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 $\lambda\left(x\right)$ --

Classic Neyman-Pearson approach demo

It notes here that N-P approach does not utilize the information in the accurate p value. Actually, at the time N-P approach was firstly devised, the accurate p value was not available usually. Now almost all statistic softwares provide accurate p values and the N-P approach becomes obsolete. Wilkinson & APA TFSI (1999) recommended to report the accurate p value rather than just significance/insignificance, unless p is smaller than any meaningful precision.

--

Wilkinson, L. & APA TFSI (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604.

The tail(s) of p value

For any given $H_0$ vs $H_1$, the p value of any given point x is $\underset{\theta\in H_{0}}{sup}P\left(\left\{ z|L\left(z\right)\ge L\left(x\right)\right\} |\theta\right)$, Where $L\left(x\right)\equiv\frac{\underset{\theta\in H_{1}}{sup}\left[f\left(x|\theta\right)\right]}{\underset{\theta\in H_{0}}{sup}\left[f\left(x|\theta\right)\right]}$

-- See R. Weber's Statistics Note (Chap 6.2 & 7.1)

I made some wrong comment on the pdf Null Ritual (Gigerenzer, Krauss, & Vitouch, 2004) Where three types of significance level (rather than p value) were discussed. I had written the comment to note that the chapter had ignored the role of $H_1$ in definition of p value. In almost every textbook, the two-tail p vs single-tail p are differentiated. Usually, the two-tail p is defined by $H_1$ like $\mu\neq0$.

Here I demonstrate a three-tail p value case on R platform.

``` z=(-1000:1000)*0.02; f=0.5 * dchisq(abs(z),df=5); h=dchisq(10,df=5)*.5; plot(z,f,type="h",col=c("black","grey")[1+(f>h)]); lines(c(-20,20),c(h,h)); ## $H_0$ is $\chi^2(5)$ * binomial(-1 vs 1) ##```

Do you agree the region nearby zero under the "V" curve (which is below the horizontal line) should be the 3rd tail? I think so, if only $H_1$ includes all other possible distributions in the same shape.

You'll also agree there will be two asymmetrical tails if $H_1$ includes just two asymmetrical curves, for example, $\mu=-2$ and $\mu=1$ ($\sigma^2\equiv1$) while $H_0$ is the standardized normal distribution.