# Model selection using p-values - tree inference

Suppose I have some i.i.d. normal observations from $$\mathbb{R}^f$$ with parameters $$(\mu, \Sigma)$$ and $$\Sigma$$ is known to be the identity matrix.

I have the following hypotheses: $$H_0^i$$: $$\mu_i = 0$$

I calculate the statistics $$T_i = \frac{1}{\sqrt{n}} \sum_j^n x_{ji}$$ which are $$N(0, 1)$$ under $$H_0^i$$, and the corresponding p-values.

I combine the test statistics/p-values in some way and test the null-hypothesis $$H_0 = \bigcap_i H_0^i$$.

If I can't reject, I declare $$\mu = 0$$. If I'm able to reject, I choose the $$T_i$$ with the lowest p-value, say $$i = 3$$, and declare $$\mu = [0, 0, \frac{1}{n} \sum_j^n x_{j3}, 0, \dots ]$$.

This seems like a bad idea:

etc.

But maybe it's a good idea? https://stats.stackexchange.com/a/207396/142710

I'm reading a famous paper "Unbiased Recursive Partitioning: A Conditional Inference Framework" by Hothorn, Hornik and Zeileis, and if I'm reading it correctly, this is exactly what they're advocating for.

My question is, is it sometimes acceptable to compare p-values for model selection? What is the right way to think about them in this case?