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

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

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

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

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

with ${w}_{\alpha ,j}$ as defined above. It is easy to see that the sum in $\stackrel{^}{M}\left(\alpha \right)$ 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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