Bai, Z. D. ; Chen, X. R. ; Miao, B. Q. ; Radhakrishna Rao, C.
(1990)
*Asymptotic theory of least distances estimate in multivariate linear models*
Statistics, 21
(4).
pp. 503-519.
ISSN 0233-1888

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Official URL: http://www.tandfonline.com/doi/abs/10.1080/0233188...

Related URL: http://dx.doi.org/10.1080/02331889008802260

## Abstract

We consider the multivariate linear model Y_{i}=X'_{i}β_{0} + εi, i = 1,...,n where Y_{i} is a p-vector random variable, X_{i} is a q × p matrix, βÀ_{0} is an unknown q-vector parameter and {s_{i}} is a sequence of iid p-vector random variable with median vector zero. The estimate βÀ_{n} of βÀ_{0} such that min βΣ^{n}_{i=1}||Y_{i}-X'_{i}β||=Σ^{n}_{i=1}||Y_{i}-X'_{i}β|| is called the least distances (LD) estimator. It may be recalled that the least squares (LS) estimator is obtained by minimizing the sum of norm squares. In this paper, it is shown that the LD estimator is unique, consistent and has an asymptotic q-variate normal distribution with mean β_{0} and co variance matrix V which depends on the distribution of the error vectors {ε_{i}}. A consistent estimator of V is proposed which together with βÀ_{n} enables an asymptotic inference on β_{0}. In particular, tests of linear hypotheses on β_{0} analogous to those of analysis of variance in the GATJSS-MARKOFF linear model are developed. Explicit expressions are obtained in some cases for the asymptotic relative efficiency of the LD compared to the LS estimator.

Item Type: | Article |
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Source: | Copyright of this article belongs to Taylor and Francis Group. |

Keywords: | GAUSS-MARKOFF Model; Least Distance Estimator; Least Squares Estimator; Multivariate Linear Model; Multivariate Median; Spatial Median |

ID Code: | 71868 |

Deposited On: | 28 Nov 2011 04:15 |

Last Modified: | 28 Nov 2011 04:15 |

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