Higher order asymptotic deals with two sorts of closely related things. First, there are questions of approximation. One is concerned with expansions or inequalities for a distribution function. Second, there are inferential issues. These involve, among other things, the application of the ideas of the study of higher order efficiency, admissibility and minimaxity. In the matter of expansions, it is as important to have usable, explicit formulas as a rigorous proof that the expansions are valid in the sense of truly approximating a target quantity up to the claimed degree of accuracy.
Classical asymptotics is based on the notion of asymptotic distribution, often derived from the central limit theorem (see Central Limit Theorems), and usually the approximations are correct up to O(n − 1 ∕ 2), where nis the sample size. Higher order asymptotics provides refinements based on asymptotic expansions of the distribution or density function of an estimator of a parameter. They are rooted in...
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Abril, J.C. (2011). Asymptotic, Higher Order. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_124
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