Windham_populationinverse {scorematchingad}

Inverse Transform for the Population Parameters Under Windham Weights

Description

Returns the matrix which reverses the effect of weights on a population for certain models.

Usage

Windham_populationinverse(cW)

Windham_populationinverse_alternative(newtheta, previoustheta, cW, cWav)

Arguments

cW

A vector of tuning constants for the Windham robustification method performed by Windham().

newtheta

The parameter vector most recently estimated

previoustheta

The parameter vector estimated in the previous step

cWav

The value of the non-zero elements of cW. That is cW have elements that are zero or equal to cWav.

Details

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.

Value

A diagonal matrix with the same number of columns as cW.

Functions

• Windham_populationinverse: The matrix with diagonal elements 1/(1+c_i)

• Windham_populationinverse_alternative: The transform implemented as described by Scealy et al. (2024). It is mathematically equivalent to multiplication by the result of Windham_populationinverse() in the situation in Scealy et al. (2024).