Create a CGF object for the sum of n i.i.d. random variables.

Given a base CGF object cgf describing a random variable \(X\), this function returns a new CGF object for the sum of n i.i.d. copies of \(X\), i.e. \(Y = X_1 + \cdots + X_n\).

Usage

sumOfiidCGF(cgf, n, block_size = NULL, iidReps = NULL, ...)

Arguments

cgf

A CGF object describing the base random variable \(X\).

n

Either a positive scalar, or a function/adaptor mapping \(\theta \mapsto n(\theta)\).

block_size

Optional. Positive integer giving the dimension of one observation of \(Y\).

iidReps

Optional. Either NULL, "any", or a positive integer.

...

Additional arguments passed to CGF creation.

Value

A CGF object for \(Y = \sum_{i=1}^{n(\theta)} X_i\).

Details

$$K_Y(t;\theta) = n(\theta)\,K_X(t;\theta).$$

For every order \(r \ge 1\), $$K_Y^{(r)}(t;\theta) = n(\theta)\,K_X^{(r)}(t;\theta).$$

• n is the number of summands inside each \(Y\).

• iidReps/block_size (if supplied) describe i.i.d. replication of \(Y\) across observations \(Y_1,\ldots,Y_B\).

Replication (iidReps / block_size):

• If both iidReps and block_size are NULL, no replication is applied.

• If block_size is provided but iidReps is NULL, we set iidReps = "any" and infer the number of blocks from length(tvec) / block_size at evaluation time.

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

• If iidReps is a positive integer, block_size may be NULL, though providing it is safer.

Examples

if (FALSE) { # \dontrun{
base_cgf <- NormalCGF
sum5 <- sumOfiidCGF(base_cgf, n=5)
theta <- c(1.2, 0.8)
tvec  <- c(0.1, -0.2, 0.0)

all.equal(sum5$K(tvec, theta), 5*base_cgf$K(tvec, theta))
all.equal(sum5$K1(tvec, theta), 5*base_cgf$K1(tvec, theta))
} # }