Breakdown in nonlinear regression

Arnold J. Stromberg; David Ruppert

doi:10.1080/01621459.1992.10476254

Breakdown in nonlinear regression

Arnold J. Stromberg, David Ruppert

Dr. Bing Zhang Department of Statistics

Research output: Contribution to journal › Article › peer-review

59 Scopus citations

Abstract

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 (Equation presented). For monotonic regression functions of the type g(α + βx), where g is bounded above and/or below, we establish upper and lower bounds for the breakdown points that depend on the data.

Original language	English
Pages (from-to)	991-997
Number of pages	7
Journal	Journal of the American Statistical Association
Volume	87
Issue number	420
DOIs	https://doi.org/10.1080/01621459.1992.10476254
State	Published - Dec 1992

Bibliographical note

Funding Information:
* Arnold J. Stromberg is Assistant Professor, Department of Statistics, University of Kentucky, Lexington, KY 40506-0027. David Ruppert is Professor, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853. This work was started while Stromberg was at the Department of Applied Statistics at the University of Minnesota. It presentsand extends material from Stromberg's Ph.D. dissertation. Stromberg was supported by NSF Grant DMS-8701201. Ruppert was supported by NSF Grants DMS-870120l and DMS-8800294 and by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell. The authors thank Ray Carroll for his helpful discussions. The referees provided useful comments, and the examples of one referee provided convincing evidence that our earlier notion of breakdown was not quite satisfactory.

Keywords

Breakdown function
Invariant to reparameterization
Least median of squares estimator
Least squares estimator
Least trimmed sum of squares estimator

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1080/01621459.1992.10476254

Cite this

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abstract = "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{\textquoteright}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 (Equation presented). For monotonic regression functions of the type g(α + βx), where g is bounded above and/or below, we establish upper and lower bounds for the breakdown points that depend on the data.",

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N2 - 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 (Equation presented). For monotonic regression functions of the type g(α + βx), where g is bounded above and/or below, we establish upper and lower bounds for the breakdown points that depend on the data.

AB - 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 (Equation presented). For monotonic regression functions of the type g(α + βx), where g is bounded above and/or below, we establish upper and lower bounds for the breakdown points that depend on the data.

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Breakdown in nonlinear regression

Abstract

Bibliographical note

Keywords

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