### Abstract

Let (Equation presented) be two independent samples from a location-scale family, and suppose that the interest lies in robust M estimation of the location difference θ_{2} − θ_{1}. It will be assumed that the two scale parameters are equal but unknown. The usual robust M estimator for the location shift is the difference of the robust M estimators for the two location parameters. In this article we propose an alternative method that uses a single estimating equation to generate the robust estimator of the location shift. This estimating equation is derived from a reparametrization of the two-sample problem introduced by Neyman and Scott. When the same Ψ function is used, the new and the usual M estimators of the location shift have the same influence functions and asymptotic variance, and hence the two types of M estimators share the same class of optimal Ψ functions. However, all members of the new class of optimal robust M estimators for the location shift have sample-wise breakdown point .5 and thus are preferable over the usual M estimators in the asymmetric case. (In the symmetric case the usual M estimators also have sample-wise breakdown point .5.) The essential idea is to use the orthogonality of the Neyman-Scott parametrization, and then to use a maximum breakdown point (but possibly low-efficiency) estimator for the nuisance parameter. Some theoretical breakdown points and numerical illustrations are given.

Original language | English (US) |
---|---|

Pages (from-to) | 201-204 |

Number of pages | 4 |

Journal | Journal of the American Statistical Association |

Volume | 86 |

Issue number | 413 |

DOIs | |

State | Published - Jan 1 1991 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

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*Journal of the American Statistical Association*, vol. 86, no. 413, pp. 201-204. https://doi.org/10.1080/01621459.1991.10475020

**Robust M estimation in the two-sample problem.** / Akritas, Michael G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Robust M estimation in the two-sample problem

AU - Akritas, Michael G.

PY - 1991/1/1

Y1 - 1991/1/1

N2 - Let (Equation presented) be two independent samples from a location-scale family, and suppose that the interest lies in robust M estimation of the location difference θ2 − θ1. It will be assumed that the two scale parameters are equal but unknown. The usual robust M estimator for the location shift is the difference of the robust M estimators for the two location parameters. In this article we propose an alternative method that uses a single estimating equation to generate the robust estimator of the location shift. This estimating equation is derived from a reparametrization of the two-sample problem introduced by Neyman and Scott. When the same Ψ function is used, the new and the usual M estimators of the location shift have the same influence functions and asymptotic variance, and hence the two types of M estimators share the same class of optimal Ψ functions. However, all members of the new class of optimal robust M estimators for the location shift have sample-wise breakdown point .5 and thus are preferable over the usual M estimators in the asymmetric case. (In the symmetric case the usual M estimators also have sample-wise breakdown point .5.) The essential idea is to use the orthogonality of the Neyman-Scott parametrization, and then to use a maximum breakdown point (but possibly low-efficiency) estimator for the nuisance parameter. Some theoretical breakdown points and numerical illustrations are given.

AB - Let (Equation presented) be two independent samples from a location-scale family, and suppose that the interest lies in robust M estimation of the location difference θ2 − θ1. It will be assumed that the two scale parameters are equal but unknown. The usual robust M estimator for the location shift is the difference of the robust M estimators for the two location parameters. In this article we propose an alternative method that uses a single estimating equation to generate the robust estimator of the location shift. This estimating equation is derived from a reparametrization of the two-sample problem introduced by Neyman and Scott. When the same Ψ function is used, the new and the usual M estimators of the location shift have the same influence functions and asymptotic variance, and hence the two types of M estimators share the same class of optimal Ψ functions. However, all members of the new class of optimal robust M estimators for the location shift have sample-wise breakdown point .5 and thus are preferable over the usual M estimators in the asymmetric case. (In the symmetric case the usual M estimators also have sample-wise breakdown point .5.) The essential idea is to use the orthogonality of the Neyman-Scott parametrization, and then to use a maximum breakdown point (but possibly low-efficiency) estimator for the nuisance parameter. Some theoretical breakdown points and numerical illustrations are given.

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U2 - 10.1080/01621459.1991.10475020

DO - 10.1080/01621459.1991.10475020

M3 - Article

AN - SCOPUS:0039574786

VL - 86

SP - 201

EP - 204

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 413

ER -