A comparison between nonparametric estimators for finite population distribution functions 4. Design-based propertiesA comparison between nonparametric estimators for finite population distribution functions 4. Design-based properties

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$$E_d\left({\left|\tilde{F}\left(t\right)-F_N\left(t\right)\right|}^2\right)\mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{,}j\in U}}\frac{{\pi }_{i\mathrm{,}j}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\left[I\left(y_i\le t\right)-{\overline{G}}_i\left(t\right)\right]\left[I\left(y_j\le t\right)-{\overline{G}}_j\left(t\right)\right]+o\left(n^{-1}\right)\mathrm{,}$$

where $E_d(\cdot )$ denotes expectation with respect to the sample design, ${\pi }_{i\mathrm{,}j}$ denotes the joint sample inclusion probability for units $i$ and $j$ (it is understood that ${\pi }_{i\mathrm{,}i}\mathrm{<mo>=</mo>}{\pi }_i),$ and where

$${\overline{G}}_i\left(t\right)\mathrm{<mo>:=</mo>}{\displaystyle \sum _{j\in U}{\overline{w}}_{i\mathrm{,}j}I\left(y_j\le t\right)}\mathrm{.}$$

The regression weights ${\overline{w}}_{i\mathrm{,}j}$ in the definition of ${\overline{G}}_i\left(t\right)$ refer to the whole finite population $U$ and are given by

$${\overline{w}}_{i\mathrm{,}j}\mathrm{<mo> :</mo>\; <mo>=</mo>}\frac{1}{N\lambda }K\left(\frac{x_i-x_j}{\lambda }\right)\frac{{\overline{M}}_{\mathrm{2,}s}\left(x_i\right)-\left(\frac{x_i-x_j}{\lambda }\right){\overline{M}}_{\mathrm{1,}s}\left(x_i\right)}{{\overline{M}}_{\mathrm{2,}s}\left(x_i\right){\overline{M}}_{\mathrm{0,}s}\left(x_i\right)-{\overline{M}}_{\mathrm{1,}s}^2\left(x_i\right)}\mathrm{,}$$

where

$${\overline{M}}_{r\mathrm{,}s}\left(x\right)\mathrm{<mo>:=</mo>}{\displaystyle \sum _{k\in U}}\frac{1}{N\lambda }K\left(\frac{x-x_k}{\lambda }\right){\left(\frac{x-x_k}{\lambda }\right)}^r\mathrm{,}r\mathrm{<mo>=</mo>\; 0,1,2.}$$

Moreover, according to Breidt and Opsomer (2000),

$$\tilde{V}\left(\tilde{F}\left(t\right)\right)\mathrm{<mo>:=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{,}j\in s}}\frac{{\pi }_{i\mathrm{,}j}-{\pi }_i{\pi }_j}{{\pi }_{i\mathrm{,}j}{\pi }_i{\pi }_j}\left[I\left(y_i\le t\right)-{\tilde{G}}_i\left(t\right)\right]\left[I\left(y_j\le t\right)-{\tilde{G}}_j\left(t\right)\right]$$

is a consistent estimator for the design mean square error of $\tilde{F}\left(t\right).$

Unfortunately the results in Breidt and Opsomer (2000) cannot be applied to the generalized difference estimator ${\tilde{F}}^{\mathrm{<mo>*</mo>}}\text{}\left(t\right)$ as well, since the latter estimator does not fall into the class of local polynomial regression estimators due to the presence of the regression function estimators ${\tilde{m}}_i$ and ${\tilde{m}}_j$ inside the indicator functions in the fitted values ${\tilde{G}}_i^{*}\text{}\left(t\right).$ However, the results for $\tilde{F}\left(t\right)$ suggest that in large samples ${\tilde{G}}_i^{*}\left(t\right)$ and

$${\overline{G}}_i^{*}\left(t\right)\mathrm{<mo>:=</mo>}{\displaystyle \sum _{j\in U}{\overline{w}}_{i\mathrm{,}j}I\left(y_j-{\overline{m}}_j\le t-{\overline{m}}_i\right)}\mathrm{,}$$

where ${\overline{m}}_i\mathrm{<mo>:=</mo>}{\displaystyle {\sum }_{j\in U}{\overline{w}}_{i\mathrm{,}j}{y}_j},$ are approximately the same, and that

$$E_d\left({\left|{\tilde{F}}^{*}\left(t\right)-F_N\left(t\right)\right|}^2\right)\mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{,}j\in U}}\frac{{\pi }_{i\mathrm{,}j}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\left[I\left(y_i\le t\right)-{\overline{G}}_i^{*}\left(t\right)\right]\left[I\left(y_j\le t\right)-{\overline{G}}_j^{*}\left(t\right)\right]+o\left(n^{-1}\right)$$

Based on this conjecture, we tested

$$\tilde{V}\left({\tilde{F}}^{\mathrm{<mo>*</mo>}}\left(t\right)\right)\mathrm{<mo>:=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{,}j\in s}}\frac{{\pi }_{i\mathrm{,}j}-{\pi }_i{\pi }_j}{{\pi }_{i\mathrm{,}j}{\pi }_i{\pi }_j}\left[I\left(y_i\le t\right)-{\tilde{G}}_i^{*}\left(t\right)\right]\left[I\left(y_j\le t\right)-{\tilde{G}}_j^{*}\left(t\right)\right]\mathrm{.}$$

as estimator for the design mean square error of the generalized difference estimator ${\tilde{F}}^{\mathrm{<mo>*</mo>}}\text{}\left(t\right)$ in the simulation study of the following section.

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