Expectation of a square of a sum

Let $$X_1,\cdots,X_n$$ be i.i.d. random variables with density $$f$$ and let $$\hat{f}$$ be an estimator of $$f$$. Is the following inequality direct from the standard properties of expectation and sup-norm?

$$\mathbb{E}\left[ \lVert \hat{f}- \mathbb{E}(\hat{f})\rVert_\infty + \lVert \mathbb{E}(\hat{f})-f\rVert_\infty \right]^2 \leq 2\mathbb{E}\lVert \hat{f}- \mathbb{E}(\hat{f})\rVert_\infty^2 + 2\lVert \mathbb{E}(\hat{f})-f\rVert_\infty^2.$$

This inequality is found in page 44 of Tsybakov's book.

• You are right. I missed to apply the expectation. – Celine Harumi Mar 24 at 12:36

Since$$(a+b)^2=a^2+2ab+b^2\le a^2+\overbrace{(a^2+b^2)}^{\ge 2ab}+b^2=2a^2+2b^2$$ $$\mathbb{E}\left[ \lVert \hat{f}- \mathbb{E}(\hat{f})\rVert_\infty + \lVert \mathbb{E}(\hat{f})-f\rVert_\infty \right]^2 \leq 2\Bbb E[\lVert \hat{f}- \mathbb{E}(\hat{f})\rVert_\infty]^2 + 2\lVert \mathbb{E}(\hat{f})-f\rVert_\infty^2.$$
• Hint:$$a^2+b^2\ge 2ab$$is equivalent to$$a^2+b^2-2ab\ge 0$$and then... – Xi'an Mar 24 at 13:53