A divide and conquer approach to least-squares estimation

The problem of estimating parameters θ which determine the mean μ(θ) of a Gaussian-distributed observation X is considered. It is noted that the maximum-likelihood (ML) estimate, in this case the least-squares estimate, has desirable statistical properties but can be difficult to compute when μ(θ) is a nonlinear function of θ. An estimate formed by combining ML estimates based on subsections of the data vector X is proposed as a computationally inexpensive alternative. The main result is that this alternative estimate, termed here the divide-and-conquer (DAC) estimate, has ML performance in the small-error region when X is appropriately subdivided. As an example application, an inexpensive range-difference-based position estimator is derived and shown by means of Monte-Carlo simulation to have small-error-region mean-square error equal to the Cramer-Rao lower bound