CdFromDensities {RVCompare}

The dominance rate of X_A over X_B given the density functions.

Description

Returns a real number in the interval [0,1] that represents the dominance rate of X_A over X_B. Basically, we are measuring the amount of mass of X_A in which the cumulative distribution of X_A is higher minus the amount of mass of X_B in which the cumulative distribution of X_B is higher.

Usage

CdFromDensities(densityX_A, densityX_B, xlims, EPSILON = 0.001)

Arguments

densityX_A

The probability density function of the random variable X_A.

densityX_B

The probability density function of the random variable X_B.

xlims

an interval that represents the domain of definition the density functions.

EPSILON

(optional, default = 1e-3) minimum difference between two values.

Value

Returns the dominance rate of X_A over X_B.

See Also

CpFromDensities

Examples

# If two symmetric distributions are centered in the same point (x = 0 in
# this case), then their Cd will be 0.5.

densityX_A <- normalDensity(0,1)
densityX_B <- uniformDensity(c(-2,2))
CdFromDensities(densityX_A, densityX_B, c(-5,5))

### Example 2 ###
# If two distributions are equal, Cd will be 0.5.  Cd(X_A,X_A) = 0.5
CdFromDensities(densityX_A, densityX_A, c(-10,10))


### Example 3 ###
# example on https://etorarza.github.io/pages/2021-interactive-comparing-RV.html
tau <- 0.11
densityX_A <- normalDensity(0.05,0.0015)
densityX_B <- mixtureDensity(c(normalDensity(0.05025,0.0015),
                               normalDensity(0.04525, 0.0015)),
                               weights = c(1 - tau, tau))
plot(densityX_A, from=0.03, to=0.07, type="l",  col="red", xlab="x", ylab="probability density")
curve(densityX_B, add=TRUE, col="blue", type="l", lty=2)
Cd <- CdFromDensities(densityX_A, densityX_B, c(.03,.07))
mtext(paste("Cd(X_A, X_B) =", format(round(Cd, 3), nsmall = 3)), side=3) # add Cd to plot as text
legend(x = c(0.0325, 0.045), y = c(200, 250),legend=c("X_A", "X_B"),
                                             col=c("red", "blue"),
                                             lty=1:2,
                                             cex=0.8) # add legend


### Example 4 ###
# The dominance factor ignores the mass of the probability where the
# distribution functinos are equal.
densityX_A <- uniformDensity(c(0.1, 0.3))
densityX_B <- uniformDensity(c(-0.2,0.5))
CdFromDensities(densityX_A, densityX_B, xlims = c(-2,2))

densityX_A <- mixtureDensity(c(uniformDensity(c(0.1,0.3)), uniformDensity(c(-1,-0.5))))
densityX_B <- mixtureDensity(c(uniformDensity(c(-0.2,0.5)), uniformDensity(c(-1,-0.5))))
CdFromDensities(densityX_A, densityX_B, xlims = c(-2,2))