How to calculate the Moments of a new PDF

[consuli]

Hello!

I have developed the PDF (probability density function) of a new distribution.

I want to calculate their moments (mean, sd, skewness, kurtosis).

How can I do this?

Consuli

[Prometheus]

Do you know about moment generating functions?

[consuli]

No, I dont. I am just an applied statistician, not a methematical statistician. Thus, the easierst (and maybe mathematical inelegant) solution is asked for. ;-)

The pdf has 4 parameters: construction offset, construction sd, kurtosis and skewness parameter.

Consuli

Edited April 16, 2018 by consuli

[Prometheus]

By develop the pdf do you mean you have developed a closed form for it? If so, calculating the moment generating function would be the best (most precise) way. It's not too hard to calculate assuming the pdf isn't some beast of a function - it's an integral of the pdf exponentiated. Post it here, i'm sure a few people would have a go.

If you have an empirical distribution rather than the closed form there are estimators for skew/kurtosis - something like R should have a function, or at least a package, to do this.

If you have a closed form and really don't fancy the integral you could build an empirical distribution by taking samples from your distribution and then use an estimator for skew/kurtosis. Should converge to the true values for a sufficient number of samples, assuming the estimators aren't biased.

 

[consuli]

I have heard, one could more easily calculate the 4 moments of a new distribution by solving the following integrals:

E(x) = Integral f(x) * x dx

Var(x) = Integral f(x) * (x-E(x))² dx

s= Var(x)^0.5

M3= Integral f(x) * ((x-E(x))/s)³ dx

M4= Integral f(x) * ((x-E(x))/s)⁴ dx

Where fx) refers to the pdf of the distribution.

Is that true?

Consuli

 

Edited April 17, 2018 by consuli

[Prometheus]

Yes. This is essentially the same as working out the first four terms in a moment generating function. -3 from the last one if you want the excess kurtosis rather than vanilla kurtosis for some reason. This page might be helpful.

[consuli]

Brilliant.

Is there some kind of official source, where I can cite these integral formulas from?

Consuli

Edited April 17, 2018 by consuli

[Prometheus]

You can find moment generating functions in just about any book on fundamental statistics, something like A First Course in Probability by Sheldon Ross. I'm not sure of a specific book with the specific equations you want. There are plenty of books on google you can check - they'de be under moment generating functions or MGFs section (or maybe under a thorough chapter on expectation).

[consuli]

Because the direct google search for one or more of the formalas is not aim leading, as google - due to different typo of mathematical formulas - is currently not really able to handle them correctly ?

Consuli

Edited April 18, 2018 by consuli

[Prometheus]

Try this link. It's an extract from a book (seems OK); you should be able to find the details of it for a full reference. I googled 'equation for fourth standardised moment'. 

[studiot]

One thing you haven't told us.

 

Is your pdf given by a formula or is it a tabulation of values?

 

You should also look up sheppard's correction

 

https://www.google.com/search?q=sheppards+correction&ie=utf-8&oe=utf-8&client=firefox-b