A study of nonparametric estimation of location using L-, M- and R-estimators

Nonparametric procedures use weak assumptions such as continuity of the distribution so that they are applicable to a large class F of underlying distributions. Statistics that are distribution-free over F may be constructed to be estimators of location. Such estimators are derived from rank tests called R-estimators. They are robust estimators. The concept of robust estimation is based on a neighborhood of parametric models called "gross error models". The M-estimator, which is a maximum likelihood type estimator, arose from such investigations using the normal distribution. A third big class of estimators is the class of linear combinations of order statistics called L-estimators. They are constructed as an average of quantiles. Examples are the sample mean and the sample median.In this thesis, some definitions and results involving these three basic classes of estimates are provided. For each class, an example of a robust estimator is presented. Numerical values are given to assess the robustness of each estimator in terms of breakdown point and gross error sensitivity. Further, the U-statistics which are unbiased estimators of location parameters, are used to obtain asymptotically efficient R-estimates.Department of Mathematical SciencesThesis (M.A.