A Survey of Lack-of-fit Tests Based on Sums of Ordered Residuals

Christensen and Lin (2014), henceforth C-L, 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 extend this research by proposing additional tests based on partial sums of residuals. The asymptotic distribution for each test statistic is found so that the P value can be efficiently approximated. To assess their strengths and weaknesses, the C-L tests and the new tests are compared in different scenarios by simulation. We propose new tests based on partial sums of absolute residuals. Previous partial sums of residuals test have used signed residuals whose values when summed can cancel each other out. The use of absolute residuals ,which requires small sample normality, allows detection of lack of fit that was previously not possible with partial sums of residuals.