Some test statistics based on the martingale term of the empirical distribution function (Q1112506)

Suppose \(U_ 1,U_ 2,...,U_ n\) are independent uniform (0,1) random variables, and let \(F_ n\) be their empirical distribution function. The martingale term of \(F_ n\) is defined to be \[ Z_ n(t)=n^{1/2}(F_ n(t)-\int^{t}_{0}[1-F_ n(s)](1-s)^{-1}ds). \] \textit{E. V. Khmaladze} [Teor. Verojatn. Primen. 26, 246-265 (1981; Zbl 0454.60049); English translation in Theory Probab. Appl. 26, 240-257 (1981)] proved that the process \(Z_ n(t)\) is a martingale and converges to a Wiener process W in \(L_ 2[0,1]\). The present author shows that \(Z_ n\) converges weakly to W, in D[0,1], and proposes a number of goodness-of- fit statistics based on \(Z_ n\). In particular, Neyman's smooth test [\textit{J. Neyman}, Skand. Aktuarie Tidskr. 20, 149-199 (1937; Zbl 0018.03403)] can be written as a function of \(Z_ n\). Also, tests of the Kolmogorov-Smirnov and Cramér-von Mises types are proposed and their Bahadur efficiencies are calculated.