We initiate the study of goodness-of-fit testing when the data consist of positive definite matrices. Motivated by the recent appearance of the cone of positive definite matrices in numerous areas of applied research, including diffusion tensor imaging, models of the volatility of financial time series, wireless communication systems, and the analysis of polarimetric radar images, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and the corresponding covariance operator. We show that the eigenvalues of the operator satisfy an interlacing property, and we apply our test to some financial data. Moreover, we establish the consistency of the test against a large class of alternative distributions and we derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We establish the Bahadur and Pitman efficiency properties of the test statistic and we show the validity of a modified Wieand condition.

中文翻译：

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拟合优度测试中的积分变换方法，II：Wishart 分布

当数据由正定矩阵组成时，我们开始研究拟合优度检验。由于最近在许多应用研究领域（包括扩散张量成像、金融时间序列波动性模型、无线通信系统和极化雷达图像分析）中出现了正定矩阵锥，我们应用了矩阵参数的 Hankel 变换，以开发具有给定形状参数和未知尺度矩阵的 Wishart 分布的拟合优度检验。我们获得了检验统计量的极限零分布和相应的协方差算子。我们证明算子的特征值满足隔行特性，并将我们的测试应用于一些金融数据。而且，我们针对一大类替代分布建立测试的一致性，并在一系列连续替代方案下推导出测试统计量的渐近分布。我们建立了检验统计量的 Bahadur 和 Pitman 效率属性，并证明了修改后的 Wieand 条件的有效性。