Improved confidence regions based on Edgeworth expansions

Let ŵ be a consistent estimate of w in RpRp satisfying the standard cumulant expansion in powers of n−1n−1 with asymptotic covariance n−1V. Then n1/2(ŵ−w) has the standard Edgeworth expansion about Np(0,V). We obtain from this the Edgeworth expansions for Tn(V)=n(ŵ−w)′V−1(ŵ−w) about χp2 and for its Studentized version, Tn(V̂). So, we obtain a confidence region for w of level α+O(n−2)α+O(n−2).