Minimum Hellinger distance estimation for a semiparametric location-shifted mixture model

ABSTRACT

In this article, we propose a minimum Hellinger distance estimation (MHDE) for a semiparametric two-component mixture model where the two components are unknown location-shifted symmetric distributions $f(x-{\mu }_1)$ and $f(x-{\mu }_2)$. In the construction of MHDE, an appropriate estimation of the unknown nuisance parameter f is required. We propose to use the inversion formula given in Bordes et al. to estimate f based on current available sample from the mixture. To obtain the MHDE, an algorithm is presented to ease the numerical calculation. We also propose a simple but intuitive and robust initial estimator of the parameters. To assess its performance, we carry out a simulation study with comparison with a minimum profile Hellinger distance estimator (MPHDE) given in Wu et al. We use the proposed estimator to analyse the Old Faithful Geyser data in order to demonstrate its application. Through the numerical studies, we observe that our proposed MHDE for this semiparametric mixture model inherits the desired robustness and efficiency properties of that for parametric models. The proposed MHDE is very competitive with the MPHDE when there is no data contamination, whereas it performs better than the MPHDE in terms of bias when data is contaminated with outliers. Moreover, the MHDE reduces significantly the computing time of the MPHDE.

KEYWORDS:

• Minimum Hellinger distance
• semiparametric mixture model
• location-shifted mixture
• robustness
• inversion formula

Acknowledgements

The authors are thankful to the editor, the associate editor and the reviewers for their constructive comments that have led to great improvement of this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by a grant from the Natural Sciences and Engineering Research Council of Canada [grant no. RGPIN 355970-2013].