The Taylor series linearization (TSL) method is used with variance estimation for statistics that are vastly more complex than mere additions of sample values.

Two factors that complicate variance estimation are complex sample design features and the nonline-arity of many common statistical estimators from complex sample surveys. Complex design features include stratification, clustering, multi-stage sampling, unequal probability sampling, and without replacement sampling. Nonlinear statistical estimators for complex sample surveys include means, proportions, and regression coefficients. For example, consider the estimator of a subgroup total,

, where wi is the sampling weight, yi is the observed value, and di is a zero/one subgroup membership indicator for the ith sampling unit. This is a linear estimator because the estimate is a linear combination of the observed values yi ...
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