A ready-to-use CGF object for the subunitary multinomial model.
Details
Parameter vector: c(N, pi_1, ..., pi_d), where pi_i are the probabilities of the 'active' categories and sum(pi) <= 1. The omitted category has probability 1 - sum(pi) and is assumed to have count 0.
Technically, this is the (d+1)-category multinomial CGF evaluated at \(t_{d+1} = -Inf\), which introduces a multiplicative factor for the event \(\{Y_{d+1}=0\}\). This is why the K() differs from the standard multinomial family by an additive constant N*log(sum(pi)).
Examples
if (FALSE) { # \dontrun{
set.seed(1)
d <- 3
N <- 10
pi_vec <- c(0.2, 0.3, 0.1) # sum < 1 (subunitary)
tt <- rnorm(d)
param <- c(N, pi_vec)
# K1 (and all higher t-derivatives) match the standard multinomial family
SubunitaryMultinomialCGF$K1(tt, param)
MultinomialCGF$K1(tt, param)
# K differs by N*log(sum(pi_vec)) per iid block:
SubunitaryMultinomialCGF$K(tt, param) - MultinomialCGF$K(tt, param)
# should be ~ N * log(sum(pi_vec))
# With multiple iid blocks,
# the difference scales with the number of blocks:
t2 <- c(tt, tt)
SubunitaryMultinomialCGF$K(t2, param) - MultinomialCGF$K(t2, param)
# should be ~ 2*N*log(sum(pi_vec))
} # }