## Unexpectedly, the theoretically best reject-region of T-test is bounded.

$f_{t}\left(x;\mu,df\right)\equiv C\left(df\right)\left(1+\frac{\left(x-\mu\right)^{2}}{df}\right)^{-\frac{df+1}{2}}$

$\lambda\left(x;\mu_{0},\mu_{1},df\right)\equiv\frac{f_{t}\left(x;\mu_{1},df\right)}{f_{t}\left(x;\mu_{0},df\right)}=\left(\frac{v+\left(x-\mu_{1}\right)^{2}}{v+\left(x-\mu_{0}\right)^{2}}\right)^{-\frac{df+1}{2}}{\longrightarrow\atop x\rightarrow\infty}1$

For NHST $H_{0}:T\sim t_{df}$ vs $H_{1}:T-1\sim t_{df}$, theoretically, $p\left(t\right)=\int_{\left\{ x:\lambda\left(x\right)\ge\lambda\left(t\right)\right\} }f_{t}\left(x,\mu_{0},df\right)dx$ is s.t. $\lim_{t\rightarrow\infty}p\left(t\right)=\frac{1}{2}$ , rather than zero. Nevertheless, pratically a large t, rejecting both $H_{0}$ and $H_{1}$, should not be counted as any evidence to retain or reject $H_{0}$.

To verify the shape of $\lambda\left(x\right)$ --

## Confidence Region and Not-reject Region

Either Confidence Interval (CI) or Null Hypothesis Significance Test (NHST) has the same business, to advise whether some sample $X\equiv\left(X_{1},X_{2},\dots,X_{n}\right)$ is or is not disliked by some hypothesized parameter $\vartheta$.

NHST.com manages a database. For each Miss $\vartheta$, NHST spies out all she dislikes. Mr X logs in NHST.com and inputs a girl name and his credit card number, to bet his luck whispering-- Does she dislike me?

CI.com manages a database too. For each Mr X, CI only needs his credit card with his name X on it, then serves him a full list of available girls.

NHST.com has been historically monopolizing the market. Nevertheless, somebody prefer visiting CI.com and find that the two may share database in most cases.

Not-reject Region of $\vartheta$ is defined as $A\left(\vartheta\right)=\left\{ x:\vartheta\; doesn't\;dislike\;x\right\}$.

Confidence Region of x is defined as $S\left(x\right)\equiv\left\{ \vartheta:\vartheta\; doesn't\;dislike\;x\right\}$.

$\theta\in S\left(X\right)\Leftrightarrow \theta\,does\,not\,dislike\,X$ $\Leftrightarrow\,X\in\,A\left(\theta\right)$

So, $Pr_{\vartheta}\left(\vartheta\in S\left(X\right)\right)\ge1-\alpha,\forall\vartheta\Longleftrightarrow Pr_{\vartheta}\left(X\notin A\left(\vartheta\right)\right)\le\alpha,\forall\vartheta$