Third and fourth central moment of a random variable

[Ad VanderVen]

TL;DR

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.








Inhibition in Speed and Concentration Tests:
The Poisson Inhibition Model

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 T_n (formula 23) and for the variance of T_n (formula 24). Now I would like to know what the formulas look like for T_n 's third and fourth central moment.

 

[mathman]

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 $E((X-\mu)^n)$.

 

[Stephen Tashi]

Now I would like to know what the formulas look like for T_n '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.

 

[Ad VanderVen]

Stephen Tashi The probability distribution is not given in the paper.

 

[Stephen Tashi]

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?

 

[Ad VanderVen]

Stephen Tashi Yes indeed.

 

[mjc123]

T_n (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.

 

[Ad VanderVen]

mjc123 Thanks a lot for your comment. I am going to look at equation (eq.15).

 

[mjc123]

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 Y_nA - 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.

 

[Ad VanderVen]

mjc123 Sorry for my late reply. It could well be the case that Y_{n A} - Y_{(n-1) A} is symmetric. But the question is abour the central moments of T_n.

 

[mjc123]

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.

 

[Ad VanderVen]

Thank you very much for this important information.

 

[Ad VanderVen]

Dear mjc123,

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