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Linear representation of M‐estimates in linear models
Author(s) -
Rao C. Radhakrishna,
Zhao L. C.
Publication year - 1992
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315607
Subject(s) - mathematics , linear regression , combinatorics , order (exchange) , representation (politics) , least absolute deviations , function (biology) , linear model , regular polygon , statistics , regression , geometry , finance , evolutionary biology , politics , political science , law , economics , biology
Consider the linear regression model, y i = x i β 0 + e i , i = l,…,n, and an M‐estimate β of β o obtained by minimizing Σρ(y i — x i β), where ρ is a convex function. Let S n = ΣX i X i X i and r n = S n ½ (β — β 0 ) — S n 2 Σx i h(e i ), where, with a suitable choice of h(.), the expression Σ x i x(e,) provides a linear representation of β. Bahadur (1966) obtained the order of r n as n→ ∞ when β o is a one‐dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of r n as n → ∞ for a general M‐estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.