Taylor Series Expansion - DRAFT VERSION

The roots of the score functions constitute sets of estimating functions (Godambe, 1960). Godambe (1960) proves an optimal property of maximum likelihood estimating functions in simple random samples, and Godambe and Thompson (1986) show that the standard p - weighted estimating function retain this optimal property in unequal probability samples. Thus, the log-likelihood function and its derivatives at each observation can be multiplied by the inverse probability of inclusion in the sample, yielding optimal estimating functions. Binder (1983) has extended the utility of this result by providing a general approach to approximating the variances of the estimates from general estimating functions. He has shown that under very general conditions, the Taylor series approximation of the sampling variance of estimates can be applied to any arbitrary estimating function, and is approximately This is popularly known as Binder’s sandwich estimator. In this case, the outer terms are approximated by evaluated at the converged parameter values. The variance term in the center is the estimated variance/covariance matrix of the first derivatives. Note that this is simply the variance/covariance matrix of a set of population totals (the summed first derivatives). In a stratified, clustered, unequally weighted sample, one can estimate this as the B estimator of the stratified, between-PSU variance.