2
plot_discrete_cdf(x=0:x_max,
                  fx=dBerG(x=0:x_max, mu=3, sigma=3),
                  col="tomato",
                  main="CDF for BerG",
                  lwd=3)
legend("bottomright", legend="mu=3, sigma=3",
       col="tomato", lty=1, lwd=2, cex=0.8)


#plot3
plot_discrete_cdf(x=0:x_max,
                  fx=dBerG(x=0:x_max, mu=5, sigma=5),
                  col="green4",
                  main="CDF for BerG",
                  lwd=3)
legend("bottomright", legend="mu=5, sigma=5",
       col="green4", lty=1, lwd=2, cex=0.8)


# Example 3
# Comparing the random generator output with
# the theoretical probabilities

x_max <- 15
probs1 <- dBerG(x=0:x_max, mu=0.5, sigma=5)
names(probs1) <- 0:x_max

x <- rBerG(n=1000, mu=0.5, sigma=5)
probs2 <- prop.table(table(x))

cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
              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 <- 1
sigma <- 2
p <- seq(from=0, to=1, by=0.01)
qxx <- qBerG(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 DBerG(mu=1, sigma=2)")