Breakdown in Nonlinear Regression
作者:
ArnoldJ. Stromberg,
David Ruppert,
期刊:
Journal of the American Statistical Association
(Taylor Available online 1992)
卷期:
Volume 87,
issue 420
页码: 991-997
ISSN:0162-1459
年代: 1992
DOI:10.1080/01621459.1992.10476254
出版商: Taylor & Francis Group
关键词: Breakdown function;Invariant to reparameterization;Least median of squares estimator;Least squares estimator;Least trimmed sum of squares estimator
数据来源: Taylor
摘要:
The breakdown point is considered an important measure of the robustness of a linear regression estimator. This article addresses the concept of breakdown in nonlinear regression. Because it is not invariant to nonlinear reparameterization, the usual definition of the breakdown point in linear regression is inadequate for nonlinear regression. The original definition of breakdown due to Hampel is more suitable for nonlinear problems but may indicate breakdown when the fitted values change very little. We introduce breakdown functions, which measure breakdown of the fitted values. Using the breakdown functions, we introduce a new definition of the breakdown point. For the linear regression model, our definition of the breakdown point coincides with the usual definition for linear regression as well as with Hampel's definition. For most nonlinear regression functions, we show that the breakdown point of the least squares estimator is 1/n. We prove that for a large class of unbounded regression functions, the breakdown point of the least median of squares or the least trimmed sum of squares estimator is close to ½. For monotonic regression functions of the typeg(α + βx), wheregis bounded above and/or below, we establish upper and lower bounds for the breakdown points that depend on the data.
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