The DMOLBE distribution

These functions define the density, distribution function, quantile function and random generation for the Discrete Marshall–Olkin Length Biased Exponential DMOLBE distribution with parameters \(\mu\) and \(\sigma\).

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

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

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

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

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.

Value

dDMOLBE gives the density, pDMOLBE gives the distribution function, qDMOLBE gives the quantile function, rDMOLBE

generates random deviates.

Details

The DMOLBE distribution with parameters \(\mu\) and \(\sigma\) has a support 0, 1, 2, ... and mass function given by

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

with \(\mu > 0\) and \(\sigma > 0\)

References

Aljohani HM, Ahsan-ul-Haq M, Zafar J, Almetwally EM, Alghamdi AS, Hussam E, Muse AH (2023). “Analysis of Covid-19 data using discrete Marshall-Olkinin Length Biased Exponential: Bayesian and frequentist approach.” Scientific Reports, 13(1), 12243.

Author

Olga Usuga, olga.usuga@udea.edu.co

Examples

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

x_max <- 20
probs1 <- dDMOLBE(x=0:x_max, mu=0.5, sigma=0.5)
probs2 <- dDMOLBE(x=0:x_max, mu=5, sigma=0.5)
probs3 <- dDMOLBE(x=0:x_max, mu=1, sigma=2)

# 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 DMOLBE",
     ylim=c(0, 0.80))
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=0.5, sigma=0.5",
                "mu=5, sigma=0.5",
                "mu=1, sigma=2"))


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

x_max <- 20
cumulative_probs1 <- pDMOLBE(q=0:x_max, mu=0.5, sigma=0.5)
cumulative_probs2 <- pDMOLBE(q=0:x_max, mu=5, sigma=0.5)
cumulative_probs3 <- pDMOLBE(q=0:x_max, mu=1, sigma=2)

plot(x=0:x_max, y=cumulative_probs1, col="dodgerblue",
     type="o", las=1, ylim=c(0, 1),
     main="Cumulative probability for DMOLBE",
     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=0.5, sigma=0.5",
                "mu=5, sigma=0.5",
                "mu=1, sigma=2"))


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

x_max <- 15
probs1 <- dDMOLBE(x=0:x_max, mu=5, sigma=0.5)
names(probs1) <- 0:x_max

x <- rDMOLBE(n=1000, mu=5, sigma=0.5)
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 <- qDMOLBE(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 DMOLBE(mu = 3, sigma = 3)")