Calculate cumulant function and up to three derivatives for
families known to the package.
Usage
cumulant(theta, fam, deriv = 0, delta)
Value
a list containing some of the following components:
zeroth
the value of the cumulant function at \(\theta\).
first
the value of the first derivative at \(\theta\), a
vector having the same dimension as \(\theta\).
second
the value of the second derivative at \(\theta\), a
\(d \times d\) matrix if \(d\) is the dimension
of \(\theta\) or a scalar if \(\theta\) is scalar.
third
the value of the third derivative at \(\theta\), a
\(d \times d \times d\) array if \(d\) is the dimension
of \(\theta\) or a scalar if \(\theta\) is scalar.
Arguments
theta
canonical parameter value.
fam
an object of class "astfam" produced by one of
the family functions (see families) specifying the
exponential family.
deriv
the number of derivatives wanted. Must be nonnegative
integer less than or equal to three.
delta
direction in which limit is taken. Cumulant
function is for family that is limit of family specified, limit
being for distributions with parameter
\(\theta + s \delta\),
the limit being as \(s \to \infty\). May be missing,
in which case \(\delta = 0\) is assumed, which is the same
as no limit being taken.