# median and mean of the sample mean of i.i.d. log-normal

Let $y:=\frac1n\sum_{i=1}^n x_i$, where $\{x_i\}_{i=1}^n$ is a set of i.i.d. random variables, and every $x_i$ has a lognormal distribution $x_i \sim\text{Lognormal}(\mu,\sigma^2)$. Let $\text{Med}[y]$ be the median of $y$. Is the following inequality true $\forall (n,\mu,\sigma)$? $$\text{Med}[y]<\mathbf E[y]$$

• One could show that the third moment skewness is positive easily enough, but that doesn't automatically imply that the mean will exceed the median (I certainly believe the mean will exceed the median here but we'd need to prove that it does). What's this for? – Glen_b -Reinstate Monica May 31 '17 at 0:38
• @Glen_b: I am computing the sample mean of the lognormal random variables via Monte Carlo. The sample mean seems tend to concentrate below the mean for large $\sigma$. I am wondering whether this is true for all cases. – Hans May 31 '17 at 0:56
• Incidentally, somewhat related (at least in the limit): Peter Hall, (1980), "On the Limiting Behaviour of the Mode and Median of a Sum of Independent Random Variables", Ann. Probab. Volume 8, Number 3, 419-430. Open Access at Project Euclid – Glen_b -Reinstate Monica May 31 '17 at 1:06
• I mean "no mgf" in the sense expressed here. By "other distributions with the same moment sequence exist" I mean there are distributions with the same sequence of moments as the lognormal / the lognormal is an example of a distribution not uniquely defined by its moments: if $f$ is a standard lognormal density ($\mu=0$, $\sigma=1$) and $g(x)=f(x)\cdot (1+\epsilon\sin[2\pi k\log(x)])$ then the contribution of $g-f$ to the $n$th moment is $0$ for each $n=1,2,...$. This extends to the general lognormal (indeed to the 3 parameter case) – Glen_b -Reinstate Monica May 31 '17 at 2:53
• Oh, this may have been it; it's more a generalization of the above case to a broader class: ... stats.stackexchange.com/questions/25010/… – Glen_b -Reinstate Monica May 31 '17 at 8:43