This paper develops tools to characterize how species are affected by environmental variability, based on a functional single index model relating a response such as growth rate to environmental conditions. In ecology the curvature of such responses are used, via Jensen’s inequality, to determine whether environmental variability is harmful or beneficial, and differing nonlinear responses to environmental variability can contribute to the coexistence of competing species. Here, we address estimation and inference for these models with observational data on individual responses to environmental conditions. Because nonparametric estimation of the curvature (second derivative) in a nonparametric functional single index model requires unrealistic sample sizes, we instead focus on directly estimating the effect of the nonlinearity by comparing the average response to a variable environment with the response at the expected environment, which we call the Jensen Effect. We develop a test statistic to assess whether this effect is significantly different from zero. In doing so we reinterpret the SiZer method of Chaudhuri and Marron (J. Amer. Statist. Assoc. 94 (1999) 807–823) by maximizing a test statistic over smoothing parameters. We show that our proposed method works well both in simulations and on real ecological data from the long-term data set described in Drake (Proc. R. Soc. Lond., B Biol. Sci. 272 (2005) 1823–1827).

中文翻译：

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詹森效应和功能性单指数模型：估计非线性反应范式的生态意义

本文基于功能性单指标模型，该模型将响应（如增长率与环境条件）联系起来，从而开发出表征物种如何受到环境变异性影响的工具。在生态学中，通过詹森不等式使用这种响应的曲率来确定环境变异性是有害的还是有益的，并且对环境变异性的不同非线性响应可以促进竞争物种的共存。在这里，我们通过对环境条件的个体响应的观测数据来处理这些模型的估计和推断。由于非参数功能单指标模型中的曲率（二阶导数）的非参数估计需要不切实际的样本大小，詹森效应。我们开发了一项测试统计数据，以评估这种影响是否显着不同于零。在这样做时，我们重新解释Chaudhuri和马龙的SIZER方法（J.阿米尔。统制。协会。 94通过在平滑参数最大化测试统计（1999）807-823）。我们证明了我们提出的方法效果很好无论是在模拟和从长期数据真实生态数据集德雷克描述（PROC。R. SOC。林斯顿。，B生物学，科学， 272（2005）1823-1827）。