A relative weighting method for estimating parameters and variances in multiple data sets
We are given multiple data sets and a nonlinear model function
for each data value.
Each data value is the sum of its error and its
model function evaluated at an unknown parameter vector.
The data errors are mean zero, finite variance,
and independent, are not necessarily normal,
and are identically distributed within each data set.
We consider the problem of estimating
the data variance as well as the parameter vector
via an extended least-squares technique motivated by
maximum likelihood estimation.
We prove convergence of an
algorithm that generalizes a standard
successive approximation algorithm from nonlinear programming.
This generalization reduces the estimation problem to a sequence of
linear least-squares problems.
It is shown that the parameter and variance estimators converge
to their true values as the number of data values goes to infinity.
Moreover, if the constraints are not active,
the parameter estimates converge in distribution.
This convergence does not depend on the data errors
being normally distributed.