dHYPERPO: The hyper-Poisson distribution

In DiscreteDists: Discrete Statistical Distributions

The hyper-Poisson distribution

Description

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

Usage

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.

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.

Value

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

Author(s)

Freddy Hernandez, [e-mail]

References

\insertRef

saez2013hyperpoDiscreteDists

See Also

HYPERPO.

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)")

DiscreteDists documentation built on Sept. 14, 2024, 1:07 a.m.