The Edgeworth Expansion

The EE in general is an expansion about the Gaussian distribution and therefore an approximation scheme specially adapted for the computation of nearly Gaussian distributed density functions. We show that the application of the EE (lEE) technique leads to excellent results for the approximation of a lognormal- and Xio-p (see section (3.3), (3.2), (4.2) and (4.3)). 

we follow that the EE and lEE approach performs accurately even for the approximation of far-from-normal -like pdf s or cdf s, respectively. 

The stochastic dynamics usually applied in finance literature is generated by lognormal- or close-to-lognormal -distributed random variables. Leip-nik (1991) shows that the series expansion of order M of a Gog) characteristic function in terms of the cumulants diverges for M oo. Hence, the 

We show that the application of the EE is admissible leading to accurate results, even in the case of lognormal-distributed random variables. This good-natured behavior of the EE, firstly comes from the fact that the volatility t5 ically occurring in bond markets is rather low, generating more close-to-normal -distributed random variables. Secondly, the series expansion of the (log) characteristic function in terms of the cumulants can be practically applied for M lower than a critical order Me. 

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More complex mathematical treatment is necessary when the thermal motion is very large, as for hydrogen atoms in a room-temperature neutron structure analysis, or when it is curvilinear as in a hindered-rotor. The Uy second-rank tensor does not adequately describe the nuclear or electron-scattering density when the motion is far from harmonic or when it deviates from the familiar ellipsoidal probability form. Tb deal with such examples, more complex mathematic expressions using Gram Charlier or Edgeworth expansions are available [210]. 

Therefore, we term this generalized series expansion the Integrated Edgeworth Expansion (lEE). The EE originally is derived to approximate density functions instead of probabilities. 

Approximating the distribution of a random variable using its moments of finite order is a well known problem in statistics, and various approximations such as the Pearson, Johnson and Burr systems, the Edgeworth and Cornish-Fisher expansions were developed (Stuart and Ord 1987). Their applications in structural reliability have been examined by Winterstein (1988), Grigoriu (1983), Hong (1996). In the present study, the fourth moment standardization function will be used. For the standardized performance function,... 

A second statistical expansion that may be used to describe the atomic probability distribution is due to Edgeworth (Kendal and Stuart 1958, Johnson 1969). It expresses a distribution in terms of its cumulants k. If D is the differential operator, P(u) is described as... 

The similar Edgeworth-Cramer asymptotic expansion may offer abetter approximation [4] usually it is truncated like... 

Shapes of molecular electronic bands are studied using the methods of the statistical theory of spectra. It is demonstrated that while the Gram-Charlier and Edgeworth type expansions give a correct description of the molecular bands in the case of harmonic-oscillator-like potentials, they are inappropriate if departure from harmonicity is considerable. The cases considered include a set of analytically-solvable model potentials and the numerically exact potential of the hydrogen molecule. 

Hence, by assuming nearly Gaussian distributed random variables we are able to compute the exercise probabilities Tip [.ST] directly by performing tbe lEE approach instead of miming a generalized Edgeworth series expansion. Overall, the lEE approach (4.2) can be seen as an equivalent to tbe generalized EE tecbnique, especially adapted to compute tbe cdf s used in finance theory. 

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