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The Taylor Series Linearization (TL) Method
The Taylor Series linearization method is used to account for the variability in estimating \(\widehat{X}\) due to differential sample weights, stratification and clustering effects induced by complex survey designs as
$$var(\widehat{X})\cong{\sum}_{h=1}^{H}\frac{t_{h}}{t_{h}-1}{\sum}_{\alpha=1}^{t_{h}}(Z_{h\alpha}^{t_{h}}-\overline{Z}_{h})(Z_{h\alpha}^{t_{h}}-\overline{Z}_{h})^{T},$$
where \(Z_{h{\alpha}}={\sum}_{i=1}^{n_{h{\alpha}}}w_{h{\alpha}i}z_{h{\alpha}i},\) and \(\overline{Z}_{h}=\frac{1}{t_{h}}{\sum}_{\alpha=1}^{t_{h}}Z_{h{\alpha}}.\) \(Z_{h{\alpha}i}\) is the Taylor deviate of \(\widehat{X}\) for the individual -hαi and can be derived by taking the derivative of \(\widehat{X}\) with respect to the sample weight (Li and Graubard 2012),
$$z_{h{\alpha}i}=\frac{\partial\widehat{X}}{\partial{w}_{h{\alpha}i}}={\sum}_{j=1}^{J}\left(\frac{\partial\widehat{X}}{\partial\widehat{p}_{j}}\times\frac{\partial\widehat{X}}{\partial\widehat{p_{j}}}+\frac{\partial\widehat{X}}{\partial{w}_{h{\alpha}i}}\times\frac{\partial\widehat{\mu}_{j}}{\partial{w}_{h{\alpha}i}}\right)$$
with
$$\frac{\partial\hat{p}_j}{\partial{w}_{h{\alpha}i}}=\frac{{\delta_{h{\alpha}i}^{j}\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_h}{\sum}_{i=1}^{n_{h\alpha}}w_{h{\alpha}i}}{\left({\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_h}{\sum}_{i-1}^{n_{h\alpha}}w_ {h\alpha{i}}\right)^{2}}=\frac{\delta_{h\alpha{i}}^{j}-\hat{p}_{j}}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_h}{\sum}_{i-1}^{n_{h\alpha}}w_ {h\alpha{i}}},$$
$$\frac{\partial\hat{p}_j}{\partial{w}_{h{\alpha}i}}=\frac{{\delta_{h{\alpha}i}^{j}}\left(y_{h{\alpha}i}-\hat{\mu}_{j}\right)}{{\sum}_{h=1}^{H}{\sum}_{\alpha=1}^{t_h}{\sum}_{i-1}^{n_{h\alpha}}w_ {h\alpha{i}}}.$$
The remaining two partial derivatives needed for calculating zhαi, i.e. \(\frac{\partial\widehat{X}}{\partial\widehat{p}_{j}}\) and \(\frac{\partial\widehat{X}}{\partial\widehat{\mu}_{j}}\), depend on the disparity measure of interest and will be introduced separately for each of the 13 disparity measures. For more details about the variance estimation, see Li et al 2018 and Yu et al (in press).
The 95% confidence intervals of based on the normal assumption are:
$$\widehat{X}\pm1.96\times\sqrt{var(\widehat{X}}).$$
Simple Designs
For designs without stratification, H=1 and index h is removed from the formula above.
$$var(\widehat{X})=\frac{t}{t-1}{\sum}_{{\alpha}=1}^{t}(_{\alpha}-\overline{Z})^{2}$$
where \(Z_{\alpha}={\sum}_{i=1}^{n_{\alpha}}w_{{\alpha}i}z_{{\alpha}i}\) and \(\overline{Z}=\frac{1}{t}{\sum}_{{\alpha}=1}^{t}Z_{\alpha}.\)
For designs without clusters (i.e., PSUs), \(n_{h{\alpha}}=1\) and index a is removed from the formula above.
$$var(\widehat{X})={\sum}_{h=1}^{H}\frac{t_{h}}{t_{h}-1}{\sum}_{i=1}^{T_{h}}\left(w_{hi}z_{hi}-\overline{Z}_{h}\right)^{2},$$
where
$$\overline{Z}_{h}=\frac{1}{t_{h}}{\sum}_{i=1}^{t_{h}}w_{hi}z_{hi}.$$
