Frontiers of Mathematics in China >

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DOI: https://doi.org/10.1007/s11464-013-0286-x

RESEARCH ARTICLE

General relative error criterion and M-estimation

  • Ying YANG ,
  • Fei YE
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  • Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received date: 13 Aug 2012

Accepted date: 09 Jan 2013

Published date: 01 Jun 2013

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
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Abstract

Relative error rather than the error itself is of the main interest in many practical applications. Criteria based on minimizing the sum of absolute relative errors (MRE) and the sum of squared relative errors (RLS) were proposed in the different areas. Motivated by K. Chen et al.’s recent work [J. Amer. Statist. Assoc., 2010, 105: 1104-1112] on the least absolute relative error (LARE) estimation for the accelerated failure time (AFT) model, in this paper, we establish the connection between relative error estimators and the M-estimation in the linear model. This connection allows us to deduce the asymptotic properties of many relative error estimators (e.g., LARE) by the well-developed M-estimation theories. On the other hand, the asymptotic properties of some important estimators (e.g., MRE and RLS) cannot be established directly. In this paper, we propose a general relative error criterion (GREC) for estimating the unknown parameter in the AFT model. Then we develop the approaches to deal with the asymptotic normalities forM-estimators with differentiable loss functions on ℝ or ℝ\{0} in the linear model. The simulation studies are conducted to evaluate the performance of the proposed estimates for the different scenarios. Illustration with a real data example is also provided.

Key words： Relative error; accelerated failure time model; M-estimation; asymptotic normality; general loss function

Cite this article

Ying YANG , Fei YE . General relative error criterion and M-estimation[J]. Frontiers of Mathematics in China, 0 , 8(3) : 695 -715 . DOI: 10.1007/s11464-013-0286-x

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