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Special Distribution

# Gamma distribution

Firstly, let's introduce how to generate
${\mathcal G}a(a,\beta)$
where
$a$
is an integral (Robert and Casella (2013)).
Attention! The parameter in (2.2.1) is scale parameter, thus the code to sample is
## sample from Ga(a, beta)
function rgamma_int(a::Int, beta)
u = rand(a)
return(-1.0 / beta * sum(log.(u)))
end
Then when considering the general
${\mathcal G}a(\alpha,\beta)$
, we can use
${G}a(a, b)$
as instrumental distribution in Accept-Rejection algorithm as mentioned in Robert and Casella (2013):
## sample from Ga(a, beta)
function rgamma_int(a::Int, beta)
u = rand(a)
return(-1.0 * beta * sum(log.(u)))
end
â€‹
## density of Ga(alpha, beta)
include("func_gamma.jl")
function dgamma(x, alpha, beta)
return(beta^alpha / lanczos_gamma(alpha) * x^(alpha-1) * exp(-1*beta*x))
end
â€‹
## accept-reject algorithm
function rgamma(alpha = 1.5, beta = 2.1)
a = Int(floor(alpha))
b = a * beta / alpha
M = exp(a * (log(a) - 1) - alpha * (log(alpha) - 1))
while true
x = rgamma_int(a, b)
u = rand()
cutpoint = dgamma(x, alpha, beta)/(M*dgamma(x, a, b))
if u <= cutpoint
return x
end
end
end
â€‹
# example
println(rgamma())
where func_gamma.jl is to calculate gamma function via Lanczos Approximation because there is not gamma function in Julia.
If
$\beta=1$
, we can do some further calculations,
and would get more simpler expression,