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Abstract Let $$f:[0,1]^{d}\to{\mathbb{R}}$$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $$\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$$ have a non-negative local discrepancy (NNLD) everywhere in $$[0,1]^{d}$$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $$f$$ over $$[0,1]^{d}$$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $$b\geqslant 2$$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $$b\geqslant 2$$ and any dimension $$d\geqslant 1$$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $$d\geqslant 3$$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $$d\geqslant 2$$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate. 

Abstract With climate change threatening agricultural productivity and global food demand increasing, it is important to better understand which farm management practices will maximize crop yields in various climatic conditions. To assess the effectiveness of agricultural practices, researchers often turn to randomized field experiments, which are reliable for identifying causal effects but are often limited in scope and therefore lack external validity. Recently, researchers have also leveraged large observational datasets from satellites and other sources, which can lead to conclusions biased by confounding variables or systematic measurement errors. Because experimental and observational datasets have complementary strengths, in this paper we propose a method that uses a combination of experimental and observational data in the same analysis. As a case study, we focus on the causal effect of crop rotation on corn (maize) and soybean yields in the Midwestern United States. We find that, in terms of root mean squared error, our hybrid method performs 13% better than using experimental data alone and 26% better than using the observational data alone in the task of predicting the effect of rotation on corn yield at held-out experimental sites. Further, the causal estimates based on our method suggest that benefits of crop rotations on corn yield are lower in years and locations with high temperatures whereas the benefits of crop rotations on soybean yield are higher in years and locations with high temperatures. In particular, we estimated that the benefit of rotation on corn yields (and soybean yields) was 0.85 t ha⁻¹(0.24 t ha⁻¹) on average for the top quintile of temperatures, 1.03 t ha⁻¹(0.21 t ha⁻¹) on average for the whole dataset, and 1.19 t ha⁻¹(0.16 t ha⁻¹) on average for the bottom quintile of temperatures. This association between temperatures and rotation benefits is consistent with the hypothesis that the benefit of the corn-soybean rotation on soybean yield is largely driven by pest pressure reductions while the benefit of the corn-soybean rotation on corn yields is largely driven by nitrogen availability.