I am reading the article of Hothorn, Hornik and Zeileis : An unbiased Recursive Partitioning : A conditional Inference Framework.
I am interested in using this paper with an objective of regression tree. The null hypothesis $H_0$ of the test is : $$H_0^j : D(Y|X_j)=D(Y)$$ The test statistic becomes : $$T_j=\sum_{i=1}^{n}g_j\left( X_{ji}\right)Y_i$$ where:
$g_j\left( X_{ji}\right)$ = $ X_{ji}$ if $X_{ji}$ is continuous and $g_j\left( X_{ji}\right)$ = $e_k\left( X_{ji}\right)$ if if $X_{ji}$ is qualitative with $k$ modalities.
In the article, I cannot see the law of $T_j$ under $H_0$. It is said that it asymptotically tends to a normal distribution of mean $\mu$ and variance $\Sigma$. Is it true only when $H_0$ is true? Or is it only the parameters of the law that are defined this way when $H_0$ is true?
Best regards,
Pierre
In this article, it is Source : http://statmath.wu-wien.ac.at/~zeileis/papers/Hothorn+Hornik+Zeileis-2006.pdf