I'm studying all statistics by Wasserman and on page 156 he defines p-values:
10.11 Definition. Suppose that for every $\alpha \in (0,1)$ we have a size $\alpha$ test with rejection region $R_{\alpha}$. Then,
$$\text{p-value} = \inf\{\alpha: T(X^n)\in R _{\alpha}\}$$
That is, the p-value is the smallest level at which we can reject $H_0$.
I've already seen some answers here to this definition in Wasserman book, but my question is more basic. I want to know what is this exponent $n$. What does $X^n$ mean?
This is an unfortunate use of poorly-chosen notation. Presumably, $X^n$ is intended to indicate a sample of size $n$; i.e., $X^n = (X_1, X_2, \ldots, X_n)$, from which a test statistic defined by some function $T$ of the sample is calculated. This notation is probably inherited from expressions like $\mathbb R^n$. But for obvious reasons, one should probably avoid writing things this way.