Precise asymptotics for the first moment of the error variance estimator in linear models

where an, bn > 0 and ∑ n an = ∞. Then there exist a threshold α, such that f( ) = ∞ for < α, while f( ) < ∞ for > α. The so-called precise asymptotics problem consists in finding, under appropriate moment conditions, an elementary function g( ) > 0, > α, such that lim ↓α g( ) = 0 and lim ↓α g( )f( ) = l = 0,∞, i.e., in establishing that f( ) ∼ l/g( ) as ↓ α. For almost exhaustive references on this area, see Gut and Spǎtaru(2000; a, b). Consider the linear model Yi = X ′ iβ+ei, i = 1, ..., n, where β is a q-dimensional unknown parametric vector, and {ei} is a sequence of i.i.d. trial errors with Ee1 = 0 and 0 < σ 2 = Ee1 < ∞. By ordinary least squares and the characteristic of linear models, the estimator of σ always takes the following form: