RelDists (version 1.0.2)

dMCWEI: The Modified Cosine-Weibull distribution (MCWEI)

Description

Density function, cumulative distribution function, quantile function, random generation and hazard function for the Modified Cosine-Weibull distribution with parameters mu, sigma and nu.

Usage

dMCWEI(x, mu = 2.2, sigma = 1.2, nu = 0.5, log = TRUE)
pMCWEI(q, mu = 2.2, sigma = 1.2, nu = 0.5, lower.tail = TRUE, log.p = FALSE)
qMCWEI(p, mu = 2.2, sigma = 1.2, nu = 0.5, lower.tail = TRUE, log.p = FALSE)
rMCWEI(n, mu = 2.2, sigma = 1.2, nu = 0.5)
hMCWEI(x, mu = 2.2, sigma = 1.2, nu = 0.5)

Value

dMCWEI gives the density, pMCWEI gives the cumulative distribution function, qMCWEI gives the quantile function, rMCWEI

generates random deviates and hMCWEI gives the hazard function.

Arguments

x, q

vector of quantiles.

mu

parameter representing the shape parameter \(\phi\) (mu > 0).

sigma

parameter representing the scale parameter \(\tau\) (sigma > 0).

nu

parameter representing the additional modified cosine parameter \(\sigma\) (nu != 0).

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 observations.

Author

Juan Andres Henao Arias,

Details

The Modified Cosine-Weibull with parameters mu, sigma and nu has density given by

\( f(x | \mu, \sigma ,\nu) = \frac{\pi \mu \sigma \nu x^{\mu -1} e^{-\sigma x^{\mu}} \cos(\frac{\pi}{2} e^{-\sigma x^{\mu}})\sin(\frac{\pi}{2} e^{-\sigma x^{\mu}})}{e^{\nu} -1} \cdot e^{\nu \left( 1- \cos^2(\frac{\pi}{2} e^{-\sigma x^{\mu}}) \right) } \)

for \(x\ge 0\), \(\mu>0\), \(\sigma>0\) and \(\nu \neq 0\).

References

Su, R., Aloraini, N. M., Alkhathami, A. A., Alshanbari, H. M., & Khalifa, H. A. E. W. (2025). A new statistical distribution: Its empirical exploration using the reliability and lifespan data in fashion industry. Alexandria Engineering Journal, 116, 660-671.

See Also

BS.

Examples

# Example 1: Plotting the density function (dMCWEI) for different values.
mu_ <- c(2.8, 4.4)
sigma_ <- c( 0.1, 0.01)
nu_ <- c( 0.1, 0.01)
X_ <- seq(0.0001, 5, length=350)

curve(dMCWEI(x, log = FALSE), type = "l", col= "royalblue", lwd=2.5,
     from = 0.001, to = 5, ylim=c(0,3), xlim = c(0,4), 
     xlab = "x", ylab = "f(x)")
title("Density Function")

colors <- c("red", "green")

for(k in seq(1, 2)) {
  curve(
    dMCWEI(x, mu=mu_[k], sigma=sigma_[k], nu = nu_[k], log = FALSE ),
    type = "l", col = colors[k], lwd=2, lty = 3, 
    from = 0.001, to=5, add = TRUE)
}

legend("topright",
       legend = c(
         expression(mu==2.2 ~ sigma==1.2 ~ nu==0.50),
         expression(mu==2.8 ~ sigma==0.1 ~ nu==0.10),
         expression(mu==4.4 ~ sigma==0.01 ~ nu==0.01)
       ),
       lwd=2,
       col=c("royalblue",colors),
       lty = c(1,3,3),
       bty="n")

# Example 2: Plotting the Cumulative Distribution function 
# (pMCWEI) for differente values.

parameters <- data.frame(
  mu=c(0.5, 1.5, 2.5),
  sigma=c(1,0.5,1.2),
  nu=c(0.8,1,1.5))

colors <- c("royalblue", "red", "green")
curve(pMCWEI(x, mu=parameters$mu[1], 
             sigma = parameters$sigma[1], 
             nu = parameters$nu[1]),
      lwd=2.5, col=colors[1], 
      from = 0.001, to=5, 
      xlab = "x", ylab = "F(x)")
title("Cumulative Probability")

for(k in seq(2, 3)) {
curve(pMCWEI(x, 
             mu=parameters$mu[k], 
             sigma = parameters$sigma[k], 
             nu = parameters$nu[k]),
      lwd=2, col=colors[k], add = TRUE, lty=3)
  
}

legend(
  "bottomright",
  legend = c(
    expression(mu==0.5 ~" "  ~sigma==1 ~ " " ~ nu==0.8),
    expression(mu==1.5 ~" " ~  sigma==0.5 ~  " " ~ nu==1.0),
    expression(mu==2.5 ~" "~ sigma==1.2 ~ " "~nu==1.5)
  ),
  lwd = 2, lty = c(1,3,3), bty="n")


# Example 3
# The quantile function
p <- seq(from=0, to=0.999, length.out=100)
plot(x=qMCWEI(p, mu=2.3, sigma=1.7, nu=1.3), y=p, xlab="Quantile",
     las=1, ylab="Probability", main="Quantile function ")
curve(pMCWEI(x, mu=2.3, sigma=1.7, nu=1.3), 
      from=0, add=TRUE, col="tomato", lwd=2.5)

# Example 4: Generating a Random Sample for the distribution.

set.seed(5)
hist(rMCWEI(200), freq=FALSE, col = "orange", 
     xlab = "x", ylab = "y", 
     main = "Theoretical values V.S Experimental values")
curve(dMCWEI(x, log = FALSE), add = TRUE, 
      from = 0.0001, to=5, col="royalblue", lwd=3)

legend("topright",
       legend = c(expression(f(x))),
       lwd = 2, lty = 1, bty="n", col="royalblue")