Using the fact that some depth functions characterize certain family of distribution functions, and under some mild conditions, distribution of the depth is continuous, we have constructed several new multivariate goodness of fit tests based on existing univariate GoF tests. Since exact computation of depth is difficult, depth is computed with respect to a large random sample drawn from the null distribution. It has been shown that test statistic based on estimated depth is close to that based on true depth for a large random sample from the null distribution. Some two sample tests for scale difference, based on data depth are also discussed. These tests are distribution-free under the null hypothesis. Finite sample properties of the tests are studied through several numerical examples. A real data example is discussed to illustrate usefulness of the proposed tests.
翻译:利用某些深度函数对分布函数的某些类别具有特征这一事实,在某种温和条件下,深度分布是连续的,我们根据现有的单象体GOF测试建立了几项新的适合试验的多变量性能。由于精确的深度计算是困难的,因此根据从无效分布中抽取的大型随机样本计算深度。已经证明,根据估计深度得出的测试统计数据接近于根据从无效分布中抽取的大型随机样本的真实深度得出的测试统计数据。还讨论根据数据深度对比例差异进行的两个抽样测试。这些测试在无效假设下是无分布的。通过几个数字例子研究这些测试的精度样本特性。讨论了一个真实的数据实例,以说明拟议测试的有用性。