$T_n^*:=sup\{t|\sum ψ(x_i;t)\gt 0\}$
$T_n^{**}:=inf\{t|\sum ψ(x_i;t)\lt 0\}$
As it's seen in the above figure, $-\infty \lt T_n^{*} \le T_n^{**} \lt +\infty$
Then, how to write the two followings ?
$$\{T_n^{*}\lt t\} \subset \{\sum ψ(x_i;t)\le 0\} \subset \{T^*_n \le t\} $$
and
$$\{T_n^{**}\lt t\} \subset \{\sum ψ(x_i;t)\lt 0\} \subset \{T^{**}_n \le t\} $$
please help me explain these two inequality? thank you
the first equation means, that $ T_n^* < t \Rightarrow \sum \psi(x_i ; t) \leq 0 $ and $\sum \psi(x_i ; t) \leq 0 \Rightarrow T_n^* \leq t$. Both these implications can be proved by contradiction using the definition of the supremum:
Assume $\sum \psi(x_i;t)>0$, then $t$ is in the set $A:=\{t| \sum \psi(x_i;t) >0\}$, hence $t \leq T_n^*=sup A$
For the other implication you need to use, that $\sum \psi(x_i;t)$ is a non-increasing function, like in the picture. Assume $T_n^* >t$, then it follows that $t \in A \Rightarrow \sum \psi(x_i;t) >0$
the second equation means, that $ T_n^{**} < t \Rightarrow \sum \psi(x_i ; t) < 0 $ and $\sum \psi(x_i ; t) < 0 \Rightarrow T_n^{**} \leq t$. This can be proven directly (using the definition of infimum), it works similar like in the first case, except we need to use "non-increasing" in the first implication.