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 F ^ R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaaaa@38EA@  with quantiles based on the expectile based estimator F ^ R M , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGnbaaaOGaaiilaaaa@3A77@  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 F ^ N M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaad6eaaeaacaWGnbaaaaaa@39B9@  is also part of the R-package expectreg. The calculation of F ^ R M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGnbaaaaaa@39BD@  is, however, more demanding as the calculation of F ^ R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaaaa@38EA@  because it involves three steps: First, we calculate the weighted expectiles as described in Section 3; second, we estimate F ^ R N , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGobaaaOGaaiilaaaa@3A78@  and in a third step, we derive F ^ R M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGnbaaaaaa@39BD@  from F ^ R N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGobaaaaaa@39BE@  (see Section 4). In the Log-Normal-Simulation it takes about 2-3 seconds for N = 1,000 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaiaai2 dacaqGXaGaaeilaiaabcdacaqGWaGaaeimaaaa@3C22@  to calculate F ^ R M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja Waa0baaSqaaiaadkfaaeaacaWGnbaaaaaa@39BD@  whereas the computational effort of F ^ R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqWqpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmOrayaaja WaaSbaaSqaaiaadkfaaeqaaaaa@38EA@  is barely noticeable.

Acknowledgements

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

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