We propose a new omnibus goodness-of-fit test based on trigonometric moments of probability-integral-transformed data. The test builds on the framework of the LK test introduced by Langholz and Kronmal [J. Amer. Statist. Assoc. 86 (1991), 1077-1084], but fully exploits the covariance structure of the associated trigonometric statistics. As a result, our test statistic converges under the null hypothesis to a $χ_2^2$ distribution, even in the presence of nuisance parameters, yielding a well-calibrated rejection region. We derive the exact asymptotic covariance matrix required for normalization and propose a unified approach to computing the LK normalizing scalar. The applicability of both the proposed test and the LK test is substantially expanded by providing implementation details for 11 families of continuous distributions, covering most commonly used parametric models. Simulation studies demonstrate accurate empirical size, close to the nominal level, and strong power properties, yielding fully plug-and-play procedures. Further insight is provided by an analysis under local alternatives. The methodology is illustrated using surface temperature forecast errors from a numerical weather prediction model.

翻译：本文提出了一种基于概率积分变换数据三角矩的新型综合性拟合优度检验方法。该检验建立在Langholz与Kronmal [J. Amer. Statist. Assoc. 86 (1991), 1077-1084]提出的LK检验框架之上，但充分挖掘了相关三角统计量的协方差结构。因此，即使在存在冗余参数的情况下，我们的检验统计量在原假设下收敛于$χ_2^2$分布，从而得到校准良好的拒绝域。我们推导了标准化所需的精确渐近协方差矩阵，并提出了一种计算LK标准化标量的统一方法。通过为11类连续分布族（涵盖最常用的参数模型）提供具体实现细节，本文所提检验与LK检验的适用性均得到显著扩展。模拟研究显示，该方法具有接近名义水平的精确经验尺度与较强的检验功效，形成了完全即插即用的检验流程。通过对局部备择假设的分析提供了进一步的理论洞见。最后，利用数值天气预报模型的地表温度预报误差数据对方法进行了实证演示。