Local Likelihood Estimation for Nonstationary Random Fields

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Stationary random fields play a fundamental role in both theoretical and applied spatial statistics. Unfortunately, stationarity is often violated when working with real data. This presents a challenge for the spatial statistician who is interested in estimating and modeling dependence structure in random fields. The investigator and his collaborator will study, and subsequently apply, a recently developed version of local likelihood estimation for estimating parameters that govern the local dependency in nonstationary random fields. The application is to use local likelihoods for the estimation of the local distortion of the cosmic microwave background (CMB) due to gravitational lensing. Detecting and estimating this lensing is important for two reasons. First, measuring the distortion gives an indirect measurement of matter in the Universe which can provide detailed maps of the distribution of Dark Matter. Secondly, one can use the estimates of gravitational lensing to obtain a more accurate measurement of the un-lensed CMB. This will provide scientists with a deeper probe of fundamental cosmological questions. Specific aims of this proposal include: 1) develop computational techniques which will make local likelihood estimation applicable to large data sets such as the CMB; 2) develop the theory of estimating equations for local weight construction with a particular focus on mitigating bias and automatically adjusting for boundaries and uneven observation locations; 3) construct estimates of uncertainty in the local likelihood estimates of the parameter function; 4) use the principle irregular term to develop flexible and universal local models for general nonstationary random fields. Nonstationary random fields are becoming a ubiquitous feature in the recent data deluge and high resolution sensing. Unfortunately, the techniques for modeling, estimating and predicting nonstationary random fields have yet to be fully developed and analyzed. This proposal will make steps towards the goal of developing a complete set of methodological techniques for analyzing nonstationary random fields. Moreover, the tools thus constructed will be useful, not only for Astronomy and Cosmology, but other branches of science and technology as well. This makes the broader impact of the scientific consequences of this proposal two fold. On the one hand, accurately estimating gravitational lensing in the CMB has far reaching consequences for the understanding of cosmic structure and the beginnings of the Universe. On the other hand, the tools from this project are expected to have broad use in other areas of real world application. Recently developed technologies such as fMRI and diffusion tensor imaging are two examples of other scientific areas that will benefit from the methodologies developed for nonstationary random fields.