Given $p$-variate normal $X \sim \mathcal{N}\left(\mu,\Sigma\right)$, consider the random variable $$ t_* = \frac{X_1}{\sqrt{\sum_{2\le i \le p} X_i^2}}. $$ For some values of $\Sigma$, this generalizes the doubly noncentral t-distribution (which generalizes the singly noncentral t-, which generalizes the central t-distribution). Is this a known distribution? In particular, I would like to know how to compute the CDF, and the first two moments, say.

I suspect that some transformation takes this $t_*$ to a form which is amenable to analysis, but I am not seeing it.

  • $\begingroup$ I don't think this will have a DF that can be written without a ton of integrals floating around. I don't even think an asymptotic distribution can be obtained without some very contrived stipulations about $\Sigma$, akin to Lindeberg-Feller conditions. $\endgroup$ – AdamO Jan 7 '14 at 18:21

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