# Modeling Expected Value Using Quantile Regression as an Ensemble

I'm trying to find a primer on a topic that I'm sure must have been studied, but I can't find anything on. Suppose we'd like to do regression in a supervised learning setting to learn the expected value of $$y$$ given $$x$$ i.e. $$\mathbb{E}_{p(y|x)}[y]$$. Rather than learning the expected value directly, suppose I instead perform quantile regression to learn $$Q$$ uniformly distributed quantiles and then combine the quantiles to estimate the desired expected value. Has anyone studied this algorithm? Does it have a name? Is there an advantage to be had by taking this circuitous route? For instance, each quantile may be biased, but their combination might have smaller variance?

• @whuber , you've been tremendously helpful. Would you happen to know? Jun 23, 2020 at 20:04
• Do you mean $\mathbb{E}(y|x)$ instead of $\mathbb{E}(p(y|x))$? Or otherwise, how is the latter the expected value of $y$ given $x$? Also, what does the subscript $_x$ stand for in the expectation? Jun 23, 2020 at 20:31
• I think I meant $\mathbb{E}_{p(y|x)}[y]$ Jun 23, 2020 at 20:33
• Thank you for the clarification. Jun 23, 2020 at 20:34

It has been established that the model-averaged as well as the composite estimator can beat the OLS estimator of $$\beta$$ in a linear regression model in terms of asymptotic efficiency for certain error distributions. Actually, the composite estimator with equal weights has some nice efficiency guarantees; given some assumptions, the asymptotic relative efficiency of the composite quantile estimator with equal weights is at least ~70% that of an OLS estimator, but it can be much more efficient than it (Zou & Yuan "Composite quantile regression and the oracle model selection theory" (2008)).