Notes on Pearson residuals and weighted likelihood estimating equations

In these notes we show that the Pearson residuals (PR) [Lindsay, B.G., 1994. Efficiency versus robustness: the case for minimum Hellinger distance and related methods. Ann. Statist. 22, 1018-1114.] have a natural asymptotic lower bound under the gross error model which can be used in the problem of choosing a root when multiple roots are present in the weighted likelihood estimating equations. Further, we show through an example how the minimum of the PR plays an important role in the robust estimation approach based on weighted likelihood.