dHYPERPO function

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.

Returns

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.

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)")

References

\insertRef saez2013hyperpoDiscreteDists

See Also

HYPERPO .

Author(s)

Freddy Hernandez, fhernanb@unal.edu.co