Small area estimation methods under cut-off sampling
Section 3. Basic direct estimators

• Previous
• Next

We first consider basic direct estimators, obtained using only the $n_i$ observations of the variable of interest from the target area. In the absence of cut-off sampling, these estimators are design-consistent as the domain sample size $n_i$ increases. Moreover, they are nonparametric in the sense that do not require any model assumption. However, they may have unacceptable sampling errors in small domains. In addition, as we shall see below, under cut-off sampling, their design-bias might be substantial.

The usual expansion estimator (Horvitz and Thompson, 1952) of $Y_i$ obtained ignoring that the sample $s_i$ is drawn only from $U_{iI}$ is given by ${\widehat{Y}}_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in s_i}}{w}_{ij}{y}_{ij}.$ Under cut-off sampling, ${\widehat{Y}}_i$ actually estimates the total in the included strata, $Y_{iI}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U_{iI}}}{y}_{ij},$ rather than the overall total $Y_i\mathrm{<mo>=</mo>}{Y}_{iI}+Y_{iE},$ where $Y_{iE}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U_{iE}}}{y}_{ij}.$ Indeed, $E_{\pi }\left({\widehat{Y}}_i\right)\mathrm{<mo>=</mo>}{Y}_{iI},$ where $E_{\pi }$ denotes expectation under repeated sampling, since the sampling weights $w_{j|i}\mathrm{<mo>=</mo>}{\pi }_{j|i}^{-1}$ in ${\widehat{Y}}_i$ expand to $U_{iI}$ instead of $U_i.$ No one would use this estimator since its bias, $B_{\pi }\left({\widehat{Y}}_i\right)\mathrm{<mo>=</mo>}{E}_{\pi }\left({\widehat{Y}}_i\right)-Y_i\mathrm{<mo>=</mo>}-Y_{iE},$ given in relative terms by the proportion of the total represented by the excluded population, ${\text{RB}}_{\pi }\left({\widehat{Y}}_i\right)\mathrm{<mo>=</mo>}-Y_{iE}/Y_i,$ can be substantial.

When auxiliary information is not available, it makes more sense to use the Hájek estimator (Hájek, 1971) for the mean ${\overline{Y}}_i,$ given by ${\widehat{\overline{Y}}}_i^{\text{HA}}\mathrm{<mo>=</mo>}{\widehat{Y}}_i/{\widehat{N}}_i,$ where ${\widehat{N}}_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in s_i}}{w}_{ij}.$ The corresponding estimator for the total is ${\widehat{Y}}_i^{\text{HA}}\mathrm{<mo>=</mo>}{N}_i{\widehat{\overline{Y}}}_i^{\text{HA}},$ considering that the means in the included and excluded strata are equal. Indeed, ignoring the ratio bias (of lower order) and noting that $E_{\pi }\left({\widehat{N}}_i\right)\mathrm{<mo>=</mo>}{N}_{iI},$ the asymptotic (as $n_i\to \infty )$ design-bias of ${\widehat{Y}}_i^{\text{HA}}$ is given in absolute and relative terms by

$$B_{\pi }\left({\widehat{Y}}_i^{\text{HA}}\right)\cong N_{iE}\left({\overline{Y}}_{iI}-{\overline{Y}}_{iE}\right)\mathrm{,}{\text{RB}}_{\pi }\left({\widehat{Y}}_i^{\text{HA}}\right)\cong \frac{N_{iE}}{N_i}\frac{{\overline{Y}}_{iI}-{\overline{Y}}_{iE}}{{\overline{Y}}_i}\mathrm{,}(3.1)$$

where ${\overline{Y}}_{iI}\mathrm{<mo>=</mo>}{Y}_{iI}/N_{iI}$ and ${\overline{Y}}_{iE}\mathrm{<mo>=</mo>}{Y}_{iE}/N_{iE}$ are the true means of the sets of included and excluded units from area $i$ respectively (Haziza et al., 2010). For the mean, the bias of ${\widehat{\overline{Y}}}_i^{\text{HA}}$ is obtained dividing by $N_i$ in (3.1). For a domain $i$ with $U_{iE}\ne \varnothing ,$ the above bias vanishes only when ${\overline{Y}}_{iI}\mathrm{<mo>=</mo>}{\overline{Y}}_{iE},$ which is unlikely in the real cases where cut-off sampling is applied, see e.g., Haziza et al. (2010) or Section 9. In the next section, we briefly describe calibration techniques as a mean of reducing the cut-off sampling bias.

Remark 3.1. The Hájek estimator of ${\overline{Y}}_i$ is a special case of the customary ratio estimator. In many monthly business surveys, parameters of interest are actually the changes over time of certain totals, such as ${\theta }_{it}\mathrm{<mo>=</mo>}{Y}_i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}/Y_i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>},$ where $Y_i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$ is the total of the target variable at time $t$ within domain $i.$ The ratio estimates of change are actually reported instead of the actual totals because it is often believed that such ratios are not affected by cut-off sampling bias. Let ${\widehat{\theta }}_{it}\mathrm{<mo>=</mo>}{\widehat{Y}}_i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}/{\widehat{Y}}_i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>}$ be the basic direct estimator of ${\theta }_{it}.$ As we have seen above, the bias of the ratio estimator due to cut-off sampling tends to be much smaller than that of the absolute totals ${\widehat{Y}}_i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$ and ${\widehat{Y}}_i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>\; .}$ However, as we have also seen, the cut-off sampling bias of ratio estimators vanishes only under strong assumptions. Indeed, ignoring the ratio bias, which is negligible for large $n_i,$ the bias of ${\widehat{\theta }}_{it}$ is given by

$$B_{\pi }\left({\widehat{\theta }}_{it}\right)\cong \frac{Y_{iI}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}{Y_{iI}\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>}}-\frac{Y_i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}}{Y_i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>}}\text{,}$$

where $Y_{iI}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}$ denotes the corresponding total for the included units only. This bias is zero only if the ratios for the population $Y_i\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}/Y_i\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>}$ are the same as those for the included units $Y_{iI}\mathrm{<mo\; stretchy="false">(</mo>}t\mathrm{<mo\; stretchy="false">)</mo>}/Y_{iI}\mathrm{<mo\; stretchy="false">(</mo>}t-\mathrm{1\; <mo\; stretchy="false">)</mo>}.$

• Previous
• Article main page
• Next