Asymptotic expansions of functions of statistics

Let ₙ\ be a sequence of statistics such that E|Tₙ-0|2(q+1)=O(n-(q+1)). Let $g=g(t,n)$ be a real function defined on R× N. In the paper it is shown that under some assumptions concerning g, the expectation Eg(Tₙ,n) (the variance var g(Tₙ,n)) may be expressed in terms of the derivatives of g and the moments E(Tₙ-0)ʲ, j=1, …, q(j=1,…, 2q), the remainder term being O(n-(q+1/2) (O(n-(q+2/2))). Similar results for vector T'ₙs are also obtained. Applications in reliability theory are given.