Quantile estimation and
quantile regression have seen a number of new developments in recent years with
Koenker (2005) as a central reference. The principle idea is thereby to
estimate an inverted cumulative distribution function, generally called the
where the 0.5 quantile
the median, plays a central role. For survey data tracing
from an unequal probability sample with known probabilities of inclusion Kuk
(1988) shows how to estimate quantiles taking the inclusion probabilities into
account. The central idea is to estimate a distribution function of the
variable of interest and invert this in a second step to obtain the quantile
function. Chambers and Dunstan (1986) propose a model-based estimator of the
distribution function. Rao, Kovar and Mantel (1990) propose a design-based
estimator of the cumulative distribution function using auxiliary information.
Bayesian approaches in this direction have recently been proposed in Chen,
Elliott, and Little (2010) and Chen, Elliott, and Little (2012).
Quantile estimation results
from minimizing an
loss function as demonstrated in Koenker (2005). If the
loss is replaced by the
loss function one obtains so called expectiles
as introduced in Aigner, Amemiya
and Poirier (1976) or Newey and Powell (1987).
this leads to the expectile function
which, like the quantile function
uniquely defines the cumulative distribution
. Expectiles are relatively
easy to estimate and they have recently gained some interest, see e.g.,
Schnabel and Eilers (2009), Pratesi, Ranalli, and Salvati (2009), Sobotka and Kneib
(2012) and Guo and Härdle (2013). However since expectiles lack a simple interpretation
their acceptance and usage in statistics is less developed than quantiles, see
Kneib (2013). Quantiles and expectiles are connected in that a unique and
invertible transformation function
exists so that
see Yao and Tong (1996) and De Rossi and
Harvey (2009). This connection can be used to estimate quantiles from a set of
fitted expectiles. The idea has been used in Schulze Waltrup, Sobotka, Kneib
and Kauermann (2014) and the authors show empirically that the resulting
quantiles can be more efficient than empirical quantiles, even if a smoothing
step is applied to the latter (see Jones 1992). An intuitive explanation for
this is that expectiles account for all the data while quantiles based on the
empirical distribution function only take the left (or the right) hand side of
the data into account. That is, the median is defined by the 50% left (or 50%
right) part of the data while the mean (as 50% expectile) is a function of all
data points. In this note we extend these findings and demonstrate how
expectiles can be estimated for unequal probability samples and how to obtain a
fitted distribution function from fitted expectiles.
The paper is organized as
follows. In Section 2 we give the necessary notation and discuss quantile
regression in unequal probability sampling. This is extended in Section 3 towards expectile estimation. Section 4 utilizes the connection between
expectiles and quantiles and demonstrates how to derive quantiles from fitted
expectiles. Section 5 demonstrates in simulations the efficiency gain in
quantiles derived from expectiles and a discussion concludes the paper in
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