A new class of depth-based statistics with same attractor

Data depth has emerged as a valuable nonparametric measure for ranking multivariate samples. The main foundation of this paper is the Q Statistics (Liu and Singh 1993), a quality index. Unlike traditional methods, data depth does not require the assumption of normality distributions and adheres to four fundamental properties: affine invariance, maximality at the center, monotonicity relative to the deepest point, and vanishing at infinity (Zou and Serfling 2000, Liu and Singh 1993). Many existing two-sample homogeneity tests, which assess mean and/or scale changes in distributions, are limited with relatively low statistical power or indeterminate asymptotic distributions. Addressing these limitations, we proposed three novel depth-based test statistics, two of which share a common attractor and are applicable across all depth functions. Our approach extends the concept of same attractive depth functions, rooted in Q statistics, to encompass both sum and product statistics. We proved the asymptotic distribution of these statistics for one-dimensional cases under Euclidean depth, along with the minimum statistics applicable across all depths. These proposed statistics use three depth functions: Mahalanobis depth (Liu and Singh 1993), Spatial depth (Brown 1958, Gower 1974), and Projection depth (Liu 1992), all of which are implemented in the R package ddalpha. Through two-sample simulations, we demonstrate the superior performance of power of sum and product statistics, utilizing a strategized permutation algorithm and benchmarking against established methods in literature. Our tests are further validated through real data analysis on spectrum, underscoring the effectiveness of the proposed tests.