We apply the method of Hankel transforms to develop goodness-of-fit tests for gamma distributions with given shape parameters and unknown rate parameters. We derive the limiting null distribution of the test statistic as an integrated squared Gaussian process, obtain the corresponding covariance operator and oscillation properties of its eigenfunctions, show that the eigenvalues of the operator satisfy an interlacing property, and make applications to two data sets. We prove consistency of the test, provide numerical power comparisons with alternative tests, study the test statistic under several contiguous alternatives, and obtain the asymptotic distribution of the test statistic for gamma alternatives with varying rate or shape parameters and for certain contaminated gamma models. We investigate the approximate Bahadur slope of the test statistic under local alternatives, and we establish the validity of the Wieand condition under which approaches through the approximate Bahadur and the Pitman efficiencies are in accord.

中文翻译：

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拟合优度测试中的积分变换方法，I：伽马分布

我们应用 Hankel 变换的方法来开发具有给定形状参数和未知速率参数的伽马分布的拟合优度测试。我们将检验统计量的极限零分布推导出为积分平方高斯过程，得到其特征函数对应的协方差算子和振荡性质，证明该算子的特征值满足交错性质，并应用于两个数据集。我们证明了检验的一致性，提供了与替代检验的数值功效比较，研究了几个连续替代方案下的检验统计量，并获得了具有不同速率或形状参数的 gamma 替代方案以及某些受污染的 gamma 模型的检验统计量的渐近分布。