The COMPO distribution

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

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)

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.

Value

dCOMPO gives the density, pCOMPO gives the distribution function, qCOMPO gives the quantile function, rCOMPO generates random deviates.

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 Comway–Maxwell–Poisson distribution. Journal of the Royal Statistical Society Series C: Applied Statistics, 54(1), 127-142.

See also

COMPO.

Author

Freddy Hernandez, fhernanb@unal.edu.co

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)")