p-value of two side alternative

Question

In the hypotehses testing, I would like to see an intuition how the operator $\min$ and the constant $2$ in the computation of $p$-value of a two side alternative $p = 2\min(F(T_0), 1-F(T_0))$ can be derived and understood for a statistics $T_0$. Here $F$ is the c.d.f.

Answer 1

$p$-value is used to assess how likely we shall get an outcome extremer than the current statistic. Let the parameter to be estimated by $\theta$. Here, since we are given with a two-sided alternative, we can take $p$-value to be the probability for statistic $T$ such that $$|T-\theta|\geq|T_0-\theta|.$$ For one-sided alternative, the absolute value is not needed.

Now consider an ideal case: the normal curve, and assume, WLOG, that $T_0>\theta$ locates at the right of the mean. Then the probability for getting values even greater than $T_0$ is $1-F(T_0)$. For the symmetric cdf, the $p$-value here is $$2\cdot(1-F(T_0)).$$

Likewise, if $T_0<\theta$, then the probability of getting a smaller value is $F(T_0)$, so the $p$-value can be assigned with $$2\cdot F(T_0).$$

Note that in the first case $1-F(T_0)<0.5<F(T_0)$ and in the second case $F(T_0)<0.5<1-F(T_0)$, so we shall use the minimum operator when we want to combine them in one formula.