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