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▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s ❛♥❞ t❤r❡s❤♦❧❞❡❞ ▲❛ss♦ ❢♦r ❤✐❣❤ ❞✐♠❡♥s✐♦♥❛❧ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ❋❛❝✉❧t② ♦❢ ❆♣♣❧✐❡❞ ■♥❢♦♠❛t✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝s✱ ❲❛rs❛✇ ❯♥✐✈❡rs✐t② ♦❢ ▲✐❢❡ ❙❝✐❡♥❝❡s ✭❙●●❲✮ ❇➛❞❧❡✇♦ ✶✳✶✷✳✷✵✶✻ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❖✉t❧✐♥❡ ▲✐♥❡❛r ♠♦❞❡❧ ✇✐t❤ ❙✉❜❣❛✉ss✐❛♥ ❡rr♦rs❀ ❚✇♦ st❡♣s s❡❧❡❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❚▲❙❉❀ ❙t❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡❀ ❚❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ s❡❧❡❝t✐♦♥ ♣r♦❝❡❞✉r❡❀ ❙✐♠✉❧❛t✐♦♥ st✉❞②✳ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

▲✐♥❡❛r ▼♦❞❡❧ Y = X β + ε ✱ ( ǫ i ) ✐✳✐✳❞✳✱ E ( ǫ ✶ ) = ✵✱ Var ( ǫ ✶ ) = σ ✷ ✱ ǫ ✶ ✐s ❙✉❜❣❛✉ss✐❛♥ ✇✐t❤ ❛ ❝♦♥st❛♥t σ > ✵ ✐✳❡✳ E ( exp ( u ǫ ✶ )) ≤ exp ( σ ✷ u ✷ / ✷ ) ❢♦r ❛❧❧ u ∈ R X ✲❞❡t❡r♠✐♥✐st✐❝ ♠❛tr✐① ♦❢ ❞✐♠❡♥s✐♦♥ n ① p ❀ β ∈ R p ❀ p = p ( n ) ❃ ❃ n ❀ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r ♦♥❧② s♣❛rs❡ ♠♦❞❡❧s ✭✜♥✐t❡ ♥✉♠❜❡r ♦❢ β i ✐s ❞✐✛❡r❡♥t ❢r♦♠ ✵✮ ❛♥❞ ♠❛tr✐① X s❛t✐s✜❡s s♦♠❡ r❡❣✉❧❛r ❝♦♥❞✐t✐♦♥s✳ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

▲❡❛st ❆❜s♦❧✉t❡ ❙❤r✐♥❦❛❣❡ ❛♥❞ ❙❝r❡❡♥✐♥❣ ❖♣❡r❛t♦r ✭▲❆❙❙❖✮ ▲❆❙❙❖ β L = argmin β ∈ R p � ✶ � � ✷ n � Y − X β � ✷ ✷ + λ n � β � ✶ ❢♦r s♦♠❡ λ n ❃✵ ❚❤r❡s❤♦❧❞❡❞ ▲❆❙❙❖ ✭❚▲✮ �� � � � � � β TL , j = � �� β L , j ✶ β L , j � ≥ δ n ❢♦r j = ✶ , ..., p ❢♦r s♦♠❡ t❤r❡s❤♦❧❞ δ n ❃✵❀ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❙♣❛rs❡ ▼♦❞❡❧ ❙♣❛rs❡ ▼♦❞❡❧ I ✵ = { j : β j � = ✵ } , I ✶ = { j : β j = ✵ } | I ✵ | = p ✵ < n p ✵ ✲✐s ❝♦♥st❛♥t✱ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ n ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❚✇♦ st❡♣s s❡❧❡❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❚▲❙❉ ❆t t❤❡ ✜rst st❡♣ ( TL ) � � � � � � �� ✇❡ ❝❤♦♦s❡ ❛ s❡t ♦❢ ✈❛r✐❛❜❧❡s S ✶ = ✶ ≤ j ≤ p : β L , j � ≥ δ n ❆t t❤❡ s❡❝♦♥❞ st❡♣ ✇❡ ✉s❡ ❛ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡ ( SD ) ♦❢ ♠✉❧t✐t❡st✐♥❣ ′ ( h ✵ ) ❍ i ✿ β i = ✵ ✈s✳ H i ✿ β i � = ✵ ❞❧❛ i ∈ S ✶ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❙t❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡ ′ ❉❡✜♥❡ ♣✲✈❛❧✉❡ ❢♦r ❤②♣♦t❤❡s✐s t❡st✐♥❣ H i ✈❡rs✉s H i ❛s π i = ✷ ( ✶ − Φ( | t i | )) ❢♦r i ∈ S ✶ ✱ ✇❤❡r❡ Φ ✐s t❤❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ N ( ✵ , ✶ ) . � � ❚❡st ❙t❛t✐st✐❝s t i = � � β ols , i / se β ols , i ✱ ✇❤❡r❡ � � − ✶ � ′ ′ X S ✶ X S ✶ X S ✶ Y β ols = � σ √ m i , i � � ✐❢ σ ✐s ❦♥♦✇♥ � β ols , i = se ❙ √ m i , i ✐❢ σ ✉♥❦♥♦✇♥ � � − ✶ X S ′ ✶ X S ✶ S ✲s♦♠❡ ❝♦♥s✐st❡♥t ❡st✐♠❛t♦r ♦❢ σ ❛♥❞ = ( m i , j ) i , j ∈ S ✶ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❙t❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡ ▲❡t s ✶ = | S ✶ | ❲❡ ❤❛✈❡ ♣✲✈❛❧✉❡s ♦r❞❡r❡❞ π ( ✶ ) ≤ ... ≤ π ( s ✶ ) ❛♥❞ r❡s♣❡❝t✐✈❡❧② ♥✉❧❧ ❤②♣♦t❤❡s❡s H ( ✶ ) ≤ ... ≤ H ( s ✶ ) α ✶ ≤ ... ≤ α s ✶ ✲s♦♠❡ ❝♦♥st❛♥ts ✭♠❛② ❞❡♣❡♥❞ ♦♥ n ✮ ■❢ π ( ✶ ) > α ✶ ✱ t❤❡♥ ✇❡ ❞♦ ♥♦t r❡❥❡❝t ❛♥② ❤②♣♦t❤❡s✐s H i ❀ ( h ✶ ) ♦t❤❡r✇✐s❡ ✐❢ π ( ✶ ) ≤ α ✶ , ..., π ( r ) ≤ α r t❤❡♥✱ ✇❡ r❡❥❡❝t H ( ✶ ) , ..., H ( r ) ✱ ✇❤❡r❡ r ✐s t❤❡ ❧❛r❣❡st ♥✉♠❜❡r s❛t✐s❢②✐♥❣ ( h ✶ ) . ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❊①❛♠♣❧❡s ♦❢ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡ ❊①❛♠♣❧❡s ♦❢ ❙❉ ♣r♦❝❡❞✉r❡s q n ❛ ) Holm α j = s ✶ + ✶ − j ([ γ j ]+ ✶ ) q n ❜ ) UHolm α j = s ✶ +[ γ j ]+ ✶ − j ❞❧❛ ✵ ≤ γ ≤ ✶ α j = jq n ❝ ) BH s ✶ α j = q n ❞ ) Bonferroni s ✶ ❢♦r s♦♠❡ q n → ✵ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❈♦♥❞✐t✐♦♥s ❢♦r α i ❢♦r st❡♣❞♦✇♥ ♣r♦❝✳ ( A ✶ ) α s ✶ → ✵ ❛s n → ∞ ( A ✷ ) ( ✶ / n ) ❧♦❣ ( ✶ /α ✶ ) → ✵ ❛s n → ∞ ❘❡♠❛r❦✳ ❋♦r q n = ✶ / ( n ❧♦❣ ( n )) t❤❡ ❝♦♥❞✐t✐♦♥s ( A ✶ ) − ( A ✷ ) ❛r❡ s❛t✐s✜❡❞ . ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❘❡❣✉❧❛r ❝♦♥❞✐t✐♦♥s ❘❡❣✉❧❛r ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ♠❛tr✐① X ( B ✶ ) � x j � ✷ / √ n = ✶ ❢♦r j = ✶ , ..., p ✱ ✇❤❡r❡ x j ✲❥t❤ ❝♦❧✉♠♥ ♦❢ ♠❛tr✐① X ( B ✷ ) let ✈ I ✶ = { v i : i ∈ I ✶ } ✱ v I ✵ = { v i : i ∈ I ✵ } ❀ C ( I ✵ ; ✸ ) = { v ∈ R p : � v I ✶ � ✶ ≤ ✸ � v I ✵ � ✶ } ❀ ❢♦r s♦♠❡ γ > ✵ t❤❡ ❜❡❧♦✇ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ′ X ′ X v ≥ γ � v � ✷ ( ✶ / n ) v ✷ ❢♦r ❡✈❡r② v ∈ C ( I ✵ ; ✸ ) ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❘❡❣✉❧❛r ❝♦♥❞✐t✐♦♥s r❡❣✉❧❛r ❝♦♥❞✐t✐♦♥s ❢♦r ♠♦❞❡❧ ❛♥❞ ❡st✐♠❛t♦rs ( B ✸ ) ♠✐♥ j ∈ I ✵ | β j | ≥ C ✶ λ n ❢♦r s♦♠❡ ❝♦♥st❛♥t C ✶ > ✵ ( B ✹ ) ❈ ✷ λ n ≤ δ n ≤ λ n ( C ✶ − ✸ /γ ) ❢♦r s♦♠❡ ❝♦♥st❛♥t C ✷ > ✵ � ( B ✺ ) λ n = C λ ❧♦❣ ( p ) / n ❢♦r C λ = ✷ σ ✳ ❝♦♥s✐st❡♥❝② ❝♦♥❞✐t✐♦♥ ❢♦r S ( C ) ❙ → P σ ❛s n → ∞ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s

❚❤❡ ❝♦♥s✐st❡♥❝② ♦❢ s❡❧❡❝t✐♦♥ ♣r♦❝❡❞✉r❡ ❈♦♥s✐st❡♥❝② ❚❤❡♦r❡♠ ❆♥② ❚▲❙❉ ♣r♦❝❡❞✉r❡ s❛t②s❢②✐♥❣✿ ( A ✶ ) − ( A ✷ ) , ( B ✶ ) − ( B ✺ ) ✐❢ σ ✐s ❦♥♦✇♥ ❛♥❞ ❛❞❞✐t✐♦♥❛❧❧② ( C ) ✐❢ σ ✐s ✉♥❦♥♦✇♥✱ ✐s ❝♦♥s✐st❡♥t ❢♦r s❡❧❡❝t✐♦♥ ♣r♦❜❧❡♠ ✐♥ ❧✐♥❡❛r ♠♦❞❡❧✳ ❚❤❡ ❝♦♥s✐st❡♥❝② ♦❢ s❡❧❡❝t✐♦♥ ♣r♦❝❡❞✉r❡ � � � ❆ s❡❧❡❝t✐♦♥ ♣r♦❝❡❞✉r❡ ✐s ❝♦♥s✐st❡♥t ✐❢ P I = I ✵ → ✶ ❛s n → ∞ ✱ ✇❤❡r❡ � I ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❝❤♦s❡♥ s✐❣♥✐✜❝❛♥t ✈❛r✐❛❜❧❡s✳ ❑♦♥r❛❞ ❋✉r♠❛➠❝③②❦ ▼♦❞❡❧ s❡❧❡❝t✐♦♥ ✇✐t❤ st❡♣❞♦✇♥ ♣r♦❝❡❞✉r❡s