Well, a linear regression basically finds you the best straight line that fits a collection of points. The problem is that rank is not a simple linear function of time. Instead, rank is a function of skill, and skill, although somewhat correlated with time played, will be different for everyone. There will be episodes when people's skill gets worse and episodes when it makes some unexpected jump.
Like they say in the bottom of a company's financials, past results are no guarantee of future performance and although perhaps in the short term you can draw a line that says "I've gotten 0.3 points since last May, so if this rate continues I'll be the next level in a couple months". You can't extrapolate that over the infinite horizon, to say "at this rate I'll be better than aarm in three years". If your skill has been going down in the short term, perhaps you should be predicting what day you'll be hitting 0 again!
I suspect that the best curve is either logarithmic or one that has some asymptote, every person has their limit beyond which they'll never get better, something like rank =
15 - 100 / (time + 80).
On another note, sigma is clearly inversely correlated with time played, because sigma essentially captures the uncertainty that one's rank is known, and that uncertainty goes down the more you play.
