Let's consider a one-sided hypothesis test $H_{0}:\theta \leq \theta_{0}$ vs $H_{1}:\theta > \theta_{0}$, for a given $\theta_{0}$ in the parameter space $\Theta$. Now the p-value function is,

P - value function $p_{n} = p_{n}(\theta_{0}) = p_{n}(x,\theta_{0}) = P(T>t|\theta = \theta_{0})$.

How $p_{n}(.)$ is a cumulative distribution function for every fixed sample $X$?

(Please see http://www.stat.rutgers.edu/home/mxie/RCPapers/insr.12000.pdf, on page 9 (top -left hand corner) they mentioned that $p_{n}(.)$ is a cumulative distribution function).

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