mixdiff: Difference of mixture distributions

In RBesT: R Bayesian Evidence Synthesis Tools

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)

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_1) and x_2 \sim f_2(x_2), the density of the difference d \equiv x_1 - x_2 is given by

f_d(d) = \int f_1(u) \, f_2(u - d) \, du.

The cumulative distribution function equates to

F_d(d) = \int f_1(u) \, (1-F_2(u-d)) \, du.

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

Value

Respective density, quantile, cumulative density or random numbers.

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))

RBesT documentation built on Aug. 22, 2023, 1:08 a.m.