The purpose of this research is to consider the problem of constructing robust nonparametric confidence intervals and tests for the median when the data distribution G is unknown and the data may be contaminated. In this robust nonparametric situation we introduce a new neighborhood of an unknown model distribution Fo in order to describe the departure of G from Fo. The neighborhood is generated from a special capacity which is determined by three parameters, and it includes the commonly used neighborhoods defined in terms of gross error and total variation distance as special cases. Various new neighborhoods are obtained from changing the values of the three parameters. We give a characterization of the neighborhood, which shows that the neighborhood consists of gross erros (contaminations) of all distributions in a certain neighborhood of Fo (a gap from Fo). One of the three parameters expresses the size of contamination and the others determine the size of the gap from Fo. The introduced neighborhood is natural and intuitively understandable and that it has nice properties for developing minimax theory in robust inference. Under this neighborhood we propose a robustification of sign test and its associated confidence interval for the median of Fo. These arguments are on the basis of a new parameter determined by the three parameters, which represents the total size of the departure of the data distribution G from Fo. We also investigate the robustness and efficiency of the proposed procedures under the real contamination and gap sizes which we discriminate from the design contamination and gap sizes used for constructing the procedures. Our results refine those of Yohai and Zamar (2004) and Ando, Kakiuchi and Kimura (2009), and include all their results as special cases.
References:
Ando, M., Kakiuchi, I. and Kimura, M. (2009). Robust nonparametric confidence intervals and tests for the median in the presence of (c, γ)-contamination, J. Statist. Plann. Inference, 139, 1836-1846.
Yohai, V. and Zamar, R. H. (2004). Robust nonparametric inference for the median, Ann. Statist., 32, 1841-1857.
Keywords: robust and nonparametric inference; confidence interval for the median; sign test; maximum asymptotic length
Biography: Kimura has been a professor of statistics at Nanzan University in Japan since 1991. He graduated from Nagoya University with a BSc in Mathematics, and obtained his MSc and DSc in Mathematical Statistics from Kyushu University. His research interests are robust statistics and multivariate analysis.