TY - JOUR

T1 - Tail behavior of the least-squares estimator

AU - Jurečková, Jana

AU - Koenker, Roger

AU - Portnoy, Stephen

N1 - Funding Information:
Research of J. Jurečková and R. Koenker was supported by the international US–Czech Grant NSF 96-17206/ME-381/2000; moreover, research of J. Jurečková was supported by the Czech Republic Grant GAČR 201/99/0264 and research of S. Portnoy supported by NSF Grant 9703758.

PY - 2001/12/15

Y1 - 2001/12/15

N2 - 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 fixed design for finite n. We now consider a random design matrix Xn and the case n→∞ and study the probability P0(max1≤i≤n|xi′β̂ n|≥γn) with γn=F-1(1-1/n), a population analog of the maximal error. Unlike in the situation with fixed n and γ→∞, for n→∞ we find fairly good tail behavior of LSE for normal F, for both fixed and random designs, even under heavy-tailed distribution for Xn.

AB - 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 fixed design for finite n. We now consider a random design matrix Xn and the case n→∞ and study the probability P0(max1≤i≤n|xi′β̂ n|≥γn) with γn=F-1(1-1/n), a population analog of the maximal error. Unlike in the situation with fixed n and γ→∞, for n→∞ we find fairly good tail behavior of LSE for normal F, for both fixed and random designs, even under heavy-tailed distribution for Xn.

KW - Domain of attraction

KW - Extreme values

KW - Least-squares estimator

KW - Tail behavior

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U2 - 10.1016/S0167-7152(01)00137-7

DO - 10.1016/S0167-7152(01)00137-7

M3 - Article

AN - SCOPUS:18044400026

VL - 55

SP - 377

EP - 384

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 4

ER -