Approximating Moments by Nonlinear Transformations
We provide a methodology to calculate the expectation of a variate x in terms of the moments of a transformation of x. Apart from the intrinsic interest in such a fundamental relation that relates the moments of a variate and its nonlinear transformations, our results can be used in practice to approximate E(x) by the low-order moments of a transformation which can be chosen to give a good approximation for E(x). To obtain an accurate evaluation of the remainder, we derive results for the bounding of functions of complex variables. Our results are useful, for example, in resampling applications like bootstrap confidence intervals for fat-tailed data. They are also useful in economics and finance in quantifying the effect of taking nonlinear transformations on moment conditions and on asset prices which are formulated as expectations.
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