Symmetric regression quantile and its application to robust estimation for the nonlinear regression model

Populational conditional quantiles in terms of percentage α are useful as indices for identifying outliers. We propose a class of symmetric quantiles for estimating unknown nonlinear regression conditional quantiles. In large samples, symmetric quantiles are more efficient than regression quantiles considered by Koenker and Bassett (Econometrica 46 (1978) 33) for small or large values of α, when the underlying distribution is symmetric, in the sense that they have smaller asymptotic variances. Symmetric quantiles play a useful role in identifying outliers. In estimating nonlinear regression parameters by symmetric trimmed means constructed by symmetric quantiles, we show that their asymptotic variances can be very close to (or can even attain) the Cramer–Rao lower bound under symmetric heavy-tailed error distributions, whereas the usual robust and nonrobust estimators cannot.