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 $r ightarrow infty$. 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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机译：我们提出了一种新的测试方法，用于基于在域中的有限设置$ Mathcal {t} $的函数的一些嘈杂的函数的噪声观察，其中可以使观察结果多个$ mathcal {t} $的每个点。函数形状特性测试的传统方法之一是符合凸，单调或正函数，这取决于我们希望测试的形状特性，以最小化平方误差和的数据集计算拟合和数据集之间的平方差异（SSD）的总和。虽然传统的方法通过观察SSD作为无穷大的点数，但是，我们建议将SSD视为$ r $，每小时在$ mathcal {t中所采取的观察人数。 $，增加到无穷大。观察SSD的渐近行为的新方法导致了一个不需要估计任何其他参数的测试程序，因此易于实现。拟议的测试程序被证明可以实现规定的权力，因为$ r lightarrow idty $。数值示例说明所提出的测试成功地检测了函数的凸起/单调/正向性，以及功能的非凸起/非单调/非阳性。
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