4. Composite estimator for the growth rate

Paul Knottnerus

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Examining a composite estimator (COM) of the form

$${\widehat{g}}_{COM}=k{\widehat{g}}_{STN}+\left(1-k\right){\widehat{g}}_{OLP},(4.1)$$

it follows from minimizing $\mathrm{var}\left({\widehat{g}}_{COM}\right)$ with respect to $k$ that

$$k=\frac{\mathrm{var}\left({\widehat{g}}_{OLP}\right)-\mathrm{cov}\left({\widehat{g}}_{OLP},{\widehat{g}}_{STN}\right)}{\mathrm{var}\left({\widehat{g}}_{OLP}\right)+\mathrm{var}\left({\widehat{g}}_{STN}\right)-2\mathrm{cov}\left({\widehat{g}}_{OLP},{\widehat{g}}_{STN}\right)};(4.2)$$

see also Särndal et al. (1992, page 372). Note that, by construction, $\mathrm{var}\left({\widehat{g}}_{COM}\right)$ can not exceed $\min \left\{\mathrm{var}\left({\widehat{g}}_{STN}\right),\mathrm{var}\left({\widehat{g}}_{OLP}\right)\right\}.$

Using the linearized forms of the estimators ${\widehat{g}}_{OLP}$ and ${\widehat{g}}_{STN},$ we get for their covariance

$$\begin{array}{cc}\mathrm{cov}\left({\widehat{g}}_{OLP},{\widehat{g}}_{STN}\right)& \approx \mathrm{cov}\left(\frac{{\overline{y}}_2-G{\overline{x}}_2}{\overline{X}},\frac{{\overline{y}}_{23}-G{\overline{x}}_{12}}{\overline{X}}\right)\hfill \\ & =\frac{1}{{\overline{X}}^2}\left\{\mathrm{cov}\left({\overline{y}}_2,{\overline{y}}_{23}\right)-G\mathrm{cov}\left({\overline{y}}_2,{\overline{x}}_{12}\right)-G\mathrm{cov}\left({\overline{x}}_2,{\overline{y}}_{23}\right)+G^2\mathrm{cov}\left({\overline{x}}_2,{\overline{x}}_{12}\right)\right\}.\hfill \end{array}$$

Now using some results from Knottnerus (2003, page 377)

$$\begin{array}{cc}\mathrm{cov}\left({\overline{y}}_2,{\overline{y}}_{23}\right)& =\mathrm{var}\left({\overline{y}}_{23}\right)\left[=\left(\frac{1}{n_{23}}-\frac{1}{N}\right)S_y^2\right]\hfill \\ \mathrm{cov}\left({\overline{x}}_2,{\overline{y}}_{23}\right)& =\mathrm{cov}\left({\overline{x}}_{23},{\overline{y}}_{23}\right)\left[=\left(\frac{1}{n_{23}}-\frac{1}{N}\right)S_{xy}\right],\hfill \end{array}$$

we obtain

$$\mathrm{cov}\left({\widehat{g}}_{OLP},{\widehat{g}}_{STN}\right)\approx \frac{1}{{\overline{X}}^2}\left\{\left(\frac{1}{n_{23}}-\frac{1}{N}\right)\left(S_y^2-G{S}_{yx}\right)+\left(\frac{1}{n_{12}}-\frac{1}{N}\right)\left(G^2{S}_x^2-G{S}_{yx}\right)\right\}.(4.3)$$

In practice $k$ can be estimated by replacing all (co)variances in (4.2) by their sample estimates, yielding

$$\widehat{k}=\frac{\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{g}}_{OLP}\right)-\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\widehat{g}}_{OLP},{\widehat{g}}_{STN}\right)}{\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{g}}_{OLP}\right)+\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{g}}_{STN}\right)-2\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\widehat{g}}_{OLP},{\widehat{g}}_{STN}\right)}(4.4)$$

To illustrate this approach, consider the following example.

Example 4.1. The data are the same as for Example 2.1. Applying formulas (2.1) - (2.4) and (4.3) to these data yields

$$\begin{array}{l}{\widehat{g}}_{STN}=0.082(0.00254),{\widehat{g}}_{OLP}=0.050(0.00134),\text{ and }\\ \mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\widehat{g}}_{STN},{\widehat{g}}_{OLP}\right)=\mathrm{0.00097.}\end{array}$$

The variances are mentioned between parentheses. Substituting these estimates into (4.4) yields $\widehat{k}=0.191$ and subsequently, ${\widehat{g}}_{COM}=0.056(0.00127).$ For the ease of exposition, all (co)variances in (4.4) are estimated from overlap $s_2,$ including the estimates of $G$ and $\overline{X}$ in (2.2), (2.4) and (4.3). Furthermore, using these estimates, we found that $\mathrm{var}\left({\widehat{g}}_{STN}\right)<\mathrm{var}\left({\widehat{g}}_{OLP}\right)$ and $k>0.5$ only if $n_2\le 12$ $\left(\lambda \le 0.167\right).$

For the sake of completeness, we also give an example for the composite estimator for the parameter of absolute change (i.e., $\overline{D}=\overline{Y}-\overline{X}$ ).

Example 4.2. We use the same data as in Example 2.1. As before all estimates for the (co)variances are based on $s_2.$ Define $D_i$ by $D_i=Y_i-X_i.$ Then we have two estimators for the parameter of absolute change

$${\widehat{\overline{D}}}_{STN}={\overline{y}}_{23}-{\overline{x}}_{12}=7.35\text{ and }{\widehat{\overline{D}}}_{OLP}={\overline{d}}_2={\overline{y}}_2-{\overline{x}}_2=\mathrm{4.89.}$$

For the (co)variances of ${\widehat{\overline{D}}}_{STN}$ and ${\widehat{\overline{D}}}_{OLP}$ we get

$$\begin{array}{cc}\hfill \mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{\overline{D}}}_{STN}\right)& =\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\overline{y}}_{23}\right)+\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\overline{x}}_{12}\right)-2\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\overline{y}}_{23},{\overline{x}}_{12}\right)\hfill \\ & =\left(\frac{1}{n_{23}}-\frac{1}{N}\right)s_{y2}^2+\left(\frac{1}{n_{12}}-\frac{1}{N}\right)s_{x2}^2-2\left(\frac{\lambda \mu }{n_2}-\frac{1}{N}\right)s_{xy2}=23.58\hfill \\ \hfill \mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{\overline{D}}}_{OLP}\right)& =\left(\frac{1}{n_2}-\frac{1}{N}\right)s_{y-x,2}^2=13.11\hfill \\ \\ \hfill \mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\widehat{\overline{D}}}_{STN},{\widehat{\overline{D}}}_{OLP}\right)& =\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\overline{y}}_{23}-{\overline{x}}_{12},{\overline{y}}_2-{\overline{x}}_2\right)\hfill \\ & =\left(\frac{1}{n_{23}}-\frac{1}{N}\right)\left(s_{y2}^2-s_{xy2}^{}\right)-\left(\frac{1}{n_{12}}-\frac{1}{N}\right)\left(s_{xy2}^{}-s_{x2}^2\right)=\mathrm{9.46.}\hfill \end{array}$$

In analogy with (4.4) we now obtain

$$\widehat{k}=\frac{\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{\overline{D}}}_{OLP}\right)-\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\widehat{\overline{D}}}_{STN},{\widehat{\overline{D}}}_{OLP}\right)}{\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{\overline{D}}}_{OLP}\right)+\mathrm{v}\widehat{\mathrm{a}}\mathrm{r}\left({\widehat{\overline{D}}}_{STN}\right)-2\mathrm{c}\widehat{\mathrm{o}}\mathrm{v}\left({\widehat{\overline{D}}}_{STN},{\widehat{\overline{D}}}_{OLP}\right)}=0.206$$

and consequently, ${\widehat{\overline{D}}}_{COM}=5.40\left(12.37\right).$

Note that ${\widehat{g}}_{COM}$ can be rewritten as

$$\begin{array}{c}{\widehat{g}}_{COM}={\widehat{g}}_{OLP}+\widehat{k}\left({\widehat{g}}_{STN}-{\widehat{g}}_{OLP}\right)\\ \approx {\widehat{g}}_{OLP}+k\left({\widehat{g}}_{STN}-{\widehat{g}}_{OLP}\right),\end{array}$$

where we used a first-order Taylor series approximation of ${\widehat{g}}_{COM}.$ Therefore, the random character of estimator $\widehat{k}$ can be neglected for estimating $\mathrm{var}\left({\widehat{g}}_{COM}\right).$ The error thus introduced is of order $1/n_2$ as $n_2\to \infty$ and ${\widehat{g}}_{COM}$ is asymptotically unbiased. Recall that the standard procedure for estimating the variance of the ratio estimator or the regression estimator is based on a first-order Taylor series approximation as well.

In addition, under the same assumptions as (3.4), it can be shown that for sufficiently large $N,$

$$k={\left(1+\frac{2\lambda {\rho }_{xy}^2}{1-{\rho }_{xy}^2}\right)}^{-1};(4.5)$$

for a proof of (4.5), see Appendix A.1. From (4.5) it can be seen that $k$ is decreasing in $\lambda .$ So we have the somewhat counterintuitive result that $k$ is decreasing in $\lambda$ whereas according to (3.1), ratio $Q$ in (3.4) is a convex function of $\lambda ;$ recall that $\mathrm{var}\left({\widehat{g}}_{STN}\right)=\mathrm{var}\left({\widehat{g}}_{OLP}\right)$ and, consequently, $Q=1$ for $\lambda =1$ and $\lambda =S_{y-Gx}^2/2G{S}_{xy}.$

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