DiscreteDists (version 1.1.2)

dDPERKS: The Discrete Perks distribution

Description

These functions define the density, distribution function, quantile function and random generation for the Discrete Perks, DPERKS(), distribution with parameters \(\mu\) and \(\sigma\).

Usage

dDPERKS(x, mu = 0.5, sigma = 0.5, log = FALSE)
pDPERKS(q, mu = 0.5, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)
rDPERKS(n, mu = 0.5, sigma = 0.5)
qDPERKS(p, mu = 0.5, sigma = 0.5, lower.tail = TRUE, log.p = FALSE)

Value

dDPERKS gives the density, pDPERKS gives the distribution function, qDPERKS gives the quantile function, rDPERKS

generates random deviates.

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, \(P[X > x]\).

n

number of random values to return.

p

vector of probabilities.

Author

Veronica Seguro Varela,

Details

The discrete Perks distribution with parameters \(\mu > 0\) and \(\sigma > 0\) has a support 0, 1, 2, ... and density given by

\(f(x | \mu, \sigma) = \frac{\mu(1+\mu)(e^\sigma-1)e^{\sigma x}}{(1+\mu e^{\sigma x})(1+\mu e^{\sigma(x+1)})} \)

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

References

Tyagi, A., Choudhary, N., & Singh, B. (2020). A new discrete distribution: Theory and applications to discrete failure lifetime and count data. J. Appl. Probab. Statist, 15, 117-143.

See Also

DPERKS.

Examples

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

x_max <- 25
probs1 <- dDPERKS(x=0:x_max, mu=0.001, sigma=0.52)
probs2 <- dDPERKS(x=0:x_max, mu=0.001, sigma=0.85)
probs3 <- dDPERKS(x=0:x_max, mu=0.001, sigma=1.5)

# To plot the first k values
plot(x=0:x_max, y=probs1, type="o", lwd=2, col="green4", las=1,
     ylab="P(X=x)", xlab="X", main="Probability for DPERKS",
     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="black")
legend("topright", col=c("green4", "tomato", "black"), lwd=3,
       legend=c("mu=0.001, sigma=0.52 ",
                "mu=0.001, sigma=0.85",
                "mu=0.001, sigma=1.5"))

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

x_max <- 25
cumulative_probs1 <- pDPERKS(q=0:x_max, mu=0.001, sigma=0.52)
cumulative_probs2 <- pDPERKS(q=0:x_max, mu=0.001, sigma=0.85)
cumulative_probs3 <- pDPERKS(q=0:x_max, mu=0.001, sigma=1.5)

plot(x=0:x_max, y=cumulative_probs1, col="green4",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for DPERKS",
     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="black")
legend("bottomright", col=c("green4", "tomato", "black"), lwd=3,
       legend=c("mu=0.001, sigma=0.52 ",
                "mu=0.001, sigma=0.85",
                "mu=0.001, sigma=1.5"))

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

x_max <- 20
mu <- 2.5
sigma <- 0.4
probs1 <- dDPERKS(x=0:x_max, mu=mu, sigma=sigma)
names(probs1) <- 0:x_max

x <- rDPERKS(n=10000, mu=mu, sigma=sigma)
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 <- 0.2
sigma <- 0.2
p <- seq(from=0, to=1, by=0.01)
qxx <- qDPERKS(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 DPERKS(mu = sigma = 0.2)")