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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.

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I strongly suggest that you do not fit only the part of the data you "like". Fitting only part of the data, unless you have a very good motive for doing so, would not be correct. A very good reason could be that the entire data was already modeled, and that some data is withheld to test for accuracy of extrapolation.

Much better to fit the whole data with something you do not like but is more robust. Moreover, think about why the model is curvilinear. Data transformation may help to linearize the model.

For example, sometimes reciprocation will change the whole curve into a linear one (especially ones that look like hyperbolas). If you show your data, you may receive more directed advice.

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