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I am trying to intuitively understand a theorem on $p$-values in Wasserman's book.

Let $X$ be a random variable. The theorem says that if $T(X)$ is a statistic such that the null hypothesis is rejected for $T(X)$ large, i.e. $T(X) > c$ for some constant c, then for an observation $x$, the p-value is the supremum of $P(T(X) \ge T(x))$ among all parameter values in the null hypothesis.

So if the p-value is small, it means that the observed $T(x)$ is pretty large, i.e. it lies along the "extreme" end of the range of values of $T(X)$. Since the observed $T(x)$ is large, according to the rejection rule in the theorem above, we reject the null hypothesis.

Is my understanding correct as to why a small p-value based on the above definition implies rejecting the null hypothesis?

Edit: What I'm really confused about is : what is the importance of the rejection rule being $T(X)>c$? Why can't the rejection rule be $T(X)<c$ with this definition?

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