content_image/304614.jpg content_image/304615.jpg content_image/304616.jpg content_image/304617.jpg content_image/304618.jpg
Let ˜ π$_{0}$ ( λ ) with a tuning parameter λ ∈ [0 , 1) be an estimator of π$_{0}$ such that ˜ π$_{0}$ ( λ ) ranges in [0 , 1]. We define a “Storey-type FDR estimator” as 

<img src='content_image/304614.jpg'>

For α ∈ [0 , 1], let 

and the decision rule based on ˜ F DR$_{λ}$ 

<img src='content_image/304615.jpg'>

<img src='content_image/304616.jpg'>

The procedure defined by (1) and (3) is called a “Storey-type FDR procedure”. 

A Storey-type FDR procedure allows the use of various estimators of π$_{0}$ and is very versatile. For example, (i) it is Storey’s procedure when ˜ π$_{0}$ ( λ ) in (1) is the estimator 

<img src='content_image/304617.jpg'>

in Storey et al. (2004); (ii) it is the generalized FDR procedure in Chen and Doerge (2014), designed for multiple testing based on discrete p-values, when ˜ π$_{0}$ ( λ ) is the estimator of π$_{0}$ proposed there and further studied in Chen and Doerge (2015); (iii) it is the BH procedure in Benjamini and Hochberg (1995) when λ = 0 is set in (4) in Storey’s procedure or ˜ π$_{0}$ ( λ ) ≡ 1 is set; (iv) it is a “dynamic adaptive FDR procedure” studied in Liang and Nettleton (2012) when λ in (4) is determined from the p-values and can be a random variable. 

## 3 Stopping time property of the rejection threshold 

For notational simplicity, we will from now on write t$_{α}$ ( ˜ F DR$_{λ}$ ) as ˜ t$_{α}$ ( λ ). Define the backward filtration 

<img src='content_image/304618.jpg'>

