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I want to make joint PDF in the case when I have the mean, the variance, the skewness and kurtosis of my solution. These four moments are spatial and time variables. mean(x,t), skewness(x,t)...

Thanks

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    $\begingroup$ can you maybe clarify a bit... do you know anything else about the process? joint pdf of what? $\endgroup$
    – don-joe
    Aug 14, 2017 at 19:06
  • $\begingroup$ @don-joe Thank you for your time firstly. So the result of my process (high dimensional calculation) is that I have are four moments of my out variable that are functions in time and x-coordinate i.e. mean, variance, skewness, kurtosis. I can plot them to see how they (their value) change in time and x-coordinate. They are four matrix t by x. Now I would like to make joint PDF that will be function of my out variable, time and x-coordinate. After that, I would jump on to find probability of exceedance. $\endgroup$
    – dino23
    Aug 14, 2017 at 19:37
  • $\begingroup$ so you do not have a explicit function of e.g. the mean that read "f(x) = ... " but you have datapoints? you need to model a pdf then, rather than calculate one. when modeling you need to consider more than just the moments but infer some interpretation about your system. and modeling is never exact $\endgroup$
    – don-joe
    Aug 15, 2017 at 8:49
  • $\begingroup$ @don-joe Yes, I have 4 matrix that are t by x. I was thinking to make in matlabe cure fitting and than I would have mean(x,t)=...... and rest of moments. So inputs of my model are gaussian standard normal distribution. Because of non-linearity, I dont expect that it will be Gaussian 100%. I have read about gauss transformation in which you combine your four moments with standard normal in hermite polynomials. But in there case moments are real number not functions. $\endgroup$
    – dino23
    Aug 15, 2017 at 9:52
  • $\begingroup$ well a normal distribution is defined only by mean and variance, so the first two moments suffice for that. $\endgroup$
    – don-joe
    Aug 15, 2017 at 10:03

2 Answers 2

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If you have every moment, then you can write down the moment generating function $M_X$. With that you can write down the characteristic function $\phi_x(k) = M_X(ik)$

The characteristic function is the Fourier transform of the probability density function (PDF)

If you only know the first n moments then at least you can write down the PDF with some residual error.

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  • $\begingroup$ I have only four moments in time and space domain. I will try your suggestion. $\endgroup$
    – dino23
    Aug 14, 2017 at 18:37
  • $\begingroup$ @dino23 Did this work? I'm also trying to do something similar. The issue I see is as follows: a truncated moment series is a polynomial, and the inverse Fourier transform of a polymomial is not actually a function; it is a linear combination of derivatives of the dirac delta distribution (see, e.g., here: math.stackexchange.com/a/1343234/3060). Perhaps this could work if a low pass filter is applied to the truncated MGF before performing the inverse Fourier transform $\endgroup$
    – Nick Alger
    Oct 26, 2022 at 2:06
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This question should be posted in https://stats.stackexchange.com, and there is an answer already.

Warning: do not rely on skewness and kurtosis too much if they are calculated from samples, as high order statistics are prone to outliers in sample data. Carefully check whether the data is significantly different from normal distribution. Or you may use more robust metrics like L-moment.

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