We use the maximum q-log-likelihood estimation for Least informative distributions (LIDs) in order to estimate the parameters in probability density functions (PDFs) efficiently and robustly when data include outlier(s). LIDs are derived by using convex combinations of two PDFs. A convex combination of two PDFs is composed of an underlying distribution and a contamination. The optimal criterion is obtained by minimizing the change of maximum q-log-likelihood function when the data contain small amount of contamination.

In this paper, we make a comparison between ordinary maximum likelihood estimation, maximum q-log-likelihood estimation (MqLE) and LIDs based on MqLE for parameter estimation from data with outliers. We derive a new Fisher information matrix based on the score function for LID from M-function and use it for choice of optimal estimator in the class of MqLE. The model selection is done by the robust information criteria. We test the methods on the real data with outliers and estimate shape and scale parameters of probability distributions. As a result, we show that the LIDs based on MqLE provide the most robust and efficient estimation of the model parameters.

M. Çankaya, J. Korbel, Least informative distributions in maximum q-log-likelihood estimation, Physica A: Statistical Mechanics and its Applications, Vol 509 (2018) 140–150