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

.