Compute the Truncated Confluent Hypergeometric function from Li and Clyde (2018) which is the normalizing constant in the tcch density of Gordy (1998) with integral representation:
Arguments
- a
a > 0
- b
b > 0
- r
r >= 0
- s
arbitrary
- v
0 < v
- k
arbitrary
- log
logical indicating whether to return values on the log scale; useful for Bayes Factor calculations
Details
tr.cch(a,b,r,s,v,k) = Int_0^1/v u^(a-1) (1 - vu)^(b -1) (k + (1 - k)vu)^(-r) exp(-s u) du
This uses a more stable method for calculating the normalizing constant using R's `integrate` function rather than the version in Gordy 1998. For calculating Bayes factors that use the `trCCH` function we recommend using the `log=TRUE` option to compute log Bayes factors.
See also
Other special functions:
hypergeometric1F1(),
hypergeometric2F1(),
phi1()
Author
Merlise Clyde (clyde@duke.edu)
Examples
# special cases
# trCCH(a, b, r, s=0, v = 1, k) is the same as
# 2F1(a, r, a + b, 1 - 1/k)*beta(a, b)/k^r
k = 10; a = 1.5; b = 2; r = 2;
trCCH(a, b, r, s=0, v = 1, k=k) *k^r/beta(a,b)
#> [1] 4.74679
hypergeometric2F1(a, r, a + b, 1 - 1/k, log = FALSE)
#> [1] 4.746772
# trCCH(a,b,0,s,1,1) is the same as
# beta(a, b) 1F1(a, a + b, -s, log=FALSE)
s = 3; r = 0; v = 1; k = 1
beta(a, b)*hypergeometric1F1(a, a+b, -s, log = FALSE)
#> [1] 0.0923551
trCCH(a, b, r, s, v, k)
#> [1] 0.09235518
# Equivalence with the Phi1 function
a = 1.5; b = 3; k = 1.25; s = 400; r = 2; v = 1;
phi1(a, r, a + b, -s, 1 - 1/k, log=FALSE)*(k^-r)*gamma(a)*gamma(b)/gamma(a+b)
#> [1] 7.04733e-05
trCCH(a,b,r,s,v,k)
#> [1] 7.052186e-05