Returns the matrix which reverses the effect of weights on a population for certain models.
Windham_populationinverse(cW)Windham_populationinverse_alternative(newtheta, previoustheta, cW, cWav)
A diagonal matrix with the same number of columns as cW.
A vector of tuning constants for the Windham robustification method performed by Windham().
The parameter vector most recently estimated
The parameter vector estimated in the previous step
The value of the non-zero elements of cW. That is cW have elements that are zero or equal to cWav.
Windham_populationinverse(): The matrix with diagonal elements \(1/(1+c_i)\)
Windham_populationinverse_alternative(): The transform implemented as described by scealy2024ro;textualscorematchingad. It is mathematically equivalent to multiplication by the result of Windham_populationinverse() in the situation in scealy2024ro;textualscorematchingad.
In the Windham robustification method (Windham()) the effect of weighting a population plays a central role.
When the
the model density is proportional to \(\exp(\eta(\theta) \cdot T(u))\),
where \(T(u)\) is a vector of sufficient statistics for a measurement \(u\),
and \(\eta\) is a linear function,
Then weights proportional to
\(\exp(\eta(c \circ \theta) \cdot t(u))\),
where \(c\) is a vector of tuning constants and \(\circ\) is the Hadamard (element-wise) product,
have a very simple effect on the population parameter vector \(\theta\):
the weighted population follows a density of the same form, but with a parameter vector of
\((1 + c) \circ \theta\).
The inverse of this change to the parameter vector is then a matrix multiplication by a diagonal matrix with elements \(1/(1+c_i)\), with \(c_i\) denoting the elements of \(c\).