Martingale approach in the theory of goodness-of-fit tests

Let us consider the parametric hypothesis that the distribution function $F$ of i. i. d. random variables belongs to the given parametric family of distribution functions $\mathbf F=\{F(x,\theta),\,\theta\in\Theta\}$. It is well-known that the limiting distribution of the parametric empirical process $\sqrt n[F_n(x)-F(x,\widehat\theta)]$ depends on $F$, and therefore the «usual» testing procedures become inconvenient.
In the paper we consider the parametric empirical process as a semimartingale. By means of its Doob–Meyer decomposition we construct some martingale and show that this martingale converges weakly to the Wiener process. This fact enables us to use the distribution-free asymptotic theory of testing parametric hypotheses.