dHYPERPO {DiscreteDists} | R Documentation |
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 |
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, fhernanb@unal.edu.co
References
Sáez-Castillo AJ, Conde-Sánchez A (2013). “A hyper-Poisson regression model for overdispersed and underdispersed count data.” Computational Statistics & Data Analysis, 61, 148–157.
See Also
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)")