A High Dimensional Omnibus Regression Test

Consider regression models where the response variable Y only depends on the p×1 vector of predictors x=(x1,…,xp)T through the sufficient predictor SP=α+xTβ. Let the covariance vector Cov(x,Y)=ΣxY. Assume the cases (xiT,Yi)T are independent and identically distributed random vectors for i=1,…,n. Then for many such regression models, β=0 if and only if ΣxY=0 where 0 is the p×1 vector of zeroes. The test of H0:ΣxY=0 versus H1:ΣxY≠0 is equivalent to the high dimensional one sample test H0:μ=0 versus HA:μ≠0 applied to w1,…,wn where wi=(xi−μx)(Yi−μY) and the expected values E(x)=μx and E(Y)=μY. Since μx and μY are unknown, the test of H0:β=0 versus H1:β≠0 is implemented by applying the one sample test to vi=(xi−x¯)(Yi−Y¯) for i=1,…,n. This test has milder regularity conditions than its few competitors. For the multiple linear regression one component partial least squares and marginal maximum likelihood estimators, the test can be adapted to test H0:(βi1,…,βik)T=0 versus H1:(βi1,…,βik)T≠0 where 1≤k≤p.