Difference of mixture distributions
Density, cumulative distribution function, quantile function and random number generation for the difference of two mixture distributions.
dmixdiff(mix1, mix2, x) pmixdiff(mix1, mix2, q, lower.tail = TRUE) qmixdiff(mix1, mix2, p, lower.tail = TRUE) rmixdiff(mix1, mix2, n)
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 |
p |
vector of cumulative probabilities for which quantiles are computed |
n |
size of random sample |
If x_1 ~ f_1(x) and x_2 ~ f_2(x), the density of the difference x = x_1 - x_2 is given by the convolution
f(x) = \int f_1(x) f_2(x - u) du = (f_1 * 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.
Respective density, quantile, cumulative density or random numbers.
# 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))
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