Some notations and preliminaries first.
Let us suppose we denote a standard gaussian density as [$]p(X) \, = \, \frac{1}{2 \pi} \exp(- \, \frac{ X^2 }{2})[$]
We also have hermite polynomials and first few hermite polynomials are denoted as
[$]H_0(X) \, = \, 1[$]
[$]H_1(X) \, = \, X[$]
[$]H_2(X) \, = \, X^2 \, - 1[$]
[$]H_3(X) \, = \, X^3 \, - 3 X[$]
[$]H_4(X) \, = \, X^4 \, - 6 X^2 \, +3[$]
and so on.
And I want to share an interesting property related to following integral
[$]\int_{-\infty}^{+\infty} X^m \, H_n(X) \, p(X) dx [$] Equation(A)
We know when m is odd and n is even or m is even and n is odd, the above integral always goes to zero.
But when m is odd and n is also odd, the above integral would always go to zero when n > m.
Similarly when m is even and n is also even, the above integral would again always go to zero, when n > m.
To explain it in words with simple examples, when fourth power of X is integrated in the above integral (eq A) with zeroth hermite, 2nd hermite or fourth hermite, there will be a finite value but when fourth power of X is integrated with sixth hermite, eighth hermite or tenth hermite, it will always go to zero. Similarly for odd powers, when fifth power of X is integrated there is a finite value when we integrate it with first hermite, third hermite or fifth hermite in the equation (A), but when we integrate fifth power of X with seventh, ninth or 11th hermite or higher odd hermites, the value would always go to zero.
The above property is interesting since in our forthcoming framework when we have fixed some moment in our equation, adding higher hermites(for higher moments) would not change the lower moments at all.
Now we want to fix first few moments to our proposed probability distribution so that all the moments are perfectly matched (a good property as I mentioned earlier would be that adding further higher moments would not change our lower moments that have already been matched).
The proposed probability distribution has a very simple form (upto sixth order) as
[$] q(X) \, = \, p(X) \, \big[ \, 1 \, + \, h_1 \, H_1(X)+ \, h_2/2 \, H_2(X)+ \, h_3/6 \, H_3(X)+ \, h_4/24 \, H_4(X)+ \, h_5/120 \, H_5(X)+ \, h_6/720 \, H_6(X) \, \big] \, [$]
here [$]h_1[$] , [$]h_2[$] etc are coefficients of hermites that will be adjusted to match the moments of above probability density form to our desired moments.
1st thing we know by inspection that above probability density equation integrates to one since all hermites (other than the first term) have an expected value of zero when integrated over normal density.
Now we write the first few raw moments of the above equation that can be verified by full-fledged integration(as I have done in my notes)
[$]E[X]\,= \int_{-\infty}^{+\infty} X \, q(X) dX \, = \, h_1[$]
[$]E[X^2]\,= \int_{-\infty}^{+\infty} X^2 \, q(X) dX \, = \,1 \, + \, h_2[$]
[$]E[X^3]\,= \int_{-\infty}^{+\infty} X^3 \, q(X) dX \, = \,3 \, h_1 \, + \, h_3[$]
[$]E[X^4]\,= \int_{-\infty}^{+\infty} X^4 \, q(X) dX \, = \, 3 \, +\,6 \, h_2 \, + \, h_4[$]
[$]E[X^5]\,= \int_{-\infty}^{+\infty} X^5 \, q(X) dX \, = \, 15 \, +\,10 \, h_3 \, + \, h_5[$]
[$]E[X^6]\,= \int_{-\infty}^{+\infty} X^6 \, q(X) dX \, = \, 15 \, +\,45 \, h_2 \, + \,15 \, h_4 \, + h_6[$]
So we see that we can keep on adding more hermite polynomials to fix higher moments and it will not alter our lower moments. However higher odd/even moments take a contribution from hermite polynomial terms associated with lower odd/even moments(that have already been fixed) up to the order of moment being calculated.
It seems a better idea to try matching moments on standard gaussian as opposed to non-standard gaussian as we do when we match moments with gram-charlier or edgeworth.
Please pardon any errors.
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Some notations and preliminaries first.
Let us suppose we denote a standard gaussian density as [$]p(X) \, = \, \frac{1}{\sigma \sqrt{2 \pi}} \exp(- \, \frac{ {(X-\mu)}^2 }{2 {\sigma}^2})[$]
We also have standardized hermite polynomials and first few standardized hermite polynomials are denoted as
[$]H_0(X) \, = \, 1[$]
[$]H_1(X) \, = \, (\frac{X- \mu}{\sigma})[$]
[$]H_2(X) \, = \, {(\frac{X- \mu}{\sigma})}^2 \, - 1[$]
[$]H_3(X) \, = \, {(\frac{X- \mu}{\sigma})}^3 \, - 3 {(\frac{X- \mu}{\sigma})}[$]
[$]H_4(X) \, = \, {(\frac{X- \mu}{\sigma})}^4 \, - 6 {(\frac{X- \mu}{\sigma})}^2 \, +3[$]
Now we want to fix first few moments to our proposed probability distribution so that all the moments are perfectly matched (a good property as I mentioned earlier would be that adding further higher moments would not change our lower moments that have already been matched).
The proposed probability distribution has a very simple form (upto sixth order) as
[$] q(X) \, = \, p(X) \, \big[ \, 1 \, + \, h_1 \, H_1(X)+ \, h_2/2 \, H_2(X)+ \, h_3/6 \, H_3(X)+ \, h_4/24 \, H_4(X)+ \, h_5/120 \, H_5(X)+ \, h_6/720 \, H_6(X) \, \big] \, [$]
here [$]h_1[$] , [$]h_2[$] etc are coefficients of hermites that will be adjusted to match the moments of above probability density form to our desired moments.
1st thing we know by inspection that above probability density equation integrates to one since all hermites (other than the first term) have an expected value of zero when integrated over normal density.
Now we write the first few raw moments of the above equation that can be verified by full-fledged integration(as I have done in my notes)
[$]E[X]\,= \int_{-\infty}^{+\infty} X \, q(X) dX \, = \mu \, + \, h_1 \, \sigma[$]
[$]E[X^2]\,= \int_{-\infty}^{+\infty} X^2 \, q(X) dX \, = \, {\mu}^2 + 2 \, h_1 \mu \sigma + \, {\sigma}^2 \, + \, h_2 \, {\sigma}^2 \,[$]
[$]E[X^3]\,= \int_{-\infty}^{+\infty} X^3 \, q(X) dX \, = \, {\mu}^3 + 3 \, h_1 {\mu}^2 \sigma + 3 \mu {\sigma}^2 + 3 h_2 \, \mu {\sigma}^2 + \, 3 h_1 \, {\sigma}^3 \, \, + \, h_3 \, {\sigma}^3 \,[$]
[$]E[X^4]\,= \int_{-\infty}^{+\infty} X^4 \, q(X) dX \, = \, {\mu}^4 + 4 \, h_1 {\mu}^3 {\sigma} + 6 {\mu}^2 {\sigma}^2 + 6 h_2 \, {\mu}^2 {\sigma}^2 + \, 12 h_1 \, \mu \, {\sigma}^3 \, \, +4 \, h_3 \, \mu \, {\sigma}^3 \,+ 3 \, {\sigma}^4 \, +6 \, h_2 \, {\sigma}^4 \, + \, h_4[$]
[$]E[X^5]\,= \int_{-\infty}^{+\infty} X^5 \, q(X) dX \, = \, {\mu}^5 + 5 \, h_1 {\mu}^4 {\sigma} + 10 {\mu}^3 {\sigma}^2 + 10 h_2 \, {\mu}^3 {\sigma}^2 + \, 30 h_1 \, {\mu}^2 \, {\sigma}^3 \, \, +10 \, h_3 \, {\mu}^2 \, {\sigma}^3 \,[$]
[$]+ 15 \, \mu {\sigma}^4 \, +30 \, h_2 \, \mu {\sigma}^4 \, +5 \, h_4 \, \mu {\sigma}^4 + 15 h_1 {\sigma}^5 + 10 h_3 {\sigma}^5 \, + \, h_5 \, {\sigma}^5[$]
[$]E[X^6]\,= \int_{-\infty}^{+\infty} X^6 \, q(X) dX \, =
\, {\mu}^6 + 6 \, h_1 {\mu}^5 {\sigma} + 15 {\mu}^4 {\sigma}^2 + 15 h_2 \, {\mu}^4 {\sigma}^2 + \, 60 h_1 \, {\mu}^3 \, {\sigma}^3 \, \, +20 \, h_3 \, {\mu}^3 \, {\sigma}^3 \,[$]
[$]+ 45 \, {\mu}^2 {\sigma}^4 \, +90 \, h_2 \, {\mu}^2 {\sigma}^4 \, +15 \, h_4 \, {\mu}^2 {\sigma}^4 + 90 h_1 \, \mu {\sigma}^5 + 60 h_3 \, \mu {\sigma}^5 \, + 6 \, h_5 \, \mu {\sigma}^5 \, + 15 \, {\sigma}^6 \,\, + 45 \, h_2 {\sigma}^6 \, + 15 \, h_4 {\sigma}^6 \, + \, h_6 \, {\sigma}^6 \,[$]
So we see that we can keep on adding more hermite polynomials to fix higher moments and it will not alter our lower moments. However higher odd/even moments take a contribution from hermite polynomial terms associated with lower odd/even moments(that have already been fixed) up to the order of moment being calculated.
Though I posted some programs to calculate densities from central moments or cumulants, we can notice that changing mean/variance of base gaussian also have a great effect towards the shape of new density and fixing first six moments with different base gaussians can results in different densities though first few moments would still be matched. I think it has to be further studied by friends how to systematically choose the parameters of base gaussian towards the best matching density. Sometimes, density becomes slightly negative for parameters of base gaussian but becomes perfectly positive when parameters of base gaussian are slightly changed.
You think life is a secret, Life is only love of flying, It has seen many ups and downs, But it likes travel more than the destination. Allama Iqbal
