On the Limit Distribution of Kolmogorov-Smirnov Statistics for a Composite Hypothesis

Effect of the state of the geomagnetic field and solar activity on chromosome aberrations frequencies dynamics

“…Using the Kolmogorov-Smirnov goodness-of-t test in the Tyurin modi cation [12], which is adapted to small volume samples, with the signi cance level ¬ , with the standard competing hypothesis H 1 available, we test the main statistical hypothesis H 0 that the elements of the matrix row" N , which are speci ed by equality (1.3), are the realizations of a normal random process with zero mean value. Using the Kolmogorov-Smirnov goodness-of-t test in the Tyurin modi cation [12], which is adapted to small volume samples, with the signi cance level ¬ , with the standard competing hypothesis H 1 available, we test the main statistical hypothesis H 0 that the elements of the matrix row" N , which are speci ed by equality (1.3), are the realizations of a normal random process with zero mean value.…”

“…The realization of the second stage is completed by verifying the adequacy of the optimal periodic trend functions f j (½ ;Â j ), j 2 f1; 2g, to the corresponding processes by the use of the Kolmogorov-Smirnov goodness-of-t test in the Tyurin modi cation [12] with the signi cance level ¬ .…”

A comparison of alternative approaches to sup-norm goodness of fit tests with estimated parameters

“…Second, the assumption that the variance function is twice differentiable is satisfied in a great number of parametric models, so this is not a strong assumption. This is also assumed in the work of Tyurin (1985), and although not a formal assumption in the general theory of Piterbarg (1996), this differentiability is implicitly used in the calculation of the constants necessary for the asymptotic results given there.…”

“…The work of Tyurin (1985) was perhaps the first such derivation, but the focus here will be on the work of Fatalov (1992Fatalov ( , 1993 and Piterbarg (1996). The reason for this is that the theory underlying the result is quite general and has roots that are more clearly probabilistic -the work of Tyurin relies on the solution to boundary value problems and can usually only be applied in the same situations as Durbin's P g , while Fatalov's work allows one to compute probabilities not available via Durbin's or Tyurin's formulation of the solution.…”