We propose a goodness-of-fit test for the distribution of errors from a multivariate indirect regression model, which we assume belongs to a location-scale family under the null hypothesis. The test statistic is based on the Khmaladze transformation of the empirical process of standardized residuals. This goodness-of-fit test is consistent at the root-$n$ rate of convergence, and the test can maintain power against local alternatives converging to the null at a root-$n$ rate.
"Goodness-of-fit testing the error distribution in multivariate indirect regression." Electron. J. Statist. 13 (2) 2658 - 2685, 2019. https://doi.org/10.1214/19-EJS1591