# Ctree - law of the test statistics

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

The exact expectation $\mu$ of the linear statistic and corresponding covariance matrix $\Sigma$ can be computed from the conditional distribution, i.e., the permutation distribution under the null hypothesis that the distribution of $Y$ does not depend on a given $X$. Based on this the coin package offers two types of test statistics (maximally selected vs. quadratic form) with three types of approximations to the null distribution (exact vs. approaximated through simulation vs. asymptotic). The exact permutation distribution is only available in certain special cases, though.
More details about these test statistics are included in the documentation of the coin package, see also: What is the test statistics used for a conditional inference regression tree?