Concatenation of CGFs (concatenationCGF) • saddlepoint

concatenationCGF() builds a CGF for a concatenated random vector with independent components.

If you have independent random vectors $X^{(1)} \in \mathbb{R}^{d_1},\;\dots,\; X^{(L)} \in \mathbb{R}^{d_L},$ and define the concatenation $Y = \bigl(X^{(1)}, \dots, X^{(L)}\bigr) \in \mathbb{R}^{d_1+\cdots+d_L},$ then the CGF factorizes additively over disjoint sub-blocks of $t$ : $K_Y(t;\theta) = \sum_{i=1}^L K_i\!\left(t^{(i)};\theta\right), \qquad t = \bigl(t^{(1)},\dots,t^{(L)}\bigr).$

This differs from sumOfIndependentCGF(), where all summands share the same tvec and are added as scalars/vectors.

Replication semantics

For concatenation, the natural “one observation” dimension is fixed:

block_size = sum(component_dims).
You only need to specify iidReps (either a positive integer or "any").

Example 1: a bivariate Poisson model (exact likelihood available)

We observe $B$ i.i.d. bivariate vectors: $Y_b = (Y_{1b}, Y_{2b}), \qquad Y_{1b} \sim \text{Poisson}(\lambda_1),\; Y_{2b} \sim \text{Poisson}(\lambda_2),$ independent across coordinates and across $b$ .

The exact MLE is simply: $\hat\lambda_1 = \overline{Y}_1,\qquad \hat\lambda_2 = \overline{Y}_2.$


set.seed(123)
B <- 30
lambda_true <- c(7, 15)

# Store data as a 2 x B matrix: columns are iid replicates
Y <- rbind(rpois(B, lambda_true[1]),
           rpois(B, lambda_true[2]))

# Model CGFs:
# lambda(theta) = theta[i], i=1,2  (so theta must be positive)
cg_1 <- PoissonModelCGF(lambda = adaptor(indices = 1), iidReps = 1)
cg_2 <- PoissonModelCGF(lambda = adaptor(indices = 2), iidReps = 1)

# Concatenate into a 2D vector, then replicate across columns (iidReps="any"):
cg_pp <- concatenationCGF(
  list(cg_1, cg_2),
  component_dims = 1,
  iidReps = "any"
)

# Saddlepoint MLE
fit <- find.saddlepoint.MLE(
  observed.data  = Y,                
  cgf            = cg_pp,
  starting.theta = c(1, 1.2),
  lb.theta       = c(1e-8, 1e-3),
  method         = "two_step"
)


cbind(
  true   = lambda_true,
  exact_mle  = rowMeans(Y),
  spa_mle = fit$MLEs.theta
)
#>      true exact_mle   spa_mle
#> [1,]    7   7.60000  7.600001
#> [2,]   15  13.86667 13.866665

Use concatenationCGF() when you want to model a multivariate observation by stitching together independent subcomponents whose CGF objects already exist.