1
$\begingroup$

What is the sign (+tive/-tive) of the third order central moment of a positive linear combination of log-normal random variables?

It seems to be a common notion that the skewness of random variables with longer tails to the right is positive. Is it correct? If so, how do you prove it?

$\endgroup$
1
  • 2
    $\begingroup$ I doubt that your second question is research-level mathematics, which is the scope of this website. However, this question seems well-suited to stats.stackexchange.com. If you ask there (not now: wait a day or two) mention that you asked it here first. $\endgroup$
    – David Roberts
    May 29, 2014 at 5:56

2 Answers 2

1
$\begingroup$

OP asks:

What is the sign (+tive/-tive) of the third order central moment of a positive linear combination of log-normal random variables`

Let $Z = a X + b Y$. Then, for any random variables $X$ and $Y$ that are independent, it can be shown that:

$$\mu_3(Z) = a^3 \mu_3(X) + b^3 \mu_3(Y)$$

where $\mu_3(W) = E[(W-\acute\mu_W)^3]$ denotes the third central moment of any random variable $W$ whose moments exist, and $\acute\mu_W = E[W]$.

If $X$ is a Lognormal random variable, then:

$$\mu_3(X) = \left(e^{\sigma ^2}-1\right)^2 \left(e^{\sigma ^2}+2\right) e^{3 \mu +\frac{3 \sigma ^2}{2}}$$

is strictly positive, and similarly for $Y$.

Thus, if $a$ and $b$ are positive, it follows that $\mu_3(Z) = E[(Z-\acute\mu_Z)^3]$ is positive.

$\endgroup$
1
  • $\begingroup$ Thanks wolfies. It is clear to me that a positive linear combination of independent log normal rv's would have a positive third central moment. But I am looking for a general case. It seems to be a folk-lore that the skewness of one sided distributions (say positive sided) is positive. I am looking for a formal argument for/against it. In particular, I am interested in a positive linear combination of log normal rv's. $\endgroup$ Jun 4, 2014 at 4:42
0
$\begingroup$

I am looking for a general case. It seems to be a folk-lore that the skewness of one sided distributions (say positive sided) is positive. I am looking for a formal argument for/against it.

This looks false if interpreted perhaps more literally than intended. What if you make a positive-sided distribution that is a mixture of a negatively skewed beta distribution and a positively skewed exponential distribution with the same mean? The skew of this positive-sided mixture with support [0, $\infty$) can be positive or negative depending on the mixture coefficients.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.