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Sparse multiple testing: can one estimate the null distribution? Etienne Roquain 1 Joint work with A. Carpentier 2 , S. Delattre 3 , N. Verzelen 4 , 1 LPSM, Sorbonne Universit, France 2 Otto-von-Guericke-Universitt Magdeburg, Allemagne 3 LPSM,

Sparse multiple testing: can one estimate the null distribution? Etienne Roquain 1 Joint work with A. Carpentier 2 , S. Delattre 3 , N. Verzelen 4 , 1 LPSM, Sorbonne Université, France 2 Otto-von-Guericke-Universität Magdeburg, Allemagne 3 LPSM, Université de Paris, France 4 INRAE, Montpellier, France MMMS2 Luminy, 02/06/2020 Arxiv 1912.03109. "On using empirical null distributions in Benjamini-Hochberg procedure" To appear in AoS. "Estimating minimum effect with outlier selection " ANR “Sanssouci", ANR “BASICS", GDR ISIS “TASTY" Roquain, Etienne Sparse multiple testing 1 / 29
Introduction 1 Upper bound 2 Lower bound 3 Additional results 4 One-sided alternatives 5 Roquain, Etienne Sparse multiple testing 2 / 29
Motivation 1: null distribution unknown M67 photography, Package photutils Original Gaussian fitting Gumbel fitting ◮ Naive null distribution fitting ◮ Impact on the risk? Roquain, Etienne Sparse multiple testing Introduction 3 / 29
Motivation 1: null distribution unknown M67 photography, Package photutils Original Gaussian fitting Gumbel fitting ◮ Naive null distribution fitting ◮ Impact on the risk? Roquain, Etienne Sparse multiple testing Introduction 3 / 29
Motivation 2: null distribution wrong Figure 4 in [Efron (2008)] ◮ Empirical null [Efron (2004,2007,2008,2009)] ◮ Impact on the risk? Roquain, Etienne Sparse multiple testing Introduction 4 / 29
Motivation 2: null distribution wrong Figure 4 in [Efron (2008)] ◮ Empirical null [Efron (2004,2007,2008,2009)] ◮ Impact on the risk? Roquain, Etienne Sparse multiple testing Introduction 4 / 29
Existing work (selection) Estimation of the null: ◮ Series of work [Efron (2004,2007,2008,2009)] ◮ Minimax rate with Fourier analysis: [Jin and Cai (2007)]; [Cai and Jin (2010)] ◮ Two group mixture model: [Efron et al. (2001)]; [Sun and Cai (2009)]; [Cai and Sun (2009)]; [Padilla and Bickel (2012)]; [Nguyen and Matias (2014)]; [Heller and Yekutieli (2014)]; [Zablocki et al. (2017)]; [Amar et al. (2017)]; [Cai et al. (2019)]; [Rebafka et al. (2019)] ◮ Estimation in factor model: [Efron (2007a)]; [Leek and Storey (2008)]; [Friguet et al. (2009)]; [Fan et al. (2012)]; [Fan and Han (2017)] Impact on the risk: ◮ FDR control in symmetric, centered, one-sided case: [Barber and Candès (2015)]; [Arias-Castro and Chen (2017)] Lower bounds in multiple testing: ◮ [Arias-Castro and Chen (2017)]; [Rabinovich et al. (2017)]; [Castillo and R. (2020).] Roquain, Etienne Sparse multiple testing Introduction 5 / 29
Existing work (selection) Estimation of the null: ◮ Series of work [Efron (2004,2007,2008,2009)] ◮ Minimax rate with Fourier analysis: [Jin and Cai (2007)]; [Cai and Jin (2010)] ◮ Two group mixture model: [Efron et al. (2001)]; [Sun and Cai (2009)]; [Cai and Sun (2009)]; [Padilla and Bickel (2012)]; [Nguyen and Matias (2014)]; [Heller and Yekutieli (2014)]; [Zablocki et al. (2017)]; [Amar et al. (2017)]; [Cai et al. (2019)]; [Rebafka et al. (2019)] ◮ Estimation in factor model: [Efron (2007a)]; [Leek and Storey (2008)]; [Friguet et al. (2009)]; [Fan et al. (2012)]; [Fan and Han (2017)] Impact on the risk: ◮ FDR control in symmetric, centered, one-sided case: [Barber and Candès (2015)]; [Arias-Castro and Chen (2017)] Lower bounds in multiple testing: ◮ [Arias-Castro and Chen (2017)]; [Rabinovich et al. (2017)]; [Castillo and R. (2020).] Roquain, Etienne Sparse multiple testing Introduction 5 / 29
Existing work (selection) Estimation of the null: ◮ Series of work [Efron (2004,2007,2008,2009)] ◮ Minimax rate with Fourier analysis: [Jin and Cai (2007)]; [Cai and Jin (2010)] ◮ Two group mixture model: [Efron et al. (2001)]; [Sun and Cai (2009)]; [Cai and Sun (2009)]; [Padilla and Bickel (2012)]; [Nguyen and Matias (2014)]; [Heller and Yekutieli (2014)]; [Zablocki et al. (2017)]; [Amar et al. (2017)]; [Cai et al. (2019)]; [Rebafka et al. (2019)] ◮ Estimation in factor model: [Efron (2007a)]; [Leek and Storey (2008)]; [Friguet et al. (2009)]; [Fan et al. (2012)]; [Fan and Han (2017)] Impact on the risk: ◮ FDR control in symmetric, centered, one-sided case: [Barber and Candès (2015)]; [Arias-Castro and Chen (2017)] Lower bounds in multiple testing: ◮ [Arias-Castro and Chen (2017)]; [Rabinovich et al. (2017)]; [Castillo and R. (2020).] Roquain, Etienne Sparse multiple testing Introduction 5 / 29
Setting Observations Y = ( Y i ) 1 ≤ i ≤ n indep , Y i ∼ P i , parameter P = ( P i ) 1 ≤ i ≤ n ∈ P Gaussian null assumption: Most of the P i ’s equal N ( θ, σ 2 ) , for some unknown θ, σ Example: � � P 1 , N ( θ, σ 2 ) , P 3 , N ( θ, σ 2 ) , N ( θ, σ 2 ) , N ( θ, σ 2 ) , P 7 , N ( θ, σ 2 ) P = ◮ Ensures θ = θ ( P ) and σ = σ ( P ) uniquely defined ◮ Test H 0 , i : “ P i = N ( θ ( P ) , σ 2 ( P ))” against H 1 , i : “ P i � = N ( θ ( P ) , σ 2 ( P ))” “ item i comes from the background ” “ item i comes from signal ” Roquain, Etienne Sparse multiple testing Introduction 6 / 29
Setting Observations Y = ( Y i ) 1 ≤ i ≤ n indep , Y i ∼ P i , parameter P = ( P i ) 1 ≤ i ≤ n ∈ P Gaussian null assumption: Most of the P i ’s equal N ( θ, σ 2 ) , for some unknown θ, σ Example: � � P 1 , N ( θ, σ 2 ) , P 3 , N ( θ, σ 2 ) , N ( θ, σ 2 ) , N ( θ, σ 2 ) , P 7 , N ( θ, σ 2 ) P = ◮ Ensures θ = θ ( P ) and σ = σ ( P ) uniquely defined ◮ Test H 0 , i : “ P i = N ( θ ( P ) , σ 2 ( P ))” against H 1 , i : “ P i � = N ( θ ( P ) , σ 2 ( P ))” “ item i comes from the background ” “ item i comes from signal ” Roquain, Etienne Sparse multiple testing Introduction 6 / 29
Setting Observations Y = ( Y i ) 1 ≤ i ≤ n indep , Y i ∼ P i , parameter P = ( P i ) 1 ≤ i ≤ n ∈ P Gaussian null assumption: Most of the P i ’s equal N ( θ, σ 2 ) , for some unknown θ, σ Example: � � P 1 , N ( θ, σ 2 ) , P 3 , N ( θ, σ 2 ) , N ( θ, σ 2 ) , N ( θ, σ 2 ) , P 7 , N ( θ, σ 2 ) P = ◮ Ensures θ = θ ( P ) and σ = σ ( P ) uniquely defined ◮ Test H 0 , i : “ P i = N ( θ ( P ) , σ 2 ( P ))” against H 1 , i : “ P i � = N ( θ ( P ) , σ 2 ( P ))” “ item i comes from the background ” “ item i comes from signal ” Roquain, Etienne Sparse multiple testing Introduction 6 / 29
Criteria ◮ True null set H 0 ( P ) = { i : P satisfies H 0 , i } , n 0 ( P ) = |H 0 ( P ) | ◮ False null set H 1 ( P ) = H 0 ( P ) c , n 1 ( P ) = |H 1 ( P ) | ◮ for a procedure R ( Y ) ⊂ { 1 , . . . , n } FDP ( P , R ( Y )) = | R ( Y ) ∩ H 0 ( P ) | ’false discovery proportion’ | R ( Y ) | ∨ 1 E P [ FDP ( P , R ( Y ))] = FDR ( P , R ) ’false discovery rate’ TDP ( P , R ( Y )) = | R ( Y ) ∩ H 1 ( P ) | ’true discovery proportion’ n 1 ( P ) ∨ 1 E P [ TDP ( P , R ( Y ))] = TDR ( P , R ) ’true discovery rate’ ◮ Sparse multiple testing (enough background) n 1 ( P ) ≤ k n with k n ’small’ Roquain, Etienne Sparse multiple testing Introduction 7 / 29
Criteria ◮ True null set H 0 ( P ) = { i : P satisfies H 0 , i } , n 0 ( P ) = |H 0 ( P ) | ◮ False null set H 1 ( P ) = H 0 ( P ) c , n 1 ( P ) = |H 1 ( P ) | ◮ for a procedure R ( Y ) ⊂ { 1 , . . . , n } FDP ( P , R ( Y )) = | R ( Y ) ∩ H 0 ( P ) | ’false discovery proportion’ | R ( Y ) | ∨ 1 E P [ FDP ( P , R ( Y ))] = FDR ( P , R ) ’false discovery rate’ TDP ( P , R ( Y )) = | R ( Y ) ∩ H 1 ( P ) | ’true discovery proportion’ n 1 ( P ) ∨ 1 E P [ TDP ( P , R ( Y ))] = TDR ( P , R ) ’true discovery rate’ ◮ Sparse multiple testing (enough background) n 1 ( P ) ≤ k n with k n ’small’ Roquain, Etienne Sparse multiple testing Introduction 7 / 29
Criteria ◮ True null set H 0 ( P ) = { i : P satisfies H 0 , i } , n 0 ( P ) = |H 0 ( P ) | ◮ False null set H 1 ( P ) = H 0 ( P ) c , n 1 ( P ) = |H 1 ( P ) | ◮ for a procedure R ( Y ) ⊂ { 1 , . . . , n } FDP ( P , R ( Y )) = | R ( Y ) ∩ H 0 ( P ) | ’false discovery proportion’ | R ( Y ) | ∨ 1 E P [ FDP ( P , R ( Y ))] = FDR ( P , R ) ’false discovery rate’ TDP ( P , R ( Y )) = | R ( Y ) ∩ H 1 ( P ) | ’true discovery proportion’ n 1 ( P ) ∨ 1 E P [ TDP ( P , R ( Y ))] = TDR ( P , R ) ’true discovery rate’ ◮ Sparse multiple testing (enough background) n 1 ( P ) ≤ k n with k n ’small’ Roquain, Etienne Sparse multiple testing Introduction 7 / 29
Criteria ◮ True null set H 0 ( P ) = { i : P satisfies H 0 , i } , n 0 ( P ) = |H 0 ( P ) | ◮ False null set H 1 ( P ) = H 0 ( P ) c , n 1 ( P ) = |H 1 ( P ) | ◮ for a procedure R ( Y ) ⊂ { 1 , . . . , n } FDP ( P , R ( Y )) = | R ( Y ) ∩ H 0 ( P ) | ’false discovery proportion’ | R ( Y ) | ∨ 1 E P [ FDP ( P , R ( Y ))] = FDR ( P , R ) ’false discovery rate’ TDP ( P , R ( Y )) = | R ( Y ) ∩ H 1 ( P ) | ’true discovery proportion’ n 1 ( P ) ∨ 1 E P [ TDP ( P , R ( Y ))] = TDR ( P , R ) ’true discovery rate’ ◮ Sparse multiple testing (enough background) n 1 ( P ) ≤ k n with k n ’small’ Roquain, Etienne Sparse multiple testing Introduction 7 / 29

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