How is the CDF ("pgamma") of The Average of 10 Samples from Two Gammas Derived?

Question

Case I: One Gamma ( I can do this! )

shape<-shape*10

scale<-scale/10

p_value_average_of_10_draws<-1-pgamma(q=average_of_10_draws, shape=shape, scale=scale, lower.tail = TRUE, log.p = FALSE)

Case II: Two Gammas (I can't do this!)

shape_A<-shape_A*10

scale_A<-scale_A/10

shape_B<-shape_B*10

scale_B<-scale_B/10

pgamma_A_and_B <-

pgamma(q=average_of_10_draws, shape=shape_A, scale=scale_A, lower.tail = TRUE, log.p = FALSE)*weight_A

+

pgamma(q=average_of_10_draws, shape=shape_B, scale=scale_B, lower.tail = TRUE, log.p = FALSE)*(1-weight_A)

p_value_average_of_10_draws<-1-pgamma_A_and_B

But this is just wrong!

Because it assumes that all ten draws will be taken from just one of A or B!

Answer 1

Well, there is well-known rule how to make probability density function (PDF) of sum of two independent random variates (Z = X+Y) with their own PDFs

PDF(z) = S PDF_x(t) * PDF_y(z-t) dt

where S is integration sign. Not sure there is a generic expression for sum of gammas with any parameters. There are packages in R which does numeric convolution in the integral above.

Is that what you want?

UPDATE

K-S example test of two gammas

library(ggplot2)

fitted.pdf <- function(x, w, a1, s1, a2, s2) {
    w*dgamma(x, shape = a1, scale = s1) + (1.0-w)*dgamma(x, shape = a2, scale = s2)
}

fitted.cdf <- function(x, w, a1, s1, a2, s2) {
    w*pgamma(x, shape = a1, scale = s1) + (1.0-w)*pgamma(x, shape = a2, scale = s2)
}

p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
p <- p + stat_function(fun = function(x) fitted.pdf(x, w=0.6, a1=1.0, s1=1.0, a2=0.8, s2=1.2))
p <- p + stat_function(fun = function(x) fitted.cdf(x, w=0.6, a1=1.0, s1=1.0, a2=0.8, s2=1.2))
p <- p + xlim(0.0, 4.0) + ylim(0.0, 1.0)
print(p)

# sample 100 from exponential
x <- rexp(100)

# K-S test
q <- ks.test(x, y=function(x) fitted.cdf(x, w=0.6, a1=1.0, s1=1.0, a2=0.8, s2=1.2))
print(q)