© 1997 by Biometrika Trust
Generalised weighted Cramér-von Mises distance estimators
Department of Statistics, Ohio State University 1465 Mt. Vernon Avenue, Marion, Ohio 43302, U.S.A. e-mail: omer{at}stat.mps.ohio-state.edu
Department of Statistics, Penn State University, University Park Pennsylvania 16802, U.S.A. e-mail: tph{at}stat.psu.edu
Generalised weighted Cramér-von Mises distance estimators in an arbitrary model with a k-dimensional parameter vector are investigated. The distance function is defined as a function G of model-based residuals for a specified target model F and the empirical cumulative distribution function Fn over the real line, invoking a weight function w. It is shown that the estimator is Fisher consistent, asymptotically multivariate normal, and nearly efficient with desirable robustness properties. If the true model is equal to the target model, the residual function G does not affect the limiting distribution. The weight function w controls the asymptotic distribution and the robustness of the estimator. Three different classes of the weight functions are introduced for different outlier patterns. These weight functions produce estimators asymptotically as efficient as the maximum likelihood estimators at the true model. An alternative way of calculating the estimators is considered. Simulation results indicate that asymptotic results are useful for moderate sample sizes and that the estimators are stable at the neighbourhood of the target model.
Key Words: Bounded influence • Efficiency • Hellinger distance • Minimum distance • Neighbourhood model • Robustness