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

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

$$M\left(\alpha \right)\mathrm{=}\mathrm{arg}\underset{m}{\mathrm{min}}\left\{{\displaystyle \sum _{i\mathrm{=1}}^{N}}\text{\hspace{0.17em}}{w}_{\alpha}\left({Y}_{i}-m\right){\left({Y}_{i}-m\right)}^{2}\right\}\mathrm{.}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}(3.1)$$

Note that $M\left(\alpha \right)$ is continuous in $\alpha $ even for finite populations. Moreover $M\left(0.5\right)$ equals the mean value $\overline{Y}\mathrm{=}{\displaystyle {\sum}_{i\mathrm{=1}}^{N}}{Y}_{i}/N.$ 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.,

$$\widehat{M}\left(\alpha \right)\mathrm{=}\mathrm{arg}\underset{m}{\mathrm{min}}\left\{{\displaystyle \sum _{j\mathrm{=1}}^{n}}\frac{1}{{\pi}_{j}}{w}_{\alpha \mathrm{,}j}{\left({y}_{j}-m\right)}^{2}\right\}$$

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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