ENDOR spectroscopy is an important tool to determine the complicated three-dimensional structure of biomolecules and in particular enables measurements of intramolecular distances. Usually, spectra are determined by averaging the data matrix, which does not take into account the significant thermal drifts that occur in the measurement process. In contrast, we present an asymptotic analysis for the homoscedastic drift model, a pioneering parametric model that achieves striking model fits in practice and allows both hypothesis testing and confidence intervals for spectra. The ENDOR spectrum and an orthogonal component are modeled as an element of complex projective space, and formulated in the framework of generalized Fr\'echet means. To this end, two general formulations of strong consistency for set-valued Fr\'echet means are extended and subsequently applied to the homoscedastic drift model to prove strong consistency. Building on this, central limit theorems for the ENDOR spectrum are shown. Furthermore, we extend applicability by taking into account a phase noise contribution leading to the heteroscedastic drift model. Both drift models offer improved signal-to-noise ratio over pre-existing models.

翻译：ENDOR波谱学是确定生物分子复杂三维结构的重要工具，尤其能实现分子内距离测量。通常波谱是通过对数据矩阵取平均确定，但该方法未考虑测量过程中发生的显著热漂移。本文针对同方差漂移模型提出渐近分析——该开创性参数模型在实践中实现了惊人的数据拟合效果，并能进行波谱的假设检验和置信区间估计。我们将ENDOR波谱及其正交分量建模为复投影空间中的元素，并在广义Fr\'echet均值框架下进行表述。为此，我们扩展了集值Fr\'echet均值的两个强相合性通解，并将其应用于同方差漂移模型以证明强相合性。在此基础上，证明了ENDOR波谱的中心极限定理。此外，通过引入相位噪声分量建立异方差漂移模型，进一步拓展了方法的适用性。相较于现有模型，这两种漂移模型均能提供更优的信噪比。