Small area estimation methods under cut-off sampling
Section 9. Estimation of total sales in Spanish provinces

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Here we describe an application to the estimation of the total sales of a certain tobacco product in the Spanish provinces. The available data set contains, for $N\mathrm{<mo>=</mo>}$ 12,791 tobacco establishments (practically all of them) in $m\mathrm{<mo>=</mo>}$ 48 provinces from Spain (the Canary Islands, Ceuta and Melilla are not included), the volume of purchases made by each establishment of this product during the three months previous to November 2016 $(z_{ij},$ in euros). It also contains a variable indicating whether the establishment is supplied with a device recording all the required information about each sale. Only the establishments with larger sales are supplied with such a device. Those establishments (in total $n\mathrm{<mo>=</mo>}$ 1,842) are able to report proper data on sales and therefore the volume of sales $(v_{ij},$ in euros) of the considered product in November 2016 is also included in the data for those establishments.

We estimate the total sales $V_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^{N_i}}{v}_{ij}$ in each of the $m\mathrm{<mo>=</mo>}$ 48 provinces included in the data using the basic direct, the selected calibration estimators and a model-based estimator. Establishments $j$ with both $z_{ij}$ and $v_{ij}$ available for a province $i$ compose the set of included units $U_{iI},$ which equals the sample $s_i$ in this case (there is no sampling within $U_{iI}).$ Then, here the basic direct estimators are given by ${\widehat{V}}_i^{\text{HA}}\mathrm{<mo>=</mo>}{N}_i{\overline{V}}_{iI},$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ which have actually zero variance, but might be severely biased. Since true values in real applications are not available and therefore real biases cannot be evaluated (there is no information from $U_{iE}),$ here we will compare the estimators considering the set of establishments with sales recorded from each province as a SRSWOR from that province. Note that this is the best scenario for the basic direct estimator. Thus, for the basic direct estimator ${\widehat{V}}_i^{\text{HA}}$ considering that the actual sample $s_i\mathrm{<mo>=</mo>}{U}_{iI}$ is a SRSWOR from $U_i,$ the variance equals the MSE (we ignore the bias). A design-unbiased estimator of the MSE is then

$${\text{mse}}_{\pi }\left({\widehat{V}}_i\right)\mathrm{<mo>=</mo>}{N}_i^2\frac{s_i^2}{n_i}\left(1-\frac{n_i}{N_i}\right)\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m\mathrm{,}$$

where $s_i^2\mathrm{<mo>=</mo>}{\left(n_i-1\right)}^{-1}{\displaystyle {\sum }_{j\in s_i}}{\left(v_{ij}-{\overline{v}}_{is}\right)}^2$ is the sample variance of the sales for province $i$ and here $n_i\mathrm{<mo>=</mo>}{N}_{iI},$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m.$

For the estimators that consider a regression model, we first make a preliminary descriptive analysis of the variables. Histograms of sales $v_{ij}$ and of purchases $z_{ij}$ show right-skewed distributions for both variables. Moreover, a scatterplot of ordinary LS residuals from a linear model for $v_{ij}$ in terms of $z_{ij},$ against $z_{ij}$ reveals a mild pattern of heteroscedasticity. Transforming the sales with the squared root, that is, taking $y_{ij}\mathrm{<mo>=</mo>}{v}_{ij}^{1/2}$ as response variable and $x_{ij}\mathrm{<mo>=</mo>}{\left(\mathrm{1,}{x}_{ij}\right)}^{\prime },$ with $x_{ij}\mathrm{<mo>=</mo>}{z}_{ij}^{1/2}$ as covariate seems to minimize the problem. Accordingly, we will consider a nested error model (5.1) for the transformed sales $y_{ij}$ in terms of the transformed purchases $x_{ij},$ and EBPs of the total sales in each province, $V_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{N_i}}{v}_{ij},$ will be computed based on this model. Note that, in terms of the model responses $y_{ij},$ the total sales are given by $V_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{N_i}}{y}_{ij}^2\mathrm{<mo>=</mo>}h\left(y_i\right).$ Then, the EBP of $V_i\mathrm{<mo>=</mo>}h\left(y_i\right)$ is given by ${\widehat{V}}_i^{\text{EBP}}\mathrm{<mo>=</mo>}{E}_{m_3}\left[h\left(y_i\right)|y_{is}\mathrm{<mo>;</mo>}\widehat{\theta }\right],$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ which can be calculated analytically or approximated by Monte Carlo simulation. We estimate the model MSE of the EBP using the parametric bootstrap described in Section 7 for $H_i\mathrm{<mo>=</mo>}{V}_i,$ taking ${{\displaystyle H}}_i^{*(b)}={{\displaystyle V}}_i^{*(b)}$ and ${{\displaystyle \widehat{H}}}_i^{\text{EBP}*(b)}={{\displaystyle \widehat{V}}}_i^{\text{EBP}*(b)}$ and considering that the model holds for included and excluded units. Residuals from this model are described below.

Note that the LCAL (or GREG) estimator is not defined for a non-linear function of the values of the response variable in the population units, such as the total sales $V_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{N_i}}{y}_{ij}^2$ after the square root transformation. Hence, here we calculate the GREG according to (4.3) using $v_{ij}$ instead of $y_{ij}$ and $z_{ij}$ instead of $x_{ij},$ which is assisted by the linear model (4.10) for the untransformed sales $v_{ij}$ in terms of purchases $z_{ij}.$ As a measure of uncertainty of the GREG, to make it comparable with that of the EBP, we estimated its model MSE through the same bootstrap procedure, replacing ${\widehat{H}}_i^{\text{EBP*}\left(\text{}b\text{}\right)}$ by ${{\displaystyle \widehat{V}}}_i^{\text{GREG}*(b)}.$ The obtained bootstrap MSE estimator actually includes the error due to the fact that the correct model is the one with transformed variables.

Before comparing the estimates, we analyze the residuals from the nested error model (5.1), given by ${\widehat{e}}_{ij}\mathrm{<mo>=</mo>}{y}_{ij}-x_{ij}^{\prime }\widehat{\beta }-{\widehat{u}}_i.$ Figure 9.1 shows a scatterplot of those residuals against predicted values ${\widehat{y}}_{ij}\mathrm{<mo>=</mo>}{x}_{ij}^{\prime }\widehat{\beta }+{\widehat{u}}_i$ (left) and a histogram of residuals (right). We can see a few negative outliers on the left plot, which agrees with a slightly larger left tail in the histogram. Apart from that, the residuals do not exhibit any remarkable pattern. In fact, in the histogram they appear to be very much concentrated around zero, which indicates a high predictive power of the model.

Figure 9.2 shows the normal Q-Q plot of predicted area effects ${\widehat{u}}_i.$ This plot supports the normality of ${\widehat{u}}_i$ except for one outlier appearing at the left tail of the distribution. This point corresponds to the province with the smallest sample size $(n_i\mathrm{<mo>=</mo>}$ 3 observations), which suggests that the estimated random effect for that province, ${\widehat{u}}_i,$ is not very reliable. Thus, we consider that the nested error model fits reasonably well the available data.

We proceed now to compare the obtained estimates. Figure 9.3 left shows EBPs of the total sales of the considered tobacco product for each province against direct estimates. Province sample sizes are used as point labels. This plot indicates a great similarity of the two types of estimates except for the two provinces with the largest sample sizes, where the EBPs are slightly larger than direct estimates, which could be due to cut-off sampling bias of the direct estimator. Figure 9.3 right displays EBPs against GREG estimates. The great similarity of GREG and EBP estimates shown by this plot supports the fact that direct estimators might be actually understating the total sales in this application.

Finally, we compare the three types of estimates of the total sales for each province in Figure 9.4 left, showing the point estimates for each province (x-axis), with provinces sorted from smaller to larger sample sizes, and with sample sizes indicated in the x-axis labels. The conclusions are the same as before; that is, the three types of estimates take very similar values for all provinces except for a couple of provinces with the larger sample sizes, where the basic direct estimator takes slightly smaller values (possibly understating the total sales). Figure 9.4 (right) shows the estimated coefficients of variation (CV) obtained ignoring the bias due to cut-off sampling. EBP estimators perform uniformly better than the other estimators in terms of estimated CV, keeping the CV values below 10% for practically all provinces, whereas GREG estimator obtains CV values above 10% for the provinces with the smallest sample sizes. We can see some peaks in the estimated CVs for some provinces with not necessarily the smallest sample sizes. These larger CV values are due to the presence of zero purchases and sales of the considered product in many tobacco shops for those particular provinces (that particular product is not acquired every month). Clearly, the direct estimator performs the worst in terms of efficiency.

Table A.1 in the Appendix reports direct, LCAL and EBP estimates of province total sales of the product supplemented with their estimated CVs. This table confirms the better performance of EBP in terms of estimated CV under the nested error model, specially for those provinces with small sample sizes. Finally, the direct estimator performs poorly in terms of CV even if the bias due to cut-off sampling is not accounted for.

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