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

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