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
alternative to quantiles are expectiles. The expectile function
is thereby defined by replacing the
loss in (2.1) by the
loss leading to
is continuous in
even for finite populations. Moreover
equals the mean value
Using the sample
with inclusion probabilities
we can estimate
by replacing the sum in (2.2) by its sample
as defined above. It is easy to see that the
is a design-unbiased estimate for the sum in
The estimate itself is however not
design-unbiased like for the quantile function above. However the same
arguments as for
in (2.2) may be used to establish
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