The data generating process is f ( x ∣ θ ) = 1 θ , 0 ≤ x ≤ θLet H 1 : θ = 5 and H 2 : θ ≠ 5. Prior probabilities of 1 2 are assigned to the two hypotheses, and a sample of size 5 yields a maximum value of 4. Find the p-value here.According to the definition of p-value, p = P ( L R ≤ L R ∗ ∣ H 1 ) ,and L R ∗ = ( 4 5 ) 5 . I am not sure what to do next. Furthermore, intuitively, if the maxima is equal to 5 exactly, shall we get an extremely big p or a small one?

Question

The data generating process is  f  (  x  ∣  θ  )  =      1    θ    ,    0  ≤  x  ≤  θLet       H    1    :  θ  =  5 and       H    2    :  θ  ≠  5. Prior probabilities of       1    2   are assigned to the two hypotheses, and a sample of size   5 yields a maximum value of   4. Find the   p-value here.According to the definition of   p-value,  p  =  P  (  L  R  ≤  L      R    ∗    ∣      H    1    )  ,and   L      R    ∗    =            (              4        5            )        5  . I am not sure what to do next. Furthermore, intuitively, if the maxima is equal to   5 exactly, shall we get an extremely big   p or a small one?

Shawn Castaneda · Accepted Answer

The   p-value is the probability of getting a value as extreme as the one observed under the null hypothesis, therefore should be   P  r  (  max  {      X    1    ,  .  .  .  ,      X    5    }  ≤  4  |  θ  =  5  ). Since       H    0    :  θ  =  5 means that   X has the continuous uniform distribution   X  ∼      U    n    i    f        (    0    ,    5    )  , we expect the pvalue to be   P  r  (      X    1    ≤  4  )  ∗  .  .  .  ∗  P  r  (      X    5    ≤  4  )  =            (              4        5            )        5    =  .32768.We should get a   p-value close to   1 if the maximum is close to   5 and less than   5, and a   p-value close to   0 if the maximum is close to   0 and positive. We should get   p-values of   0 if the maximum is greater than   5, and it&#039;s impossible to get a maximum less than   0 because it&#039;s not in the support of the data-generating process.

The data generating process is f ( x ∣ θ ) =

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