TY - JOUR

T1 - Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances

AU - Del Barrio, Eustasio

AU - Gine, Evarist

AU - Utzet, Frederic

PY - 2005/2/1

Y1 - 2005/2/1

N2 - Weighted L2 functionals of the empirical quantile process appear as a component of many test statistics, in particular in tests of fit to location-scale families of distributions based on weighted Wasserstein distances. An essentially complete set of distributional limit theorems for the squared empirical quantite process integrated with respect to general weights is presented. The results rely on limit theorems for quadratic forms in exponential random variables, and the proofs use only simple asymptotic theory for probability distributions in ℝn. The limit theorems are then applied to determine the asymptotic distribution of the test statistics on which weighted Wasserstein tests are based. In particular, this paper contains an elementary derivation of the limit distribution of the Shapiro-Wilk test statistic under normality. © 2005 ISI/BS.

AB - Weighted L2 functionals of the empirical quantile process appear as a component of many test statistics, in particular in tests of fit to location-scale families of distributions based on weighted Wasserstein distances. An essentially complete set of distributional limit theorems for the squared empirical quantite process integrated with respect to general weights is presented. The results rely on limit theorems for quadratic forms in exponential random variables, and the proofs use only simple asymptotic theory for probability distributions in ℝn. The limit theorems are then applied to determine the asymptotic distribution of the test statistics on which weighted Wasserstein tests are based. In particular, this paper contains an elementary derivation of the limit distribution of the Shapiro-Wilk test statistic under normality. © 2005 ISI/BS.

KW - Distributional limit theorems

KW - Tests of fit to location-scale families

KW - Weighted L norms of the quantile process 2

KW - Weighted wasserstein distance

U2 - https://doi.org/10.3150/bj/1110228245

DO - https://doi.org/10.3150/bj/1110228245

M3 - Article

VL - 11

SP - 131

EP - 189

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

ER -