A High Dimensional Omnibus Regression Test

Consider regression models where the response variable Y only depends on the p × 1 vector of predictors x = ( x 1 , … , x p ) T through the sufficient predictor S P = α + x T β . Let the covariance vector Cov ( x , Y ) = Σ x Y . Assume the cases ( x i T , Y i ) T are independent and identically distributed random vectors for i = 1 , … , n . Then for many such regression models, β = 0 if and only if Σ x Y = 0 where 0 is the p × 1 vector of zeroes. The test of H 0 : Σ x Y = 0 versus H 1 : Σ x Y ≠ 0 is equivalent to the high dimensional one sample test H 0 : μ = 0 versus H A : μ ≠ 0 applied to w 1 , … , w n where w i = ( x i − μ x ) ( Y i − μ Y ) and the expected values E ( x ) = μ x and E ( Y ) = μ Y . Since μ x and μ Y are unknown, the test of H 0 : β = 0 versus H 1 : β ≠ 0 is implemented by applying the one sample test to v i = ( x i − x ¯ ) ( Y i − 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 H 0 : ( β i 1 , … , β i k ) T = 0 versus H 1 : ( β i 1 , … , β i k ) T ≠ 0 where 1 ≤ k ≤ p .