Talk:Noncentral t-distribution

I can't figure out why the $$\mu$$ is rendering as Roman, rather than italic in this article. Compare to, say, Variance, but there it renders as PNG rather than HTML. And I don't think logging in with special preferences is a solution. — DIV (128.250.204.118 06:04, 18 March 2007 (UTC))


 * I agree that an italic mu would look better. I wish I knew how to make that appear. Steve8675309 00:29, 20 March 2007 (UTC)


 * This can be solved by placing \,\! at the end of each formula, as specified at MetaWikiPedia:Help:Formula. --128.250.5.248 (talk) 04:56, 12 August 2009 (UTC)

I think this description of the noncentral t-distribution is incomplete and overly confusing. A scale parameter is generally included and the density can be written much more succinctly. Jurgen (talk) 20:06, 4 May 2010 (UTC)

I am puzzled by the representation of the first form of the CDF. It specifies that when (x < 0) the recursive part of the function---~F---operates on the reversed sign of x: but ~F is utterly insensitive to the sign of x, since x only appears in squared terms (i.e. in y). So why bother to assert that the sign of x changes? mu changes sign, yes, but not x. Lexy-lou (talk) 16:14, 20 April 2014 (UTC)

In Section 1.2.1 Differential equation, the sign of the first derivative of the PDF at zero seems wrong. It says that the first derivative has the opposite sign of the non-centrality parameter ($\mu$), and from the plots of the PDF in the same article, and from general knowledge, when the non-centrality parameter is positive the PDF is increasing at zero. — Preceding unsigned comment added by Niausnil (talk • contribs) 13:35, 29 January 2015 (UTC)

Mode as a function of noncentrality parameter
The subsection Noncentral t-distribution currently says


 * Moreover, the negative of the mode is exactly the mode for a noncentral t-distribution with the same number of degrees of freedom ν but noncentrality parameter −μ.


 * The mode is strictly increasing with μ when μ > 0 and strictly decreasing with μ when μ < 0.

These are mutually contradictory when μ < 0. The last part of the last sentence says that when μ is negative and increases toward 0, the mode goes in the opposite direction (away from 0). That conflicts with the first quoted sentence, which implies that the μ > 0 and μ < 0 cases both have the mode going in the same direction as μ.

Unless I hear an objection, I’ll change the last sentence to The mode is strictly increasing with μ. Loraof (talk) 20:07, 18 June 2018 (UTC)


 * Done. Loraof (talk) 18:03, 20 June 2018 (UTC)

Mode as a function of noncentrality parameter
The subsection Noncentral t-distribution gives a lower and an upper bound to the mode:
 * $$ \sqrt{\frac{\nu}{\nu+(5/2)}} < \frac{\mathrm{mode}}{\mu} < \sqrt{\frac{\nu}{\nu+1}}$$.

Then it says:

"In the limit, when μ → 0, the mode is approximated by"
 * $$\sqrt{\frac{\nu}{2}}\frac{\Gamma\left(\frac{\nu+2}{2}\right)}{\Gamma\left(\frac{\nu+3}{2}\right)}\mu;\,$$

However this aproximation is worse (by worse I mean lower) than the lower bound: $$\sqrt{\frac{\nu}{\nu+5/2}}\mu\,$$.

The article from where this was taken doesn't compare the result it obtained for the μ → 0 approximation with the bounds it provided. Does anyone know why the lower bound would not be a better approximation than the currently one?

Zaphodxvii (talk) 03:03, 31 December 2020 (UTC)

Construction of a noncentral t-random variable
In the definition of the noncentral $$t$$-distribution, it defines $$Z$$ as a normal random variable with mean $$\mu$$ and unit variance. If this variable already has mean $$\mu$$, then the construction of a noncentral $$t$$-random variable does not need $$\mu$$. — Preceding unsigned comment added by Jourdy345 (talk • contribs) 18:05, 27 May 2022 (UTC)