Minimum hellinger distance estimation for normal models

Abstract

A robust estimator introduced by Beran (1977a, 1977b), which is based on the minimum Hellinger distance between a projection model density and a nonparametric sample density, is studied empirically. An extensive simulation provides an estimate of the small sample distribution and supplies empirical evidence of the estimator performance for a normal location-scale model. While the performance of the minimum Hellinger distance estimator is seen to be competitive with the maximum likelihood estimator at the true model, its robustness to deviations from normality is shown to be competitive in this setting with that obtained from the M-estimator and the Cramér-von Mises minimum distance estimator. Beran also introduced a goodness-of-fit statisticH ₂, based on the minimized Hellinger distance between a member of a parametric family of densities and a nonparametric density estimate. We investigate the statistic H (the square root of H ₂) as a test for normality when both location and scale are unspecified. Empirically derived critical values are given which do not require extensive tables. The power of the statistic H compares favorably with four other widely used tests for normality.

Keywords:

• Minimum distance
• robustness
• efficiency
• Hellinger distance
• influence curve
• nonparametric
• goodness of fit
• power

^∗The authors wish to thank William C. Parr, William R. Schucany, and a referee for many helpful suggestions. The research of the second author was partially supported by DARPA Contract No. F19628-88-K-0042.

^∗The authors wish to thank William C. Parr, William R. Schucany, and a referee for many helpful suggestions. The research of the second author was partially supported by DARPA Contract No. F19628-88-K-0042.

Notes

^∗The authors wish to thank William C. Parr, William R. Schucany, and a referee for many helpful suggestions. The research of the second author was partially supported by DARPA Contract No. F19628-88-K-0042.