The hyper-Poisson distribution

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

dHYPERPO(x, mu = 1, sigma = 1, log = FALSE)

pHYPERPO(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rHYPERPO(n, mu = 1, sigma = 1)

qHYPERPO(p, mu = 1, 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.

Value

dHYPERPO gives the density, pHYPERPO gives the distribution function, qHYPERPO gives the quantile function, rHYPERPO generates random deviates.

Details

The hyper-Poisson distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and density given by

\(f(x | \mu, \sigma) = \frac{\mu^x}{_1F_1(1;\mu;\sigma)}\frac{\Gamma(\sigma)}{\Gamma(x+\sigma)}\)

where the function \(_1F_1(a;c;z)\) is defined as

\(_1F_1(a;c;z) = \sum_{r=0}^{\infty}\frac{(a)_r}{(c)_r}\frac{z^r}{r!}\)

and \((a)_r = \frac{\gamma(a+r)}{\gamma(a)}\) for \(a>0\) and \(r\) positive integer.

Note: in this implementation we changed the original parameters \(\lambda\) and \(\gamma\) for \(\mu\) and \(\sigma\) respectively, we did it to implement this distribution within gamlss framework.

References

saez2013hyperpoDiscreteDists

See also

HYPERPO.

Author

Freddy Hernandez, fhernanb@unal.edu.co

Examples

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

x_max <- 30
probs1 <- dHYPERPO(x=0:x_max, mu=5, sigma=0.1)
probs2 <- dHYPERPO(x=0:x_max, mu=5, sigma=1.0)
probs3 <- dHYPERPO(x=0:x_max, mu=5, sigma=1.8)

# 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 hyper-Poisson",
     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=5, sigma=0.1",
                "mu=5, sigma=1.0",
                "mu=5, sigma=1.8"))


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

x_max <- 15
cumulative_probs1 <- pHYPERPO(q=0:x_max, mu=5, sigma=0.1)
cumulative_probs2 <- pHYPERPO(q=0:x_max, mu=5, sigma=1.0)
cumulative_probs3 <- pHYPERPO(q=0:x_max, mu=5, sigma=1.8)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for hyper-Poisson",
     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=5, sigma=0.1",
                "mu=5, sigma=1.0",
                "mu=5, sigma=1.8"))


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

x_max <- 15
probs1 <- dHYPERPO(x=0:x_max, mu=3, sigma=1.1)
names(probs1) <- 0:x_max

x <- rHYPERPO(n=1000, mu=3, sigma=1.1)
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 <- 3
sigma <-3
p <- seq(from=0, to=1, by=0.01)
qxx <- qHYPERPO(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 HP(mu=3, sigma=3)")