I will show another possible solution, that is quite widely applicable, and with todays R software, quite easy to implement. That is the saddlepoint density approximation, which ought to be wider known!
For terminology about the gamma distribution, I will follow https://en.wikipedia.org/wiki/Gamma_distribution with the shape/scale parametrization, k is shape parameter and θ is scale. For the saddlepoint approximation I will follow Ronald W Butler: "Saddlepoint approximations with applications" (Cambridge UP). The saddlepoint approximation is explained here: How does saddlepoint approximation work?
here I will show how it is used in this application.
ましょ既存のmomentgenerating機能を持つ確率変数
M （S ）= EのE S Xのために存在しなければならないのゼロが含まれているいくつかの開区間で。次いでによってキュムラント生成関数定義
K （複数可）= ログM （複数可）
ことが知られているE X = K '（0 ）、ヴァー（X ）= K "（0 ）X
M(s)=EesX
sK(s)=logM(s)
EX=K′(0),Var(X)=K′′(0)。saddlepoint式は、
暗黙的に定義
Sの関数として
X（の範囲内であることが必要
X）。私たちは、この暗黙的に定義された関数を記述
S（X ）。キュムラント関数は凸であるため、サドルポイント方程式には常に1つの解しかありません。
K′(s^)=x
sxXs^(x)
次いで、密度にsaddlepoint近似のXは、で与えられる
F（X ）= 1fX
f^(x)=12πK′′(s^)−−−−−−−√exp(K(s^)−s^x)
X1,X2,…,XnXi(ki,θi)
K(s)=−∑i=1nkiln(1−θis)
defined for
s<1/max(θ1,θ2,…,θn).
The first derivative is
K′(s)=∑i=1nkiθi1−θis
and the second derivative is
K′′(s)=∑i=1nkiθ2i(1−θis)2.
In the following I will give some
R code calculating this, and will use the parameter values
n=3,
k=(1,2,3),
θ=(1,2,3). Note that the following
R code uses a new argument in the uniroot function introduced in R 3.1, so will not run in older R's.
shape <- 1:3 #ki
scale <- 1:3 # thetai
# For this case, we get expectation=14, variance=36
make_cumgenfun <- function(shape, scale) {
# we return list(shape, scale, K, K', K'')
n <- length(shape)
m <- length(scale)
stopifnot( n == m, shape > 0, scale > 0 )
return( list( shape=shape, scale=scale,
Vectorize(function(s) {-sum(shape * log(1-scale * s) ) }),
Vectorize(function(s) {sum((shape*scale)/(1-s*scale))}) ,
Vectorize(function(s) { sum(shape*scale*scale/(1-s*scale)) })) )
}
solve_speq <- function(x, cumgenfun) {
# Returns saddle point!
shape <- cumgenfun[[1]]
scale <- cumgenfun[[2]]
Kd <- cumgenfun[[4]]
uniroot(function(s) Kd(s)-x,lower=-100,
upper = 0.3333,
extendInt = "upX")$root
}
make_fhat <- function(shape, scale) {
cgf1 <- make_cumgenfun(shape, scale)
K <- cgf1[[3]]
Kd <- cgf1[[4]]
Kdd <- cgf1[[5]]
# Function finding fhat for one specific x:
fhat0 <- function(x) {
# Solve saddlepoint equation:
s <- solve_speq(x, cgf1)
# Calculating saddlepoint density value:
(1/sqrt(2*pi*Kdd(s)))*exp(K(s)-s*x)
}
# Returning a vectorized version:
return(Vectorize(fhat0))
} #end make_fhat
fhat <- make_fhat(shape, scale)
plot(fhat, from=0.01, to=40, col="red", main="unnormalized saddlepoint approximation\nto sum of three gamma variables")
resulting in the following plot:
I will leave the normalized saddlepoint approximation as an exercise.
