Journal of Probability and Statistics

Research Article

Control of the False Discovery Proportion for Independently Tested Null Hypotheses

Algorithm 2

Focused simultaneous inferences on the UPBs for the false discovery proportion and the UCB for π‘š 0 .
For any given 𝛼 and 𝛾 , choose 𝑑 0 = 0 . 0 1 / π‘š , 𝑑 ξ…ž 0 = 0 . 0 5 , πœ† 0 = 0 . 8 , πœ† ξ…ž 0 = 0 . 9 5 .
( 1 ) Compute  π‘š βˆ— 0 , 1 βˆ’ 𝛼 ( πœ† ) of (4.2) for some pre-specified πœ† 𝑖 ’s in the region ( πœ† 0 , πœ† ξ…ž 0 )
 and pre-specified 𝑑 𝑖 ’s in the region ( 𝑑 0 , 𝑑 ξ…ž 0 ) ,
 say πœ† 𝑖 = πœ† 0 + ( πœ† ξ…ž 0 βˆ’ πœ† 0 ) 𝑖 / 1 0 0 0 , 𝑑 𝑖 = 𝑑 0 + ( 𝑑 ξ…ž 0 βˆ’ 𝑑 0 ) 𝑖 / 1 0 0 0 for 𝑖 = 0 , … , 1 0 0 0 .
( 2 ) Compute  π‘š βˆ— 0 , 1 βˆ’ 𝛼 = m i n ( m i n 𝑖  π‘š βˆ— 0 , 1 βˆ’ 𝛼 ( πœ† 𝑖 ) , m i n 𝑖  π‘š βˆ— 0 , 1 βˆ’ 𝛼 ( 𝑑 𝑖 ) ) .
 This  π‘š βˆ— 0 , 1 βˆ’ 𝛼 is the 1 βˆ’ 𝛼 UCB for π‘š 0 . If  π‘š βˆ— 0 , 1 βˆ’ 𝛼 exceeds π‘š , replace it by π‘š .
( 3 ) Sort the observed 𝑃 -values such that 𝑃 ( 1 ) ≀ β‹― ≀ 𝑃 ( π‘š ) ,
 and use (4.4) to compute the 1 βˆ’ 𝛼 UPB for the false discovery proportion 𝑄 ,
 that is, for 𝑃 ( 𝑖 ) ∈ [ 𝑑 0 , 𝑑 ξ…ž 0 ]
 𝑄 βˆ— 1 βˆ’ 𝛼 ( 𝑃 ( 𝑖 ) ) = ( 1 / 𝑖 ) (  π‘š βˆ— 0 , 1 βˆ’ 𝛼 𝑃 ( 𝑖 ) + Μƒ 𝑧 βˆ— 1 βˆ’ 𝛼 (  π‘š βˆ— 0 , 1 βˆ’ 𝛼 )   π‘š βˆ— 0 , 1 βˆ’ 𝛼 𝑃 ( 𝑖 ) ( 1 βˆ’ 𝑃 ( 𝑖 ) ) ) .
 If  𝑄 βˆ— 1 βˆ’ 𝛼 ( 𝑃 ( 𝑖 ) ) exceeds 1, replace it by 1.
( 4 ) Compute 𝜏 = m a x { 𝑃 ( 𝑖 ) ∈ [ 𝑑 0 , 𝑑 ξ…ž 0  𝑄 ] ∢ βˆ— 1 βˆ’ 𝛼 ( 𝑃 ( 𝑖 ) ) ≀ 𝛾 } ,
 reject the hypotheses whose 𝑃 -values are no greater than 𝜏 ,
 which ensures that the false discovery proportion 𝑄 is not exceeding 𝛾 with
 probability 1 βˆ’ 𝛼 .