# Joint distribution of mid p-value and p-value

I have a question about the joint distribution of the mid p-value and p-value.

We know that, for right tailed test with discrete test statistic $X$ with distribution $F$, the p-value is defined as $P=Pr(X \geq observed~X)$ and the mid p-value is defined as $mid~P=Pr(X \gt observed~X)+\frac{1}{2}*P(X=Observed)$.

I would like to find out $Pr(P_{mid} \leq t~and~P \gt t)$ for some $t \in (0,1)$. Here $P$ is the p-value and $P_{mid}$ is the mid p-value.

• What test do you have in mind?
– whuber
Aug 10 '12 at 23:13
• Thanks so much for the reply. I am hoping to get a general solution for any discrete distribution F. Aug 11 '12 at 1:06
• The answer depends strongly on the test statistic, not just the distribution: that's why you need to tell us what test you're using.
– whuber
Aug 11 '12 at 21:05
• Ok. We can focus on the Fisher's exact test. Thanks. Aug 12 '12 at 2:31
• Simple translation of the question: "I would like to find out values of the hypergeometric distribution."
– whuber
Aug 14 '12 at 14:05

It is well known that the distribution of $P_{\rm mid}$ more closely approximates a uniform distribution than that of $P$.
$P$ is obviously always slightly larger than $P_{\rm mid}$. If $X=x$, then the difference between the two p-values is $$P-P_{\rm mid}=P(X=x)/2$$ On average this is $$E(P-P_{\rm mid})=\frac12 \sum_x P(X=x)^2.$$ In general, if $X$ can take on many possible distinct values, so that $\max_x P(X=x)$ becomes small, then $E(P-P_{\rm mid})$ will also be small.
If $P(X=x)$ is a decreasing function of $x$, as it usually is for $x$ sufficiently large, then $P-P_{\rm mid}$ will also decrease with $x$.