Create a Parametric Subunitary Multinomial CGF Object

Creates a CGF object for the subunitary multinomial distribution with total count N = n(theta) and active-category probabilities pi = prob_vec(theta), where sum(pi) <= 1.

Usage

SubunitaryMultinomialModelCGF(n, prob_vec, iidReps = "any", ...)

Arguments

n

A function (or adaptor) mapping theta to the multinomial total count N.

prob_vec

A function (or adaptor) mapping theta to the active-category probability vector pi (length d). These entries should be non-negative and sum(pi) should be <= 1.

iidReps

Either "any" or a positive integer specifying how many i.i.d. blocks are expected.

...

Additional named arguments passed to CGF creating functions.

Value

A CGF object.

Details

Let \(Y \sim \mathrm{Multinomial}(N,\pi_1,\dots,\pi_d,\pi_{d+1})\) with \(\sum_{i=1}^{d+1} \pi_i = 1\). Conditioning on \(\{Y_{d+1}=0\}\), we have \(W = (Y \mid Y_{d+1}=0) \sim \mathrm{Multinomial}(N,p_1,\dots,p_d) \), where $$ p_i = \frac{\pi_i}{\sum_{j=1}^d \pi_j}, \quad i=1,\dots,d. $$

The CGF:

From the perspective of restricting to \(\{Y_{d+1}=0\}\), one effectively sets \(t_{d+1}=-\infty\) in the multinomial CGF. This yields the subunitary CGF: $$ K_{\mathrm{subunitary}}(t_1,\dots,t_d) = \log\bigl\{\Pr(Y_{d+1}=0)\bigr\} \;+\; K_{W}(t_1,\dots,t_d)\,, $$ where \(\Pr(Y_{d+1}=0) = N \log\left(\!\sum_{i=1}^d \pi_i\right)\) and \(K_{W}\) is the usual multinomial CGF for \(\mathrm{Multinomial}(N,p_1,\dots,p_d)\).

In many applications (such as certain capture-recapture models), some categories are effectively impossible based on observed data, so they can be merged into a single "impossible" category with zero count. The probabilities of the \(d\) active categories then sum to less than one.

Examples

if (FALSE) { # \dontrun{
set.seed(1)
d <- 3
N <- 10
pi_vec <- c(0.2, 0.3, 0.1)  # sum < 1
theta <- c(N, pi_vec)

# Build a theta-mapped CGF: theta -> (N, pi_1,...,pi_d)
cgf <- SubunitaryMultinomialModelCGF(
  n        = function(th) th[1],
  prob_vec = function(th) th[2:(d + 1)],
  iidReps  = 2
)

# With iidReps = 2, tvec must be length 2*d:
t_block <- rnorm(d)
tvec <- c(t_block, t_block)

# Evaluate K and K1 at theta:
cgf$K(tvec, theta)
cgf$K1(tvec, theta)
} # }