Suppose $z\sim\mathcal{N}\left(\lambda^2 e_1,I_n\right)$ where $e_1$ is the first column of the $n$-dimensional identity matrix, denoted here as $I_n$. Suppose $S\sim\mathcal{W}\left(m,I_n\right)$ is a Wishart matrix on $m$ observations with parameter $I_n$. I am interested in the distribution of $$ h = \frac{e_1^{\top} S^{-1}z}{||S^{-1}z||_2}, $$ as a function of $n, m,$ and $\lambda$. Clearly $-1 \le h \le 1$. Moreover, as $m\to\inf$, some transformations of $h$ are distributed as a non-central $t$-distrbution: $\sqrt{n-1} \tan\left(\arcsin\left(h\right)\right) \sim t\left(n-1,\lambda^2\right)$. (Modulo possibly some dropped constants because it is too late at night.)

Is anything known about the distribution of $h$? Possibly in the finite $m$ case? I have looked at the 'three variable' form for generating variates like $l^{\top}S^{-1}z$, but am not able to reconcile this with the normalization in my problem.


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