A consistent test of independence and goodness-of-fit in linear regression models. Issue 7 (3rd July 2022)

• Record Type: Journal Article
• Authors: Yu, Qiqing; Liu, Ruiqi
• Is Part Of: Communications in statistics. Volume 51:Issue 7(2022)
• Page Start: 3955
• Page End: 3974
• Publication Date: 2022-07-03
• Subjects: Diagnostic plot -- Linear regression model -- Non-parametric test -- Semi-parametric model; Primary 62J05 -- Secondary 62J20; Mathematical statistics -- Periodicals; Mathematical statistics -- Data processing -- Periodicals; Digital computer simulation -- Periodicals; 519.5
• DOI: 10.1080/03610918.2020.1728316
• Languages: English
• ISSNs: 0361-0918

Abstract

Abstract: We propose a new approach to simultaneously test the assumptions of independence and goodness-of-fit for a multiple linear regression model Y = β ′ X + W, say H 0, vs. H 1 : H 0 is false. Our approach is based on the difference between the empirical distribution function of F x, Y and a consistent estimator F ̂ of F x, Y *, where F x, Y * satisfies H 0 (even if F x, Y doesn't) and F x, Y * = F x, Y iff H 0 holds. The p-value of the test is based on the resampling distribution from F ̂ . The new test is consistent, i.e., its power tends to 1 as the sample size increases, even when E ( | Y | | X ) = ∞ . On the contrary, the consistency of existing tests is proven under special cases of H 1, but not all cases under H 1 . Moreover, our simulation study suggests that existing tests e.g., the test in Sen and Sen (2014) and the quantile regression test can have powers ≤ 20 % for large sample sizes.