dHYPERPO2: The hyper-Poisson distribution (with mu as mean)

In DiscreteDists: Discrete Statistical Distributions

The hyper-Poisson distribution (with mu as mean)

Description

These functions define the density, distribution function, quantile function and random generation for the hyper-Poisson in the second parameterization with parameters \mu (as mean) and \sigma as the dispersion parameter.

Usage

dHYPERPO2(x, mu = 1, sigma = 1, log = FALSE)

pHYPERPO2(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

rHYPERPO2(n, mu = 1, sigma = 1)

qHYPERPO2(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE)

Arguments

x, q	vector of (non-negative integer) quantiles.
mu	vector of positive values of this parameter.
sigma	vector of positive values of this 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, ...

Note: in this implementation the parameter \mu is the mean of the distribution and \sigma corresponds to the dispersion parameter. If you fit a model with this parameterization, the time will increase because an internal procedure to convert \mu to \lambda parameter.

Value

dHYPERPO2 gives the density, pHYPERPO2 gives the distribution function, qHYPERPO2 gives the quantile function, rHYPERPO2 generates random deviates.

Author(s)

Freddy Hernandez, [e-mail]

References

\insertRef

saez2013hyperpoDiscreteDists

See Also

HYPERPO2, HYPERPO.

Examples

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

x_max <- 30
probs1 <- dHYPERPO2(x=0:x_max, sigma=0.01, mu=3)
probs2 <- dHYPERPO2(x=0:x_max, sigma=0.50, mu=5)
probs3 <- dHYPERPO2(x=0:x_max, sigma=1.00, mu=7)
# 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, 0.30))
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("sigma=0.01, mu=3",
                "sigma=0.50, mu=5",
                "sigma=1.00, mu=7"))

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

x_max <- 15
cumulative_probs1 <- pHYPERPO2(q=0:x_max, mu=1, sigma=1.5)
cumulative_probs2 <- pHYPERPO2(q=0:x_max, mu=3, sigma=1.5)
cumulative_probs3 <- pHYPERPO2(q=0:x_max, mu=5, 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 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("sigma=1.5, mu=1",
                "sigma=1.5, mu=3",
                "sigma=1.5, mu=5"))

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

x_max <- 15
probs1 <- dHYPERPO2(x=0:x_max, mu=3, sigma=1.1)
names(probs1) <- 0:x_max

x <- rHYPERPO2(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 <- qHYPERPO2(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 HP2(mu = sigma = 3)")

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