Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on <i>λ</i>-th power skewness and kurtosis

Temperature data, like many other measurements in quantitative fields, are usually modelled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution EPDλ(μ,σ), including tests for normality and Laplacity when <i>λ</i> is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two <i>Z</i>-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain <i>Z</i>-scores that are nearly independent and distributed as standard Gaussians, resulting in a χ22 distribution valid for any sample size (up to very high precision for n≥20). The case λ=1 leads to a powerful test of fit for the Laplace(μ,σ) distribution, whose empirical power is superior to all 39 competitors in the literature, over a wide range of 400 alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.与定量领域诸多其他测量数据类似，气温数据通常采用正态分布（normal distribution）进行建模。然而，部分分布模型能够实现更优的拟合效果，同时避免尾部事件概率被低估的问题。基于此，我们拓展了皮尔逊（Pearson）的偏度与峰度概念，针对指数幂分布（exponential power distribution）EPDλ(μ,σ)构建了一族基于拉奥得分（Rao's score）的拟合优度检验（goodness-of-fit tests）方法，其中当λ取1或2时，分别对应正态性检验与拉普拉斯分布拟合检验。我们推导了检验统计量在原假设与局部备择假设下的渐近分布：该统计量为两个Z得分（Z-score）的平方和。我们还提出了一种创新性的回归策略，用以获取近似独立且服从标准正态分布的Z得分，由此得到的χ²₂分布适用于任意样本量（当n≥20时，精度极高）。当λ=1时，该方法可用于构建拉普拉斯分布Laplace(μ,σ)的高效拟合检验，在400余种备择假设的广泛场景下，其经验功效优于现有文献中的全部39种同类检验方法。该情形下的理论证明极具挑战性且内容丰富。我们将所提出的检验方法应用于三个气温数据集，且该新型检验方法已在R语言包PoweR中实现。