# Asymptotic Confidence Interval for Ri

In what follows, let E, E_{TT} and E_{SS} denote the maximum likelihood estimator of the corresponding matrices the maximum likelihood estimator of the mutual information is then given by

Notice that in this Gaussian scenario, the maximum likelihood ratio test to assess the independence of e_{T} and *e _{S}* equals (Kendall and Stuart, 1966)

where n denotes the number of patients. The statistic (10.9) is proportional to the plug-in estimate (10.8) and, hence, from classical likelihood ratio test theory, it follows that the large-sample null distribution of (10.8) is *xp*_{q}. Using the previous distribution, a test can be constructed to evaluate the quality of the surrogate at the individual level, i.e., the hypothesis *H _{0}* :

*R*= 0.

To obtain the asymptotic distribution of (10.8) in the non-null scenario, notice first that

where the p_{i} denote the sample canonical correlations. If p_{i} are all distinct and non-zero, then pi are asymptotically independent and normally distributed with means *pi* and variances ^-(1 — *p ^{2})^{2}* (Hsu, 1941; Brillinger, 2004). It then follows that, asymptotically,

*I(e*, e

_{T}_{S}) ~ N (I(e

_{T}, e

_{S}), JA p

^{A}/n) and a confidence interval for ДЛ can be obtained using (10.7). If some of the

*pi*are equal or some of them are zero, then the asymptotic distribution becomes much more complex and a bootstrap confidence interval may be a more viable option (Hsu, 1941).