# Non-central scaled Student's t cumulative density function required (alternatively the pdf)

I need to cite the pdf(density) or cdf(distribution function) of a non-central scaled Student's t distribution.

There is an article about the non-central Student's t distribution http://en.wikipedia.org/wiki/Noncentral_t-distribution. But I have found no article that states the pdf or cdf and which also considers a scaling parameter. Note that the distribution should be scaled independently of the shape parameter $\nu$.

In particular a parametrization as in the gamlss.dist package, function TF2, would be nice - i.e. mean $\mu$, degrees of freedom $\nu$ and a scale $\sigma$ that corresponds to the standard deviation.

Has anyone seen something alike in a paper or book?

Many thanks!!! Jo

• If I'm not mistaken, the PDF of the three-parameter $t$-distribution is given on page 125 of the help file of the gamlss.dist package. The PDF is also explained here on wikipedia. – COOLSerdash Jun 22 '14 at 19:48
• Thanks! But in both cases $\sigma$ is not the standard deviation, but $SD=\sigma\sqrt{\frac{\nu}{\nu-2}}$. Of course I could shift round and plug in, but this would look pretty messy. But maybe this is the only way... – Joz Jun 22 '14 at 20:57
• That is no different to the standard deviation of a non-scaled $t$-distribution as usually used, which is $\sqrt{\frac{\nu}{\nu-2}}$, at least when $\nu \gt 2$. So $\sigma$ is the scaling parameter but not the standard deviation. – Henry Jun 22 '14 at 21:27

## 1 Answer

Take your pick:

Guenther, W. C. (1978). Evaluation of probabilities for the noncentral distributions and the difference of two T-variables with a desk calculator. Journal of Statistical Computation and Simulation, 6:199–206.

Johnson, N. L. and Kotz, S. (1970). Continuous Univariate Distributions—2. Boston, MA: Houghton Mifflin Company.

Lenth, R. V. (1989). Algorithm as 243: Cumulative distribution function of the non-central t distribution. Journal of the Royal Statistical Society. Series C (Applied Statistics), 38(1):185–189.

Owen, D. B. (1968). A survey of properties and applications of the noncentral t-distribution. Technometrics, 10(3):445–478.

Witkovsky, V. (2013). A note on computing extreme tail probabilities of the noncentral T distribution with large noncentrality parameter. arXiv, September 4:1–9.

• Thanks a lot! I could not get the first and fourth one yet, but I'm working on it. So far it seems like there are many noncentral t distributions, but no SCALED non-central t (where the scale is equal to the standard deviation) --- or i missed it... – Joz Jun 22 '14 at 18:54
• Does the inverse of parameter $\nu$ ('degrees of freedom') not satisfy your desire for a scale parameter? – Alexis Jun 22 '14 at 19:57
• This is good but not good enough :) since the variance should be modeled independently of the shape (degrees of freedom). – Joz Jun 22 '14 at 20:58