Definition p-value and find p-value in practice

Question

I have a problem that I can't solution. Let $\mathbf{X}=\{X_1,X_2,\ldots,X_n\}\sim\mathrm{Uniform}(0,\theta)$ and we have $H_0:\theta=\theta_0$ and $H_1:\theta>\theta_0$. We reject the $H_0$ when $T(\mathbf{X})=\max\mathbf{X}>c$. Find $\mathbf{p\textbf{-}value}$.
I know that $\mathbf{p\textbf{-}value}=\mathbb{Pr}_{\theta_0}(T(\mathbf{X})>T(\mathbf{x}))$ where $\mathbf{x}$ is the observed value(realization) of $\mathbf{X}$, but next I don't know what is observed value and how to find $T(\mathbf{x})$.

Answer 1

The $p$-value depends on the observations, here denoted as $\mathbf{x}=(x_1,\dots,x_n)$ (if you do the test with other observations, the $p$-value will be different) and on $\theta_0$. Intuitively, the $p$-value give you the probability that the statistic behaves the way you observe when $H_0$ holds.

Under $H_0$, the $X_i$ are independent and have a specific distribution, namely, the uniform distribution on $(0,\theta_0)$. Therefore, for fixed $x_1,\dots,x_n$ and $X_i$ i.i.d. having uniform distribution on $(0,\theta_0)$, we have to compute $$ \mathbb P\left(\max_{1\leqslant i\leqslant n}X_i>\max_{1\leqslant i\leqslant n}x_i\right). $$ At the end, we find that the $p$-value is $$1-\left(\frac{\max_{1\leqslant i\leqslant n}x_i}{\theta_0}\right)^n.$$