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Regression percentiles using asymmetric squared error loss. (English) Zbl 0822.62054
Summary: We consider the problem of estimating regression percentiles, for example the 75th conditional percentile of the response variable $y$ given the covariate vector $x$. Asymmetric Least Squares (ALS) is a variant of ordinary least squares, in which the squared error loss function is given different weight depending on whether the residual is positive or negative. ALS estimates of regression percentiles are easy to compute. They are reasonably efficient under normality conditions. There is an interesting connection between ALS estimates and absolute residual regression for detecting heteroscedasticity. Three examples are given to demonstrate the utility of estimated regression percentiles for understanding regression data, particularly when the covariate $x$ is multi-dimensional.

##### MSC:
 62J05 Linear regression 62H12 Multivariate estimation 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis)