Comparison of the parameter estimations of the main types of pearson distributions by robustness criteria

The aim of this study is to compare the performance of alternative estimators which can be used to estimate the parameters of Pearson differential equation which is used to determine a probability function that defines any data set according to certain measures. The distributions defined by the parameters of the Pearson differential equation are called Pearson distributions. Within this distribution family, there are thirteen distributions, three main types and other transition types. Pearson differential equation parameters can be obtained by an estimator without making any transformation on the data set, or an estimator obtained by transforming the mean of the data set to zero or an estimator obtained by transforming converting the mode of the data set to zero. The number of parameters to be estimated is reduced from four to three if the transformation to the data set is applied. The alternative estimators are in the M-estimator class of the robust estimators classes. For this reason, performance criteria have been determined as the sensitivity, bias, relative efficiency and influence function commonly used for these class of estimators. In order to compare the big and small sample performances of the alternative estimators, three populations consisting of Type I, Type IV and Type VI distributions, which are the main types of Pearson distributions, were handled and 10000 samples with 50, 100, 200 and 800 volumes were selected with replacement. According to the results, it is seen that there is no best estimator in all criteria. Depending on the criterion to be based, the estimator to be preferred varies.