Asymptotic expansions of functions of statistics

Abstract

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Good Practices in Free-Energy Calculations

“…Most error analyses of Δ A ,,,,, are based on this approach, even if this is not explicitly stated. The main theorem underlying the delta method states in a slightly simplified form that for a function f ( X N ) of a sample X N consisting of N independent values of a random variable, the expectation value of f ( X N ) can be written as E true[ f false( X N false) true] = f false( X̅ false) + ∑ j = 1 n f j false( X̅ false) j ! E ( X N − X̅ ) j + O true( N − false( n + 1 false) / 2 true) providing that f is bounded, and the first n + 1 derivatives exist and are also bounded. Here, X̅ is the true value of the average X and f j ( X̅ ) denotes the j th derivative of f at X̅ .…”

Beyond the delta method

“…It should also hold true in an infinite dimensional setting, which should be useful to deal with semiparametric models although there are some fine technical points to deal with (see the discussion in [23,24,42]). In the multidimensional setting, a Taylor expansion is used in [26] to compute the first two moments of functions of estimators. This might be combined easily with our approach to obtain higher moments.…”