testing for autocorrelation in unequally replicated functional measurement error models

in the ordinary linear models, regressing the residuals against lagged values has been suggested as an approach to test the hypothesis of zero autocorrelation among residuals. in this paper we extend these results to the both equally and unequally replicated functionally measurement error models. we consider the equally and unequally replicated cases separately, because in the first case the residuals of the means of replicate groups of observations in both x and y directions are functions of the same residual while in the second case we have no analogous result and so we have to deal with the residuals in each direction. we derive the asymptotic validity of these tests and we carry out a bootstrap simulation study to determine how well the asymptotic theory of the proposed test works for different size of samples.