7

How to show that, for any given left tail probability, the corresponding quantile of standardized t distribution is increasing in degree of freedom for left tail probability less than 0.5? This is obvious when we check statistical tables. But how can we show it analytically?

Can anyone share any thoughts?

kjetil b halvorsen
  • 63,378
  • 26
  • 142
  • 467
Ruth
  • 463
  • 2
  • 7
  • 1
    May well be intractable analytically. The quantile function itself is a function of the inverse of a regularized incomplete beta function, whose derivative with respect to the degrees of freedom is probably difficult to obtain. See Shaw, W.T. (2006). "Sampling Student’s T distribution – Use of the inverse cumulative distribution function." Journal of Computational Finance 9 (4): 37–73 for some results, that may already be fairly ugly. – Christoph Hanck Apr 27 '16 at 14:25
  • 1
    Could you clarify what you mean by a "standardized t distribution"? Although "standardized" usually means "scaled to have unit variance," t distributions don't even have a variance until their degrees of freedom exceed $2$. – whuber Apr 27 '16 at 14:35
  • @whuber standardized t distribution means zero mean and unit scale parameter. It is the one used in statistical table. Yes, consider degrees of freedom above 2, so its variance exists. – Ruth Apr 27 '16 at 14:41
  • Statistical tables usually do *not* standardize the Student $t$ distribution. The Student $t$ has no scale parameter (although *generalizations* of it may have a scale parameter). Thus, it's still unclear what you are asking. – whuber Apr 27 '16 at 15:00
  • 1
    @whuber you may refer to this: http://mathworld.wolfram.com/Studentst-Distribution.html. This is what I refer to as standardized t distribution. Thanks. – Ruth Apr 27 '16 at 16:09
  • 1
    Thank you--that makes your intention very clear (+1). Please note there is nothing "standardized" about this distribution. It would be a good idea to eliminate that term from your question. Consider calling it what MathWorld does: the *Student's t Distribution.* – whuber Apr 27 '16 at 16:11
  • This should in principle be a straightforward application of the [Implicit Function Theorem](https://en.wikipedia.org/wiki/Implicit_function_theorem) for $n=m=1$: setting the CDF equal to your target quantile $F_r(t)\stackrel{!}{=}q$ implicitly defines the quantile as a function of the degrees of freedom, $t(r)$. Then the theorem will give you $\frac{d}{dr} t(r)$ based on the Jacobian of $F$. The math gets ugly, though - it doesn't look like the solution is simple. – Stephan Kolassa Jun 17 '20 at 14:15
  • Asked again at [Quantile of t distribution when the degree of freedom increases](https://stats.stackexchange.com/q/463051/1352). I find it a bit surprising that there indeed doesn't seem to be any work on this. – Stephan Kolassa Jun 17 '20 at 14:27
  • Specifically, you would need derivatives of the hypergeometric function ${}_2F_1$ (which appear in the CDF) with respect to the second and fourth argument (where the degrees of freedom come in). [Ancarani & Gasaneo (2009)](https://iopscience.iop.org/article/10.1088/1751-8113/42/39/395208) give the derivative with respect to the second argument (and this is complicated indeed), while the one with respect to the fourth argument is easier and found in [the Wikipedia article](https://en.wikipedia.org/wiki/Hypergeometric_function). – Stephan Kolassa Jun 17 '20 at 15:20

0 Answers0