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I have a problem that I don't think I've met before. I have N observations of the variables v1 and v2 and I assume that there is a function f such as v2 = f(v1).

I want to know if f has a particular 'statistical' monotony (increasing or decreasing) and if it is 'statistically' convex or concave. By 'statistical' I mean that my observations may include error terms so you may encounter pairs of variable that show a monotony that is the opposite of the global monotony, if that's English.

Should I simply compute a f' and a f'' ? (derivatives of f).

If you have some thoughts on this I'd be glad to read it. Thanks,

Arthur

(btw I use Stata)

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    $\begingroup$ You need more assumptions. After all, you could regress v2 against v1 to estimate a linear function, which is guaranteed monotonic (and both convex and concave). $\endgroup$ – whuber Jul 4 '11 at 16:57
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Here is a paper that gives minimax test for this problem.

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