Improved confidence regions based on Edgeworth expansions

Let ŵ be a consistent estimate of w in ℝ p satisfying the standard cumulant expansion in powers of n -1 with asymptotic covariance n -1V. Then n 1/2(ŵ-w) has the standard Edgeworth expansion about N p(0,V). We obtain from this the Edgeworth expansions for T n (V)=n(ŵ-w)'V -1(ŵ-w) about χ p 2 and for its Studentized version, T n (V̂). So, we obtain a confidence region for w of level α+O(n -2). © 2012 Elsevier B.V. All rights reserved.