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What is the median of the non-central t distribution with non-centrality parameter $\delta \ne 0$? This may be a hopeless question because the CDF appears to be expressed as an infinite sum, and I cannot find any information about the inverse CDF function.

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You can approximate it.

For example, I made the following nonlinear fits for $\nu$ (degrees of freedom) from 1 through 20 and $\delta$ (noncentrality parameter) from 0 through 5 (in steps of 1/2). Let

$$a(\nu) = 0.963158 + \frac{0.051726}{\nu-0.705428} + 0.0112409\log(\nu),$$

$$b(\nu) = -0.0214885+\frac{0.406419}{0.659586 +\nu}+0.00531844 \log(\nu),$$

and

$$g(\nu, \delta) = \delta + a(\nu) \exp(b(\nu) \delta) - 1.$$

Then $g$ estimates the median to within 0.15 for $\nu=1$, 0.03 for $\nu=2$, .015 for $\nu=3$, and .007 for $\nu = 4, 5, \ldots, 20$.

The estimation was done by computing the values of $a$ and $b$ for each value of $\nu$ from 1 through 20 and then separately fitting $a$ and $b$ to $\nu$. I examined plots of $a$ and $b$ to determine an appropriate functional form for these fits.

You can do better by focusing on the intervals of these parameters of interest to you. In particular, if you're not interested in really small values of $\nu$ you could easily improve these estimates, likely to within 0.005 consistently.

Here are plots of the median versus $\delta$ for $\nu=1$, the hardest case, and the negative residuals (true median minus approximate value) versus $\delta$:

Non-central t median, delta from 0 to 5, nu=1

Non-central t median residuals, delta from 0 to 5, nu=1

The residuals are truly small compared to the medians.

BTW, for all but the smallest degrees of freedom the median is close to the noncentrality parameter. Here's a graph of the median, for $\delta$ from 0 to 5 and $\nu$ (treated as a real parameter) from 1 to 20.

Non-central t median versus nu and delta (in pseudo 3D)

For many purposes using $\delta$ to estimate the median might be good enough. Here is a plot of the error (relative to $\delta$) made by assuming the median equals $\delta$ (for $\nu$ from 2 through 20).

(Median - delta)/delta versus delta and nu

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    $\begingroup$ +1, awesome. How did you come up with functional form for $g$? $\endgroup$ – mpiktas Mar 2 '11 at 19:59
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    $\begingroup$ @mpiktas A combination of looking at the formulas for the CDF and plotting the medians versus $\nu$. The exponential, with two parameters, actually fits about as well as a quadratic polynomial (with three parameters). $\endgroup$ – whuber Mar 2 '11 at 20:36
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If you are interested in (degrees of freedom) ν > 2, the following asymptotic expression [derived from an interpolative approximation to the noncentral student-t quantile, DL Bartley, Ann. Occup. Hyg., Vol. 52, 2008] is sufficiently accurate for many purposes:

 Median[ t[δ,ν] ] ~ δ(1 + 1/(3ν)).

With ν > 2, the maximum magnitude of the bias of the above expression relative to the noncentral student-t median is about 2% and falls off quickly with increasing ν. The contour diagram shows the bias of the asymptotic approximation relative to the noncentral student-t median:

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