CGF for a randomly-stopped sum

Builds a CGF object for the random vector $$Y = \sum_{i=1}^{N} X_i,$$ where:

• \(N\) is a non-negative integer-valued scalar random variable with CGF count_cgf,

• \(X_i\) are i.i.d. copies of a (possibly vector-valued) summand with CGF summand_cgf,

• \(N\) is independent of all \(X_i\).

Usage

randomlyStoppedSumCGF(
  count_cgf,
  summand_cgf,
  block_size = NULL,
  iidReps = NULL,
  ...
)

Arguments

count_cgf

A CGF object for the (scalar) count variable \(N\).

summand_cgf

A CGF object for the summand \(X\).

block_size

Optional. The dimension \(d\) of one observation of \(Y\) (and hence of \(X\)). This can be a positive integer (fixed \(d\)) or NULL

iidReps

Optional. Replication count \(B\) for i.i.d. observations \(Y_1,\ldots,Y_B\). May be a positive integer, "any", or NULL.

...

Additional named arguments passed to CGF creation.

Value

A CGF object.

Details

What do block_size and iidReps mean for RSS models?

The replication arguments refer to i.i.d. replication of the variable \(Y\), not replication of summands inside the random sum.

If you have \(B\) independent observations, $$Y_b = \sum_{i=1}^{N_b} X_{b,i}, \qquad b=1,\ldots,B,$$ then:

• block_size is the dimension of one \(Y_b\) (which equals the dimension of one \(X_{b,i}\)).

• iidReps is the number of independent observations \(B\).

Why do we require block_size and/or iidReps?

Many summand CGFs in this package are vectorized: if you accidentally feed a long vector tvec into an 'untagged' RSS CGF, the summand CGF may interpret that as a higher-dimensional summand \(X\) (or multiple i.i.d. summands inside one \(X\)), which silently changes the model.

To prevent this, randomlyStoppedSumCGF() requires at least one of block_size or iidReps.

The semantics are:

• If iidReps is NULL (default), then block_size must be provided and we set iidReps="any". This means the number of observations is inferred at evaluation time as B = length(tvec)/block_size.

• If iidReps is "any", then block_size must be provided.

• If iidReps is a positive integer, then block_size may be NULL (it will be inferred as length(tvec)/iidReps), but providing block_size explicitly is encouraged.

Examples

if (FALSE) { # \dontrun{
## Scalar RSS, B observations:
K.N <- GeometricModelCGF(prob = adaptor(fixed_param = 0.4))
K.X <- BinomialModelCGF(n = adaptor(fixed_param = 1), p = adaptor(indices = 1))

# Either specify B explicitly...
K.U1 <- randomlyStoppedSumCGF(K.N, K.X, iidReps = 40)      # block_size inferred as length(tvec)/40

# ...or specify the scalar block size and let B be inferred from length(tvec):
K.U2 <- randomlyStoppedSumCGF(K.N, K.X, block_size = 1)    # iidReps defaults to "any"

## Vector RSS: summand is d-dimensional
d <- 3
K.Xv <- MultinomialModelCGF(n = adaptor(fixed_param = 1),
                            prob_vec = adaptor(indices = 1:d))
K.Y  <- randomlyStoppedSumCGF(PoissonCGF, K.Xv, block_size = d, iidReps = "any")
} # }