Lack of fit tests based on sums of ordered residuals for linear models

Christensen & Lin ( 2015 ) suggested two lack of fit tests to assess the adequacy of a linear model based on partial sums of residuals. In particular, their tests evaluated the adequacy of the mean function. Their tests relied on asymptotic results without requiring small sample normality. We propose four new tests, find their asymptotic distributions, and propose an alternative simulation method for defining tests that is remarkably robust to the distribution of the errors. To assess their strengths and weaknesses, the Christensen & Lin ( 2015 ) tests and the new tests were compared in different scenarios by simulation. In particular, the new tests include two based on partial sums of absolute residuals. Previous partial sums of residuals tests have used signed residuals whose values when summed can cancel each other out. The use of absolute residuals requires small sample normality, but allows detection of lack of fit that was previously not possible with partial sums of residuals.