dHYPERPO2 function

The hyper-Poisson distribution (with mu as mean)

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.

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.

Returns

dHYPERPO2 gives the density, pHYPERPO2 gives the distribution function, qHYPERPO2 gives the quantile function, rHYPERPO2

generates random deviates.

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.

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)")

References

\insertRef saez2013hyperpoDiscreteDists

See Also

HYPERPO2 , HYPERPO .

Author(s)

Freddy Hernandez, fhernanb@unal.edu.co