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?
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.
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.
