  Great strides have been made in the field of reconstructing past temperatures
based on models relating temperature to temperature-sensitive paleoclimate
proxies. One of the goals of such reconstructions is to assess if current
climate is anomalous in a millennial context. These regression based approaches
model the conditional mean of the temperature distribution as a function of
paleoclimate proxies (or vice versa). Some of the recent focus in the area has
considered methods which help reduce the uncertainty inherent in such
statistical paleoclimate reconstructions, with the ultimate goal of improving
the confidence that can be attached to such endeavors. A second important
scientific focus in the subject area is the area of forward models for proxies,
the goal of which is to understand the way paleoclimate proxies are driven by
temperature and other environmental variables. In this paper we introduce novel
statistical methodology for (1) quantile regression with autoregressive
residual structure, (2) estimation of corresponding model parameters, (3)
development of a rigorous framework for specifying uncertainty estimates of
quantities of interest, yielding (4) statistical byproducts that address the
two scientific foci discussed above. Our statistical methodology demonstrably
produces a more robust reconstruction than is possible by using
conditional-mean-fitting methods. Our reconstruction shares some of the common
features of past reconstructions, but also gains useful insights. More
importantly, we are able to demonstrate a significantly smaller uncertainty
than that from previous regression methods. In addition, the quantile
regression component allows us to model, in a more complete and flexible way
than least squares, the conditional distribution of temperature given proxies.
This relationship can be used to inform forward models relating how proxies are
driven by temperature.
