Chapter 2 considered the normal approximation to distribution functions. We saw that the normal approximation could be derived by inverting the characteristic function, and that many of its properties could be described in terms of the first few derivatives of the characteristic function at 0. This chapter considers higher-order approximations to cumulative distribution functions and densities. Not surprisingly, these can also be expressed as approximate inversions of the characteristic function. Two heuristic motivations for the Edgeworth series are presented. Their correctness for densities and distribution functions is demonstrated using the methods of the previous chapter. Regularity conditions are investigated and discussed. Examples are given. The standards for assessment of accuracy of these methods are discussed and criticized. The Edgeworth series is inverted to yield the Cornish-Fisher expansion. Extensions of results from. the standard application of means of independent and identically distributed continuous random variables to non-identically distributed and lattice cases are presented. Parallels in the lattice case with classical Sheppard’s corrections are developed.
KeywordsCharacteristic Function Cumulative Distribution Function Lattice Distribution Hermite Polynomial Tail Probability
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