A Third and fourth central moment of a random variable

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The discussion centers on determining the third and fourth central moments of the random variable Tn, as outlined in a referenced paper. Participants note that the probability distribution of Tn is not provided, complicating the derivation of these moments. One contributor suggests that while Tn is derived from correlated variables with gamma distributions, the symmetry of the distribution may imply that the odd central moments are zero. However, another participant raises concerns about the skewness of Tn, indicating that simulations show a positive skewness, contradicting the symmetry argument. The conversation highlights the complexities involved in calculating central moments for correlated variables.
Ad VanderVen
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In a paper published in the JOURNAL OF MATHEMATICAL PSYCHOLOGY 39, 265-274 (1995) a formula is given on page 272 for the expectation of a random variable (formula 23) and for it's variance (formula 24). Now I would like to know what the formulas look like for it's third and fourth central moment.








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JAN C. dang AND AD H. G. S. VAN DER VEN
My question is as follows. In the attached paper a formula is given on page 272 for the expectation of Tn (formula 23) and for the variance of Tn (formula 24). Now I would like to know what the formulas look like for Tn 's third and fourth central moment.
 

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General answer for any random variable ##X##. Let ##E(X)=\mu## be the mean of ##X##. Then the ##n^{th}## central moment is given by ##E((X-\mu)^n)##.
 
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Ad VanderVen said:
Now I would like to know what the formulas look like for Tn 's third and fourth central moment.
My guess is that you are asking for the third and fourth central moments of that particular ##T_n## rather than asking about the general concept of third and fourth moments. If that's the case, you are more likely to get an answer if you state the distribution of ##T_n## rather than hope that someone will read enough of the paper to figure that out.
 
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Stephen Tashi The probability distribution is not given in the paper.
 
Ad VanderVen said:
Stephen Tashi The probability distribution is not given in the paper.

Is your question about how to infer the third and fourth moments of ##T_n## from what is given in the paper?
 
Tn (eq. 22) is a constant plus the difference between two variables with the same gamma distribution (eq. 15). If those two variables were independent it would be fairly simple, but they are correlated, and I'm not sure how to do that.
 
mjc123 Thanks a lot for your comment. I am going to look at equation (eq.15).
 
Based on a qualitative symmetry argument, I would say that in the stationary regime, when Y is fluctuating about a constant mean, then although the Y's are correlated, the distribution of YnA - Y(n-1)A must be symmetrical about zero, so the odd central moments must all be zero. But I'm not sure I could prove it rigorously.
 
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mjc123 Sorry for my late reply. It could well be the case that Yn A - Y(n-1) A is symmetric. But the question is abour the central moments of Tn.
 
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If y = A + Bx, where A and B are constants, then if x is symmetrical about zero, y is symmetrical about A.
Now look at equation 22.
 
  • #12
Thank you very much for this important information.
 
  • #13
Dear mjc123,

You suggest that the variable ##T_n## has a skewness that equals zero, but simulations suggest that the skewness is positive.
 
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