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.

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.

Hayley Mccarthy · Accepted Answer

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   θ. 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  −  θ  |  ≥  |      T    0    −  θ  |  .For one-sided alternative, the absolute value is not needed.Now consider an ideal case: the normal curve, and assume, WLOG, that       T    0    &amp;gt;  θ 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  ⋅  (  1  −  F  (      T    0    )  )  .Likewise, if       T    0    &amp;lt;  θ, then the probability of getting a smaller value is   F  (      T    0    ), so the   p-value can be assigned with  2  ⋅  F  (      T    0    )  .Note that in the first case   1  −  F  (      T    0    )  &amp;lt;  0.5  &amp;lt;  F  (      T    0    ) and in the second case   F  (      T    0    )  &amp;lt;  0.5  &amp;lt;  1  −  F  (      T    0    ), so we shall use the minimum operator when we want to combine them in one formula.

In the hypotehses testing, I would like to see an intuition how the operator <mo movablelimits="

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