A Block Minorization–Maximization Algorithm for Heteroscedastic Regression

The computation of the maximum likelihood (ML) estimator for heteroscedastic regression models is considered. The traditional Newton algorithms for the problem require matrix multiplications and inversions, which are bottlenecks in modern Big Data contexts. A new Big Data-appropriate minorization-maximization (MM) algorithm is considered for the computation of the ML estimator. The MM algorithm is proved to generate monotonically increasing sequences of likelihood values and to be convergent to a stationary point of the log-likelihood function. A distributed and parallel implementation of the MM algorithm is presented, and the MM algorithm is shown to have differing time complexity to the Newton algorithm. Simulation studies demonstrate that the MM algorithm improves upon the computation time of the Newton algorithm in some practical scenarios where the number of observations is large.