# A short note on quantile and expectile estimation in unequal probability samples 6. Discussion

In Section 4 we extended the toolbox of expectiles to the estimation of distribution functions in the framework of unequal probability sampling. We defined expectiles for unequal probability samples. When comparing quantiles based on ${\widehat{F}}_{R}$ with quantiles based on the expectile based estimator ${\widehat{F}}_{R}^{M},$ we observed that the proposed estimator performs well in comparison to existing methods. The calculation of empirical expectiles is implemented in the open source software R (see R Core Team 2014) and can be found in the R-package expectreg by Sobotka, Schnabel, and Schulze Waltrup (2013). The calculation of the expectile based distribution function estimator ${\widehat{F}}_{N}^{M}$ is also part of the R-package expectreg. The calculation of ${\widehat{F}}_{R}^{M}$ is, however, more demanding as the calculation of ${\widehat{F}}_{R}$ because it involves three steps: First, we calculate the weighted expectiles as described in Section 3; second, we estimate ${\widehat{F}}_{R}^{N},$ and in a third step, we derive ${\widehat{F}}_{R}^{M}$ from ${\widehat{F}}_{R}^{N}$ (see Section 4). In the Log-Normal-Simulation it takes about 2-3 seconds for $N\mathrm{=}\text{1,000}$ to calculate ${\widehat{F}}_{R}^{M}$ whereas the computational effort of ${\widehat{F}}_{R}$ is barely noticeable.

## Acknowledgements

Both authors acknowledge financial support provided by the Deutsche Forschungsgemeinschaft DFG (KA 1188/7-1).

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