4. Anticipated variance

Piero Demetrio Falorsi and Paolo Righi

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Prior to sampling, the $y_{rk}$  values are not known and the variance expressed in formula (3.4) cannot be used for planning the sampling precision at the design phase. In practice, it is necessary to either obtain some proxy values or predict the $y_{rk}$  values based on superpopulation models that exploit auxiliary information. The increasing availability of auxiliary information (deriving by integration of administrative registers and survey frames) facilitates the use of predictions. Under a model-based inference, the $y_{rk}$  values are assumed to be the realization of a superpopulation model $M.$  The model we study has the following form:

$$\{\begin{array}{l}{y}_{rk}=f_r\left(x_k;{\beta }_r\right)+u_{rk}\hfill \\ E_M\left(u_{rk}\right)=0\forall k;E_M\left(u_{rk}^2\right)={\sigma }_{rk}^2;E_M\left(u_{rk},u_{rl}\right)=0\forall k\ne l\hfill \end{array},(4.1)$$

where $x_k$  is a vector of predictors (available in the sampling frame), ${\beta }_r$  is a vector of regression coefficients and $f_r\left(x_k;{\beta }_r\right)$  is a known function, $u_{rk}$  is the error term and $E_M(\cdot )$  denotes the expectation under the model. The parameters ${\beta }_r$  and the variances ${\sigma }_{rk}^2$  are assumed to be known, although in practice they are usually estimated. The model (4.1) is variable-specific and different models for different variables may be used and this does not create additional difficulty. As a measure of uncertainty, we consider the Anticipated Variance (AV) (Isaki and Fuller 1982):

$$\text{AV}\left({\widehat{t}}_{\left(dr\right)}\right)=E_M{E}_p{\left({\widehat{t}}_{\left(dr\right)}-t_{\left(dr\right)}\right)}^2.(4.2)$$

A general expression for the $\text{AV}$  under linear models was derived by Nedyalkova and Tillé (2008). Their formulation is obtained by considering a linear function $f_r(\cdot )$  and a unique set of auxiliary variables, $x_k,$  used for both the prediction of the $y$ values and for balancing the sample. In our context, we have introduced $x_k$  and $z_k={\pi }_k{\delta }_k,$  highlighting that the auxiliary variables can be different for prediction and balancing. The variables $x_k$  must be as predictive of $y_{rk}$  as possible, while the variables $z_k$  play an instrumental role in controlling the sample sizes for sub-populations.

In the context considered here, inserting the approximate variance (3.4) in the equation (4.2), we obtain the approximate expression of the $\text{AV}:$

$$\text{AAV}\left({\widehat{t}}_{\left(dr\right)}\right)=\left[N/\left(N-H\right)\right]{\displaystyle {\sum }_{k\in U}\left(1/{\pi }_k-1\right)E_M\left({\eta }_{\left(dr\right)k}^2\right)},(4.3)$$

where the terms ${\eta }_{\left(dr\right)k}^2$  in (3.4) are replaced by $E_M\left({\eta }_{\left(dr\right)k}^2\right).$  By defining

$${\tilde{y}}_{rk}=f_r\left(x_k;{\beta }_r\right),(4.4)$$

the equation (4.3) may be reformulated as

$$\text{AAV}\left({\widehat{t}}_{\left(dr\right)}\right)=\left[N/\left(N-H\right)\right]\left[{\displaystyle {\sum }_{k\in U}\frac{1}{{\pi }_k}\left({\tilde{y}}_{rk}^2+{\sigma }_{rk}^2\right){\gamma }_{dk}-{\displaystyle {\sum }_{k\in U}\left({\tilde{y}}_{rk}^2+{\sigma }_{rk}^2\right){\gamma }_{dk}-{\text{AAV}}_{3\left(dr\right)}}}\right],(4.5)$$

where the third variance component of $\text{AAV}\left({\widehat{t}}_{\left(dr\right)}\right)$  is

$$\begin{array}{lll}{\text{AAV}}_{3\left(dr\right)}\hfill & =\hfill & {{\displaystyle {\sum }_{k\in U}\left(1-{\pi }_k\right)a}}_{\left(dr\right)k}\left(\pi \right)\left[2{\tilde{y}}_{rk}{\gamma }_{dk}-{\pi }_k{a}_{\left(dr\right)k}\left(\pi \right)\right]\hfill \\ \hfill & +\hfill & {\displaystyle {\sum }_{k\in U}\left(1-{\pi }_k\right)}\left[2b_{\left(dr\right)k}\left(\pi \right)-{\pi }_k{c}_{\left(dr\right)k}\left(\pi \right)\right]\hfill \end{array}(4.6)$$

and $a_{\left(dr\right)k}\left(\pi \right),$   $b_{\left(dr\right)k}\left(\pi \right)$  and $c_{\left(dr\right)k}\left(\pi \right)$  are real numbers defined respectively in equations (A1.4), (A1.7) and (A1.8) of Appendix A1.

Remark 4.1. Expression (4.5) is a cumbersome formula but, for all practical purposes, calculations may be simplified by considering a slight upward approximation by setting $b_{\left(dr\right)k}\left(\pi \right)=c_{\left(dr\right)k}\left(\pi \right)=0$  in (4.6). The proof is given in Appendix A3. An upward approximation is a safe choice in this setting, since it averts from the risk of defining an insufficient sample size for the expected accuracy.

Remark 4.2. The SSRSWOR design is obtained if the planned domains define a unique partition of population (Option 1 of the example in Section 2) and the model (4.1) is specified so that the predicted values are: ${\tilde{y}}_{rk}={\overline{Y}}_{rh}$  with ${\sigma }_{rk}^2={\sigma }_{rh}^2$  (for $k\in U_h).$ The $\text{AAV}$  becomes

$$\text{AAV}\left({\widehat{t}}_{\left(dr\right)}\right)=\left[N/\left(N-H\right)\right]{\displaystyle {\sum }_{d=1}^D{\displaystyle {\sum }_{h\in H_d}{\sigma }_{rh}^2}{N}_h\left(N_h/n_h-1\right)},(4.7)$$

where $H_d$  is the set of planned domains included in $U_d$  (see Appendix A4). Note that the expression (4.7) agrees with the Result 2 of Nedyalkova and Tillé (2008), but for the term $N/\left(N-H\right).$  If $\left[N/\left(N-H\right)\right]\left(1/N_h\right)\approx 1/\left(N_h-1\right)$  the expression (4.7) would approximate the variance of the HT estimate in the SSRSWOR design. The above approximation is proved true when the number of domains $H$  remains small compared to the overall population size $N,$  and when the domain sizes $N_h$  are large.

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