Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

We propose a new method of testing for a function's convexity, monotonicity, or positivity, based on some noisy observations of the function made over a finite set $\mathcal{T}$ of points in the domain, where the observations can be made multiple times at each point in $\mathcal{T}$. One of the traditional approaches to the test of a function's shape characteristic is to fit a convex, a monotone, or a positive function, depending on the shape characteristic we wish to test for, to the data set minimizing the sum of squared errors, and to compute the sum of squared differences (SSD) between the fit and the data set. While the traditional approach proceeds by observing the SSD as the number of points in $\mathcal{T}$ increases to infinity, we propose observing the SSD as $r$, the number of observations taken at each point in $\mathcal{T}$, increases to infinity. This new way of observing the asymptotic behavior of the SSD leads to a test procedure that does not require the estimation of any additional parameters, and hence, is easy to implement. The proposed test procedure is proven to achieve a prescribed power as $r \rightarrow \infty$. Numerical examples illustrate that the proposed test successfully detects the convexity/monotonicity/positivity of a function, as well as the non-convexity/non-monotonicity/non-positivity of a function.

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