DiscreteDists (version 1.0.0)

dDBH: The Discrete Burr Hatke distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Discrete Burr Hatke distribution with parameter \(\mu\).

Usage

dDBH(x, mu, log = FALSE)
pDBH(q, mu, lower.tail = TRUE, log.p = FALSE)
qDBH(p, mu = 1, lower.tail = TRUE, log.p = FALSE)
rDBH(n, mu = 1)

Value

dDBH gives the density, pDBH gives the distribution function, qDBH gives the quantile function, rDBH

generates random deviates.

Arguments

x, q

vector of (non-negative integer) quantiles.

mu

vector of the mu 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, \(P[X > x]\).

p

vector of probabilities.

n

number of random values to return

Author

Valentina Hurtado Sepulveda,

Details

The Discrete Burr-Hatke distribution with parameters \(\mu\) has a support 0, 1, 2, ... and density given by

\(f(x | \mu) = (\frac{1}{x+1}-\frac{\mu}{x+2})\mu^{x}\)

The pmf is log-convex for all values of \(0 < \mu < 1\), where \(\frac{f(x+1;\mu)}{f(x;\mu)}\) is an increasing function in \(x\) for all values of the parameter \(\mu\).

Note: in this implementation we changed the original parameters \(\lambda\) for \(\mu\), we did it to implement this distribution within gamlss framework.

References

el2020discreteDiscreteDists

See Also

DBH.

Examples

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

plot(x=0:5, y=dDBH(x=0:5, mu=0.1),
     type="h", lwd=2, col="dodgerblue", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 1),
     main="Probability mu=0.1")

plot(x=0:10, y=dDBH(x=0:10, mu=0.5),
     type="h", lwd=2, col="tomato", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 1),
     main="Probability mu=0.5")

plot(x=0:15, y=dDBH(x=0:15, mu=0.9),
     type="h", lwd=2, col="green4", las=1,
     ylab="P(X=x)", xlab="X", ylim=c(0, 1),
     main="Probability mu=0.9")

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

x_max <- 15
cumulative_probs1 <- pDBH(q=0:x_max, mu=0.1)
cumulative_probs2 <- pDBH(q=0:x_max, mu=0.5)
cumulative_probs3 <- pDBH(q=0:x_max, mu=0.9)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for Burr-Hatke",
     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=0.1",
                "mu=0.5",
                "mu=0.9"))

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

mu <- 0.4
x_max <- 10
probs1 <- dDBH(x=0:x_max, mu=mu)
names(probs1) <- 0:x_max

x <- rDBH(n=1000, mu=mu)
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 <- 0.97
p <- seq(from=0, to=1, by = 0.01)
qxx <- qDBH(p, mu, lower.tail = TRUE, log.p = FALSE)
plot(p, qxx, type="s", lwd=2, col="green3", ylab="quantiles",
     main="Quantiles of BH(mu=0.97)")