4

Let $X$ be a bounded random variable, so that $|X| \leq R$. Suppose that you know $E[|X|^j]$, for $j \leq k$. Is there any way to obtain an upper bound on $E[ |X|^{k+1}]$ that is better than the worst-case, i.e., $E[ |X|^{k+1}] \leq R \cdot E[|X|^k]$

The context is the following: I can sample $X_1, X_2, ..., X_n$ and estimate $E[|X|^j]$, for $j \leq k$. I can say something like "with probability at least $1-\delta$, we know that $E[|X|^j] <= \frac{1}{n} \sum_{i=1}^n |X_i|^j+ {\rm{error}}(j)$ for all $j \leq k$." I can't do this for all $j > k$ because then I need to keep paying the probability cost of these finite sample bounds. So I want to have an upper bound on the $(k+1)^\text{th}$ moment as well that's more efficient than just the worse case scenario. Any ideas appreciated!

Mike Earnest
  • 84,902
Alan Chung
  • 1,426

1 Answers1

1

A general trick I like is to find some degree $k + 1$ monic polynomial $$f(x) = x^{k + 1} + \sum_{i = 0}^k a_i x^i$$ that satisfies $f(x) \leq 0$ on $[0, R]$. Then we can use $$\mathbb{E} f(|X|) \leq 0$$ to obtain $$\mathbb{E} |X|^{k + 1} \leq \sum_{i = 0}^k -a_i \mathbb{E} |X|^i.$$ For example, taking $f(x) = x^{k + 1} - R x^k$ recovers the trivial bound $$\mathbb{E} |X|^{k + 1} \leq R \mathbb{E} |X|^{k}.$$ But in particular situations there are often better choices of $f$. For example, taking $$f(x) = x (x - R / 2)^2 (x - R) = x^4 - 2R x^3 + \frac{5R^2}{4} x^2 - \frac{R^3}{4} x$$ gives $$\mathbb{E} |X|^{4} \leq 2R \mathbb{E} |X|^{3} - \frac{5R^2}{4} \mathbb{E} |X|^2 + \frac{R^3}{4} \mathbb{E} |X|.$$ which might be useful for controlling the fourth moment, particularly if you suspect that $|X|$ takes value in $\{0, R/2, R\}$. See this paper for a cool application of this approach.

abacaba
  • 11,210