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Much effort has been made to improve the famous step up test of Benjamini andHochberg given by linear critical values $\frac{i\alpha}{n}$. It is pointed outby Gavrilov, Benjamini and Sarkar that step down multiple tests based on thecritical values $\beta_i=\frac{i\alpha}{n+1-i(1-\alpha)}$ still control thefalse discovery rate (FDR) at the upper bound $\alpha$ under basic independenceassumptions. Since that result in not longer true for step up tests ordependent single tests, a big discussion about the corresponding FDR starts inthe literature. The present paper establishes finite sample formulas and boundsfor the FDR and the expected number of false rejections for multiple testsusing critical values $\beta_i$ under martingale and reverse martingaledependence models. It is pointed out that martingale methods are natural toolsfor the treatment of local FDR estimators which are closely connected to thepresent coefficients $\beta_i.$ The martingale approach also yields new resultsand further inside for the special basic independence model.

Finite sample bounds for expected number of false rejections under martingale dependence with applications to FDR