### Abstract

Consider the standard linear model y_{i}=z_{i}β+e_{i}, i=1, 2,..., n, where z_{i} denotes the ith row of an n x p design matrix, β∈ℝ^{p} is an unknown parameter to be estimated and e_{i} are independent random variables with a common distribution function F. The least absolute deviation (LAD) estimate {Mathematical expression} of β is defined as any solution of the minimization problem {Mathematical expression} In this paper Bahadur type representations are obtained for {Mathematical expression} under very mild conditions on F near zero and on z_{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 residuals e_{i} form a simple autoregressive process.

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

Pages (from-to) | 547-558 |

Number of pages | 12 |

Journal | Probability Theory and Related Fields |

Volume | 83 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 1989 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

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*Probability Theory and Related Fields*, vol. 83, no. 4, pp. 547-558. https://doi.org/10.1007/BF01845702

**Strong representations for LAD estimators in linear models.** / Babu, Gutti Jogesh.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong representations for LAD estimators in linear models

AU - Babu, Gutti Jogesh

PY - 1989/12/1

Y1 - 1989/12/1

N2 - Consider the standard linear model yi=ziβ+ei, i=1, 2,..., n, where zi denotes the ith row of an n x p design matrix, β∈ℝp is an unknown parameter to be estimated and ei are independent random variables with a common distribution function F. The least absolute deviation (LAD) estimate {Mathematical expression} of β is defined as any solution of the minimization problem {Mathematical expression} In this paper Bahadur type representations are obtained for {Mathematical expression} under very mild conditions on F near zero and on zi, i=1,..., n. These results are extended to the case, when {ei} is a mixing sequence. In particular the results are applicable when the residuals ei form a simple autoregressive process.

AB - Consider the standard linear model yi=ziβ+ei, i=1, 2,..., n, where zi denotes the ith row of an n x p design matrix, β∈ℝp is an unknown parameter to be estimated and ei are independent random variables with a common distribution function F. The least absolute deviation (LAD) estimate {Mathematical expression} of β is defined as any solution of the minimization problem {Mathematical expression} In this paper Bahadur type representations are obtained for {Mathematical expression} under very mild conditions on F near zero and on zi, i=1,..., n. These results are extended to the case, when {ei} is a mixing sequence. In particular the results are applicable when the residuals ei form a simple autoregressive process.

UR - http://www.scopus.com/inward/record.url?scp=0000318629&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000318629&partnerID=8YFLogxK

U2 - 10.1007/BF01845702

DO - 10.1007/BF01845702

M3 - Article

AN - SCOPUS:0000318629

VL - 83

SP - 547

EP - 558

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 4

ER -