📂Lemmas

Summarizing Inequalities in the Form of an Inequality

Theorem

Let $x_{1} , \cdots , x_{n}$ and positive $a_{1} , \cdots , a_{n} > 0$ along with constant $\theta \in \mathbb{R}$ be given. $$\forall i \in [n] : x_{i} < a_{i} \theta \iff \max_{i \in [n]} {{ x_{i} } \over { a_{i} }} < \theta$$

Proof

The fact that $(\implies)$ holds for all $i \in [n]$ implies that even the largest $x_{i} / a_{i}$ is smaller than $\theta$. $(\impliedby)$ states that even the largest $x_{i} / a_{i}$ being smaller than $\theta$ implies that $x_{i} / a_{i} < \theta$ holds for all $i \in [n]$.

■

Explanation

Opposite Direction

This is necessary for the proof of a theorem related to sufficient statistics. Naturally, as the opposite direction, the following theorem can be considered. $$\forall i \in [n] : x_{i} > b_{i} \theta \iff \min_{i \in [n]} {{ x_{i} } \over { b_{i} }} > \theta$$