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Strong representations for LAD estimators in linear models
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  • Published: December 1989

Strong representations for LAD estimators in linear models

  • Gutti Jogesh Babu1 

Probability Theory and Related Fields volume 83, pages 547–558 (1989)Cite this article

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Summary

Consider the standard linear modely i =z i β+e i ,i=1, 2,...,n, where zi denotes theith row of ann x p design matrix,β∈ℝp is an unknown parameter to be estimated ande i are independent random variables with a common distribution functionF. The least absolute deviation (LAD) estimate\(\tilde \beta \) of β is defined as any solution of the minimization problem

$$\sum\limits_{i = 1}^n { \left| {y_i - z_i \tilde \beta } \right| = \inf \left\{ {\sum\limits_{i = 1}^n { \left| {y_i - z_i \beta } \right|:\beta \in \mathbb{R}^p } } \right\}} .$$

In this paper Bahadur type representations are obtained for\(\tilde \beta \) under very mild conditions onF near zero and onz i,i=1,...,n. These results are extended to the case, when {e i} is a mixing sequence. In particular the results are applicable when the residualse i form a simple autoregressive process.

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Authors and Affiliations

  1. Department of Statistics, The Pennsylvania State University, 219 Pond Laboratory, 16802, University Park, PA, USA

    Gutti Jogesh Babu

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  1. Gutti Jogesh Babu
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Cite this article

Babu, G.J. Strong representations for LAD estimators in linear models. Probab. Th. Rel. Fields 83, 547–558 (1989). https://doi.org/10.1007/BF01845702

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  • Received: 10 November 1988

  • Revised: 07 July 1989

  • Issue Date: December 1989

  • DOI: https://doi.org/10.1007/BF01845702

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Keywords

  • Linear Model
  • Absolute Deviation
  • Minimization Problem
  • Unknown Parameter
  • Mathematical Biology
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