RBesT (version 1.6-3)

mixdiff: Difference of mixture distributions

Description

Density, cumulative distribution function, quantile function and random number generation for the difference of two mixture distributions.

Usage

dmixdiff(mix1, mix2, x)
pmixdiff(mix1, mix2, q, lower.tail = TRUE)
qmixdiff(mix1, mix2, p, lower.tail = TRUE)
rmixdiff(mix1, mix2, n)

Value

Respective density, quantile, cumulative density or random numbers.

Arguments

mix1

first mixture density

mix2

second mixture density

x

vector of values for which density values are computed

q

vector of quantiles for which cumulative probabilities are computed

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise P[X > x].

p

vector of cumulative probabilities for which quantiles are computed

n

size of random sample

Details

If \(x_1 \sim f_1(x)\) and \(x_2 \sim f_2(x)\), the density of the difference \(x \equiv x_1 - x_2\) is given by the convolution

$$f(x) = \int f_1(x) \, f_2(x - u) \, du = (f_1 \ast f_2)(x).$$

The cumulative distribution function equates to

$$F(x) = \int F_1(x+u) \, f_2(u) \, du.$$

Both integrals are performed over the full support of the densities and use the numerical integration function integrate.

Examples

# 1. Difference between two beta distributions, i.e. Pr( mix1 - mix2 > 0)
mix1 <- mixbeta(c(1, 11, 4))
mix2 <- mixbeta(c(1, 8, 7))
pmixdiff(mix1, mix2, 0, FALSE)

# Interval probability, i.e. Pr( 0.3 > mix1 - mix2 > 0)
pmixdiff(mix1, mix2, 0.3) - pmixdiff(mix1, mix2, 0)

# 2. two distributions, one of them a mixture
m1 <- mixbeta( c(1,30,50))
m2 <- mixbeta( c(0.75,20,50),c(0.25,1,1))

# random sample of difference
set.seed(23434)
rM <- rmixdiff(m1, m2, 1E4)

# histogram of random numbers and exact density
hist(rM,prob=TRUE,new=TRUE,nclass=40)
curve(dmixdiff(m1,m2,x), add=TRUE, n=51)

# threshold probabilities for difference, at 0 and 0.2
pmixdiff(m1, m2, 0)
mean(rM<0)
pmixdiff(m1,m2,0.2)
mean(rM<0.2)

# median of difference
mdn <- qmixdiff(m1, m2, 0.5)
mean(rM<mdn)

# 95%-interval
qmixdiff(m1, m2, c(0.025,0.975))
quantile(rM, c(0.025,0.975))