# Inequality between Expectation and Quantil

For a sample of independent observations $$X_1,X_2,...,X_n$$ on a continuous distribution $$F$$, let the ordered sample values be $$X_{(1)},X_{(2)},...,X_{(n)}$$. From the theory of order statistics, the density function $$g(x)$$ of the maximum variable $$X_{(n)}$$ is known to be $$g(x)=n\,f(x)\,[F(x)]^{n-1},$$ with $$f(x)=F'(x)$$.

On the other hand, let $$q_\alpha=F^{-1}(\alpha)$$ be the quantil function with respect to $$1/2<\alpha<1$$.

Is it possible to prove the following inequality

$$q_\alpha\leq\int_{-\infty}^\infty\,x\,g(x)\,dx\quad?$$

• For a fixed $n$, if you allow $\alpha$ to be arbitrarily closed to $1$, then you can always have a quantile $q_{\alpha}$ larger than the expected value of the sample maximum. And for a very skewed distribution, even the expected value of sample maximum maybe less than the median. – BGM Nov 16 '18 at 15:31
• My guess is that for sufficient large samples the inequality might hold. While, the sample size depends on $\alpha$ such as $n>\alpha/(1-\alpha)$ or so. Do you have a counter example? – kaffeeauf Nov 16 '18 at 15:49
• For sufficiently large $n$, the sample maximum will converge to the supremum of the support, so in other words for a fixed quantile, you can have a sample maximum with expected value larger than the quantile with sufficiently large $n$ – BGM Nov 16 '18 at 16:28

if you have F such that $$F(-B)=0.3$$, $$F(0)=0.5$$ and $$F(1)=1$$

You have $$q_\alpha>=0$$ for $$\alpha \ge 0.5$$

and $$E=\int_{-\infty}^\infty\,x\,g(x)\,dx \le -B(0.3)^n+1$$ (as you have a probability $$(0.3)^n$$ to draw all your sample bellow -B)

you can chose $$B\gt\frac{1}{(0.3)^n}$$ such that $$E \lt 0 \le q_\alpha$$

So the inequality is not always true.

• Ok. Could you help me to understand how to obtain the estimate $E\leq 1-B\, 0.3^n$ in your argument? Is $F$ supposed to be a continuous distribution in your example? – kaffeeauf Nov 16 '18 at 16:05
• expectation E of the maximum M can be express E(M)=P(A)E(M/A)+(1-P(A))E(M/not A). In my case A is the event : all sample is bellow -B. – Arnaud Mégret Nov 16 '18 at 16:11
• Actually, the idea is to let a high probability for very very negative values that have an influence on the average maximum but not on the quantile as it is only influenced by when F reach $\alpha$ and not by how the density is distributed before that point – Arnaud Mégret Nov 16 '18 at 16:23