A short note on quantile and expectile estimation in unequal probability samples 3. Expectile estimationA short note on quantile and expectile estimation in unequal probability samples 3. Expectile estimation

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An alternative to quantiles are expectiles. The expectile function $M\left(\alpha \right)$  is thereby defined by replacing the $L_1$  loss in (2.1) by the $L_2$  loss leading to

$$M\left(\alpha \right)\mathrm{<mo>=</mo>}\arg \underset{m}{\min }\left\{{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N}{w}_{\alpha }\left(Y_i-m\right){\left(Y_i-m\right)}^2\right\}\mathrm{.}(3.1)$$

Note that $M\left(\alpha \right)$ is continuous in $\alpha$ even for finite populations. Moreover $M\left(0.5\right)$ equals the mean value $\overline{Y}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N}{Y}_i/N.$ Using the sample $y_1\mathrm{,}\dots \mathrm{,}{y}_n$ with inclusion probabilities ${\pi }_1\mathrm{,}\dots \mathrm{,}{\pi }_n$ we can estimate $M\left(\alpha \right)$ by replacing the sum in (2.2) by its sample version, i.e.,

$$\widehat{M}\left(\alpha \right)\mathrm{<mo>=</mo>}\arg \underset{m}{\min }\left\{{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^n}\frac{1}{{\pi }_j}{w}_{\alpha \mathrm{,}j}{\left(y_j-m\right)}^2\right\}$$

with $w_{\alpha \mathrm{,}j}$ as defined above. It is easy to see that the sum in $\widehat{M}\left(\alpha \right)$ is a design-unbiased estimate for the sum in $M\left(\alpha \right).$ The estimate itself is however not design-unbiased like for the quantile function above. However the same arguments as for $Q_N\left(\alpha \right)$ in (2.2) may be used to establish design-consistency.

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