Generalized martingale-residual processes for goodness-of-fit inference in Cox's type regression models

In the paper a general class of stochastic processes based on the sums of weighted martingale-transform residuals for goodness-of-fit inference in general Cox's type regression models is studied. Their form makes the inference robust to covariate outliers. A weak convergence result for such processes is obtained giving the possibility of establishing the randomness of their graphs together with the construction of the formal χ²-type goodness-of-fit tests. By using the Khmaladze innovation approach, a modified version of the initial class of processes is also defined. Weak convergence results for the processes are derived. This leads to the main application which concerns the formal construction of the Kolmogorov-Smirnov and Cramér-von Mises-type goodness-of-fit tests. This is done within the general situation considered.