Is the kth central moment less than the kth raw moment for even k?

Question

If $X$ is a real-valued random variable, then the $k$th raw moment is $\mathbb{E}[X^k]$, while the $k$th central moment is $\mathbb{E}[(X-\mathbb{E}[X])^k]$. If $k$ is even, is the $k$th central moment always bounded above the $k$th raw moment?

When $k = 2$, then $\mathbb{E}[(X-\mathbb{E}[X])^2] = \mathbb{E}[X^2]-\mathbb{E}[X]^2$, and because $\mathbb{E}[X]^2$ is always positive, it follows that this is less than or equal to $\mathbb{E}[X^2]$. But I'm having trouble extending this to larger moments.

Answer 1

The statement is not true in general and it is easy to construct a counterexample. For example, we want $\mathbb{E}(X^4)\leq \mathbb{E}((X-\mathbb{E}(X))^4)$, we can let $X$ has a small probability of taking a large positive value, while keeping $\mathbb{E}(X)$ negative. Let $X$ be a random variable with $P(X=-2)=1-\epsilon$ and $P(x=M)=\epsilon$. We choose $\epsilon=1/(M+2)$ such that $\mathbb{E}(X)=-1$.

Then we have $$ \mathbb{E}(X^2)=M+2,\quad \mathbb{E}((X-\mathbb{E}(X))^2)=M+1 $$ and $$ \mathbb{E}(X^4)=M^3-2M^2+4M+8,\quad \mathbb{E}((X-\mathbb{E}(X))^4)=M^3+2M^2+2M+1. $$ Obviously, if $M$ is large enough, we have $\mathbb{E}(X^4)\leq \mathbb{E}((X-\mathbb{E}(X))^4)$.

Answer 2

Roberto Rastapopoulos' proof could be simplified by firstly proving the inequality holds when $a\ge 0$: $$a^m-(a-1)^m\ge a-1;$$

Then similarly for $X\ge0$ with $E(X)=1$, $$E(X^r)-E[(X-1)^r] = \int_{X\ge 0}[x^r-(x-1)^r]dP(\omega)\ge \int_{X\ge 0}(x-1)dP(\omega)=0.$$