DiscreteDists (version 1.1.1)

dCOMPO: The COMPO distribution

Description

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

Usage

dCOMPO(x, mu, sigma, log = FALSE)
pCOMPO(q, mu, sigma, lower.tail = TRUE, log.p = FALSE)
qCOMPO(p, mu, sigma, lower.tail = TRUE, log.p = FALSE)
rCOMPO(n, mu, sigma)

Value

dCOMPO gives the density, pCOMPO gives the distribution function, qCOMPO gives the quantile function, rCOMPO

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, \(P[X > x]\).

p

vector of probabilities.

n

number of random values to return.

Author

Freddy Hernandez,

Details

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

\(f(x | \mu, \sigma) = \frac{\mu^x}{(x!)^{\sigma} Z(\mu, \sigma)} \)

with \(\mu > 0\), \(\sigma \geq 0\) and

\(Z(\mu, \sigma)=\sum_{j=0}^{\infty} \frac{\mu^j}{(j!)^\sigma}\).

The proposed functions here are based on the functions from the COMPoissonReg package.

References

Shmueli, G., Minka, T. P., Kadane, J. B., Borle, S., & Boatwright, P. (2005). A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society Series C: Applied Statistics, 54(1), 127-142.

See Also

COMPO.

Examples

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

x_max <- 20
probs1 <- dCOMPO(x=0:x_max, mu=2, sigma=0.5)
probs2 <- dCOMPO(x=0:x_max, mu=8, sigma=1.0)
probs3 <- dCOMPO(x=0:x_max, mu=15, sigma=1.5)

# 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 COMPO",
     ylim=c(0, 0.30))
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=2, sigma=0.5",
                "mu=8, sigma=1.0",
                "mu=15, sigma=1.5"))

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

x_max <- 20
cumulative_probs1 <- pCOMPO(q=0:x_max, mu=2, sigma=0.5)
cumulative_probs2 <- pCOMPO(q=0:x_max, mu=8, sigma=1.0)
cumulative_probs3 <- pCOMPO(q=0:x_max, mu=15, sigma=1.5)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for COMPO",
     xlab="X", ylab="Probability")
points(x=0:x_max, y=cumulative_probs2, type="o", col="tomato")
points(x=0:x_max, y=cumulative_probs3, type="o", col="green4")
legend("bottomright", col=c("dodgerblue", "tomato", "green4"), lwd=3,
       legend=c("mu=2, sigma=0.5",
                "mu=8, sigma=1.0",
                "mu=15, sigma=1.5"))

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

x_max <- 50
probs1 <- dCOMPO(x=0:x_max, mu=5, sigma=0.5)
names(probs1) <- 0:x_max

x <- rCOMPO(n=1000, mu=5, sigma=0.5)
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 <- 3
sigma <- 1.5
p <- seq(from=0.01, to=0.99, by=0.01)
qxx <- qCOMPO(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 COMPO(mu = 3, sigma = 1.5)")