Comparison of the alternative parameter estimators of Pearson distributions by robustness criteria

Abstract

Pearson's differential equation is used for fitting a distribution to a data set. The differential equation has some alternative moment-based estimators (depending on the transformation to data). The estimator used when no transformation is made on the data set has 4 elements, and the estimators that require any transformation have 3 elements. We describe all elements of the estimators by corresponding vectors. One of the factors affecting the preference of an estimator is robustness. We use covariance matrix, bias, relative efficiency and influence function as our robustness criteria. Our aim is to compare the performance of the estimators of the differential equation for some specific distributions (namely Type I, Type IV, Type VI and Type III). 10,000 samples with specific sizes were selected with replacement. Also, we evaluated the performance of the estimators over real-life data. Considering the results, there is no best estimator in all criteria. Depending on the criterion to be based, the estimator to be preferred varies.