Weighted type of quantile regression and its application

In this paper we introduce a weighted composite quantile regression (CQR) estimation approach and study its application in nonlinear models such as exponential models and ARCH type of models. The weighted CQR is augmented by using a data-driven weighting scheme. With the error distribution unspecified, the proposed estimators share robustness from quantile regression and achieve nearly the same efficiency as the oracle maximum likelihood estimator(MLE) for a variety of error distributions including the normal, mixed-normal, Student's t, Cauchy distributions and etc, We also suggest an algorithm for fast implementation of the proposed methodology. Simulations are conducted to compare the performance of different estimators, and the proposed approach is used to analyze the daily S&P 500 Composite index, which endorse our theoretical results.