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Edgeworth Expansion for Near-Normal Data

​
standardized data
raw data
distribution
gamma
chi
α
3
β
3
ν
3
This Demonstration shows how you can use the Edgeworth expansion to describe standardized near-normal data. In order to do so, you can apply Mathematica's built-in function RandomVariate to data obtained from either (1) a gamma distribution, with shape parameter
α
and scale parameter
β
, or (2) a chi distribution with
ν
degrees of freedom. The built-in command Standardize shifts and rescales the elements of the generated list of data to have a zero mean and a unit sample variance. The Edgeworth expansion uses the third and fourth central moments (
μ
3
and
μ
4
) and is given by
f(x)=
-
2
x
2
e
2π
1+
μ
3
(
3
x
-3x)6+
2
μ
3
(-120+360
2
x
-120
4
x
+8
6
x
)576+(12-24
2
x
+4
4
x
)(
μ
4
-3)96
.
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