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