Find trendline for minimum (not mean) values in distribution

I would like to perform something like a linear regression on my distribution of data, but I'm interested in a trendline that estimates the minimum, not mean, value for each time bin. I'd like to do this in R.

The image below shows a scatterplot of the minimum value for each time bin. The black line is a typical linear regression, which estimates the mean. What I'd like is something like what I painted in red - an estimation of the minimum.

My data look like this:

Those are just the first few lines but you get the idea.

• The sample minimum you already know; the population minimum only exists for distributions with a restricted range & you haven't said anything about the distributional assumptions: are you asking about predicting the minimum for a newly sampled data-set? – Scortchi Nov 28 '13 at 21:52
• I am asking about predicting the population minimum based on the samples that I have. I would assume a normal distribution of the population. – Charcha Nov 28 '13 at 22:42
• To my eye, it's not clear how you'd mathematically define the minima points along your red line. You know what you want, by eye, but I can't see a clear and precise definition. You could take your standard regression and then subtract some value of the SD from it. Or do a quantile regression with a small quantile number (at some point, it will break, so you can't just use 0 or 0.01). – Wayne Nov 28 '13 at 23:16
• That's easy: the population minimum for a normally distributed variable is $-\infty$. – Scortchi Nov 28 '13 at 23:38
• Perhaps take a step back & explain what you're trying to learn from the data. It's hard to imagine why the trend in the minimum of a lot of time bins with different numbers of observations in each is of any interest. – Scortchi Nov 29 '13 at 0:09