Is $F(E[Y_n]) \approx E[F(Y_n)]$ a reasonable approximation? Studying the asymptotic distribution of order statistics I came across this approximation:
$$F \left( E \left[ Y_n^{\left(n \right)} \right] \right) \approx E \left[ F \left( (Y_n^{\left( n \right)} \right) \right] $$ 
where $Y_n^{\left( n \right) }$ denotes the sequence of the nth order statistics and $F(.)$ is the distribution function of the underlying distribution. 
I understand that because of the Probability Integral Transform, $F \left( Y_n^{ \left(n \right)} \right)$ is the largest observation from a sample of size $n$ from a uniform distribution over $(0,1)$. 
Intuitively it makes sense to claim that the area to the left of the mean of $Y_n^{ \left( n \right) } $ is approximately equal to the mean of the area to the left of $Y_n^{ \left( n \right) } $. Is that really the case though? Also does this approximation hold for other order stastistics as well? 
All help is greatly appreciated, thank you.
 A: It's the first term in a Taylor expansion - when taking only one term, it's a zero-order approximation. 
It's quite common with Taylor expansions for moments of functions of random variables to take a second-order approximation (which would be 3 terms of the expansion, but when taking expectations the second term is zero).
The suggested Taylor approximation will only be accurate in some particular circumstances. I don't think that it will typically be the case for largest order statistics, but it might happen at least sometimes. The next non-zero term in the Taylor expansion should give some sense of the size of the bias.
A: Jensen's Inequality states that for a random variable $Y$ and a convex function $\varphi$, we have
$$
\varphi(E[Y]) \leq E[\varphi(Y)].
$$
If $\varphi$ is concave, then the direction of the inequality is switched. Furthermore, equality holds for non-constant $Y$ if and only if $\varphi$ is linear.
For large $n$, we expect $Y_n^{(n)}$ to be in the right tail of the distribution, where the distribution function $F$ is approximately linear for many common distributions. I believe this is where the given approximation would come from.
