I am looking to fit some data with central multivariate t distributions. 

I understand an ECME algorithm is the most promising way to do this. I wondered if anyone knew of any existing code for fitting t distributions with ECME that would get me started (in any language)?

If you check the references below you'll find quite a bit of variation in the background, though there are some common elements.

Those numbers are at least partly based on some comments from Fisher, where he said  

(while discussing a level of 1/20)  


  It is convenient to take this point as a limit in judging
  whether a deviation is to be considered significant or not.
  Deviations exceeding twice the standard deviation are
  thus formally regarded as significant


$\quad$ Fisher, R.A. (1925) Statistical
Methods for Research Workers, p. 47

On the other hand, he was sometimes more broad:


  If one in twenty does not seem high enough odds, we
  may, if we prefer it, draw the line at one in fifty (the
  2 per cent point), or one in a hundred (the 1 per cent
  point). Personally, the writer prefers to set a low standard
  of significance at the 5 per cent point, and ignore entirely
  all results which fail to reach this level. A scientific fact
  should be regarded as experimentally established only
  if a properly designed experiment rarely fails to give
  this level of significance.


$\quad$ Fisher, R.A. (1926)
The arrangement of field experiments.
$\quad$ Journal
of the Ministry of Agriculture, p. 504

Fisher also used 5% for one of his book's tables - but most of his other tables had a larger variety of significance levels

Some of his comments have suggested more or less strict (i.e. lower or higher alpha levels) approaches in different situations.

That sort of discussion above led to a tendency to produce tables focusing 5% and 1% significance levels (and sometimes with others, like 10%, 2% and 0.5%) for want of any other 'standard' values to use.

However, in this paper, Cowles and Davis suggest that the use of 5% - or something close to it at least - goes back further than Fisher's comment.

In short, our use of 5% (and to a lesser extent 1%) is pretty much arbitrary convention, though clearly a lot of people seem to feel that for many problems they're in the right kind of ballpark. 

There's no reason either particular value should be used in general.

Further references:

Dallal, Gerard E. (2012). The Little Handbook of Statistical Practice.  - 
Why 0.05?

Stigler, Stephen (December 2008). "Fisher and the 5% level". Chance 21 (4): 12.
available here

(Between them, you get a fair bit of background - it does look like between them there's a good case for thinking significance levels at least in the general ballpark of 5% - say between 2% and 10% - had been more or less in the air for a while.)
