euMMD: efficiently computing the MMD two-sample test statistic for univariate data

Abstract The maximum mean discrepancy (MMD) test is a nonparametric kernelised two-sample that, when using characteristic kernel, can detect any distributional change between two samples. However, the total number of $$d$$ d -dimensional observations $$n$$ n , direct computation statistic $$\mathcal {O}(dn^2 )$$ O ( 2 ) . While approximations with lower computational complexity are known, more efficient methods for computing exact unknown. This paper provides an method MMD univariate case in {O}(n\log n)$$ log Laplacian kernel. Furthermore, this extended to approximate real-valued data also log-linear observations. Experiments show that have good statistical performance compared test, particularly cases where $$d> n$$ >