# Non-central t-distributions with different degrees of freedom

I have two non-central t distributions with CDF given by $t_{n,a}$ and $t_{m,a}$ with $n>m$ degrees of freedom respectively and the same non-centrality parameter $a$.

The question is: for which values of $x$ the inverse CDF of the first is greater than the inverse CDF of the second?

I.e. for which values of $x$ is $t^{-1}_{n,a}(x)>t^{-1}_{m,a}(x)$ ?

• If a numerical answer is enough, you can find it yourself inR Apr 28, 2016 at 8:09
• I know, but I'd like to have a theoretical answer, if it's possible.
– xanz
Apr 28, 2016 at 8:26
• @Xi'an the limit value for $x$ is that value for which the $t_{n,a}(x)=t_{m,a}(x)=0$? I.e. the answer is when $x<t_{n,a}(0)$?
– xanz
Apr 28, 2016 at 8:43

Since the non-centrality parameter is irrelevant for this question, I take $a=0$.

Here is a picture of several cdfs of Student's $t$-distributions:

And an even better image from Wikipedia:

As the $t_n$ distribution has fatter tails than the $t_m$ distribution when $n<m$, hence the former is be less concentrated around its mean/mode/median zero than the latter. This means the cdf of $t_n$ will be larger than the cdf of $t_m$ for large enough negative values of $x$ and smaller for large enough positive values of $x$. As you can see on the picture, all cdf's cross at zero (the slope of the cdf at zero $F(0;n)$ is the value of the pdf at zero $f(0;n)$ which is increasing with $n$).

• But in the case of a non-central t-distribution, the shift due to the non-centrality parameter does it change things?
– xanz
Apr 28, 2016 at 9:06
• The answer is right, I'm just answering my own previous comment: no, it doesn't. The CDF of two noncentral t-distributions evaluated in 0 is always equal to $\Phi(-a)$ if the noncentrality parameter $a$ is the same (check the CDF expression on wikipedia), regardless of the dofs. And if $n>m$ the tails of $m$ are fatter than the tails of $n$, thus if $x<\Phi(-a)$ then the requirement in the question is satisfied. Thank you!
– xanz
Apr 28, 2016 at 9:15
• @xanz: your answer only mentioned the same non-centrality parameter, which is indeed irrelevant for the tail comparison. If you had different non-centrality parameters then the cdfs could cross more than once. Apr 28, 2016 at 9:31
• Yeah, now I know. At first it wasn't clear to me that the noncentrality parameter is irrelevant for the number of crossings, but now I've figured it out. Thank you.
– xanz
Apr 28, 2016 at 10:26