The DsPA distribution

The function DsPA() defines the Discrete

dDsPA(x, mu, sigma, log = FALSE)

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

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

rDsPA(n, mu, sigma)

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, \(P[X > x]\).
p: vector of probabilities.
n: number of random values to return.

Value

dDsPA gives the density, pDsPA gives the distribution function, qDsPA gives the quantile function.

Details

The DsPA distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ...

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

References

Alghamdi, A. S., Ahsan-ul-Haq, M., Babar, A., Aljohani, H. M., Afify, A. Z., & Cell, Q. E. (2022). The discrete power-Ailamujia distribution: properties, inference, and applications. AIMS Math, 7(5), 8344-8360.

Author

Maria Camila Mena Romana, mamenar@unal.edu.co

Examples

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

x_max <- 30
probs1 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.5)
probs2 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.7)
probs3 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.9)

# 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 DsPA",
     ylim=c(0, 0.40))
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=1.2, sigma=0.5",
                "mu=1.2, sigma=0.7",
                "mu=1.2, sigma=0.9"))


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

x_max <- 15
cumulative_probs1 <- pDsPA(q=0:x_max, mu=1.2, sigma=0.5)
cumulative_probs2 <- pDsPA(q=0:x_max, mu=1.2, sigma=0.7)
cumulative_probs3 <- pDsPA(q=0:x_max, mu=1.2, sigma=0.9)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for DsPA",
     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=1.2, sigma=0.5",
                "mu=1.2, sigma=0.7",
                "mu=1.2, sigma=0.9"))


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

x_max <- 50
probs1 <- dDsPA(x=0:x_max, mu=1.2, sigma=0.9)
names(probs1) <- 0:x_max

x <- rDsPA(n=1000, mu=1.2, sigma=0.9)
probs2 <- prop.table(table(x))

cn <- union(names(probs1), names(probs2))
height <- rbind(probs1[cn], probs2[cn])
mp <- barplot(height, beside=TRUE, names.arg=cn,
              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 <- 1.2
sigma <- 0.9
p <- seq(from=0, to=1, by=0.01)
qxx <- qDsPA(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 DsPA(mu=1.2, sigma=0.9)")