Uniform Distribution of P-Values Under the Null Hypothesis

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Hypothesis Testing.

Your Task
Let $T(X)$ be a continuous test statistic with continuous cumulative distribution function $F_0(t)$ under the null hypothesis $H_0$. Define the one-sided $p$-value by $P = 1 - F_0(T)$.
(1) Prove by the Probability Integral Transform that under $H_0$, $P \sim \text{Uniform}(0, 1)$.
(2) Show that under any alternative hypothesis $H_1$ where $T$ is stochastically larger than under $H_0$, $\mathbb{P}_{H_1}(P \le \alpha) > \alpha$ for any significance level $\alpha \in (0, 1)$.

Constraints

• Samples are independent and identically distributed on their respective support.
• Boundary conditions are defined strictly within standard real intervals.