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Tail behavior of the least-squares estimator

✍ Scribed by Jana Jurečková; Roger Koenker; Stephen Portnoy

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
113 KB
Volume
55
Category
Article
ISSN
0167-7152

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✦ Synopsis

The tail behavior of the least-squares estimator in the linear regression model was studied in He et al. (Econometrica 58 (1990) 1195) under a ÿxed design for ÿnite n. We now consider a random design matrix Xn and the case n → ∞ and study the probability P 0 (max16i6n |x i Rn | ¿ n) with n = F -1 (1 -1=n); a population analog of the maximal error. Unlike in the situation with ÿxed n and → ∞; for n → ∞ we ÿnd fairly good tail behavior of LSE for normal F; for both ÿxed and random designs, even under heavy-tailed distribution for Xn.

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For a p-dimensional normal distribution with mean vector % and covariance matrix I p , it is known that the maximum likelihood estimator % of % with p 3 is inadmissible under the squared loss. The present paper considers possible extensions of the result to the case where the loss is a member of a g