Taylor Series Linearization (TSL)

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 and di. On the other hand, the domain mean, , is a nonlinear estimator as it is the ratio of two random variables and is not a linear combination of the observed data.

Unbiased variance estimation formulae for linear estimators are available for most complex sample designs. However, for nonlinear estimators, unbiased variance estimation formulae are often not available, and approximate methods must be used. The most common approximate methods are replication methods, such as the jackknife method or balanced repeated replication, and the TSL method.

The TSL method uses the linear terms of a Taylor series expansion to approximate the estimator by a linear function of the observed data. The variance estimation formulae for a linear estimator corresponding to the specifie sampling design can then be applied to the linear approximation. This generally leads to a statistical consistent estimator of the variance of a nonlinear estimator.

To illustrate the TSL method, let be an estimate of the parameter θ where ŷ and ○ are two linear sample statistics. For example, . Also define μy and μx to be the expected values of ŷ and ○, respectively. can be expanded in a Taylor series expansion about μy and μx so that

where ∂Fy and ∂FX are the first-order partial derivatives of F with respect to ŷ and ○ evaluated at their respective expectations, μy and μx. If the higher-order terms are negligible, then variance of can be approximated by

This approximation can easily be extended to functions of more than two linear sample statistics. An approximate estimate of the variance of is then obtained by substituting sample-based estimates of μy, μx, Var(ŷ) and Var(○) in the previous formula.

An equivalent computational procedure is formed by recognizing that the variable portion of the Taylor series approximation is so that

Because ŷ and ○ are linear estimators, the Taylor series variance approximation can be computed using the linearized values so that [Page 877] As before, substituting sample-based estimates of μy and μx, namely, ŷ and ○, in the formula for zi and then using the variance formula of a linear estimator for the sample design in question to estimate the variance of yields an approximate estimate of the variance of . This reduces the problem of estimating the variance of a nonlinear statistics to that of estimating the variance of the sum of the linearized values. As an example, the linearized values for the mean

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