# Performing a linear regression on a subset of data

I have some simulated positive ($x$, $y$) data that, when plotted, produce an asymptotic rectangular hyperbola. I would like to test the following using a subset of the data (specifically, data that make up last part of the curve, which forms essentially a straight line, hence the reason for carrying out linear regression):

$H_0: \beta_1 = 0$ vs. $H_1: \beta_1 > 0$.

The dilemma comes in choosing how many data points to include in the subset in order to ensure that $H_0$ is NOT rejected.

My data is in relation to ecological sampling. I am attempting to devise a "stopping rule" for the number of individuals needed to be sampled until we are "confident enough" that all species in an area have been observed.

My thoughts are to either: (1) use an arbitrary $number$ of data points in the calculation of curve slope (e.g. last 10 data points occurring on the curve); or, (2) use a fixed proportion of data points occurring on the last part of the curve (e.g., the last 10% of data points occurring on the curve). So, for the latter case, if there are 50 data points, only the last 5 will be used in calculating the slope of the line.

Both approaches are highly subjective.

Are there statistical arguments to choosing one approach over the other?

NOTE: My data do not fit any parametric models (and finding the best one is both tedious and expensive). So, I am appealing to nonparametric smoothing techniques to get the job done.