❆❞❛♣t✐✈❡ ❙❡♥s✐♥❣ ❢♦r ❘❡❝♦✈❡r✐♥❣ ❙tr✉❝t✉r❡❞ ❙♣❛rs❡ ❙❡ts ❊r✈✐♥ ❚á♥❝③♦s✱ ❘✉✐ ❈❛str♦ ❊✐♥❞❤♦✈❡♥ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② ❙tr✉❝t✉r❡s ❙❡♠✐♥❛r ✶✻✳✶✵✳✷✵✶✺
❙✉♣♣♦rt ❘❡❝♦✈❡r②✴❉❡t❡❝t✐♦♥ ▼♦t✐✈❛t✐♦♥ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ r❡❝♦✈❡r✐♥❣ t❤❡ s✉♣♣♦rt ✭♦r ❞❡t❡❝t✐♥❣ t❤❡ ♣r❡s❡♥❝❡✮ ♦❢ ❛♥ ✉♥❦♥♦✇♥ s✐❣♥❛❧✳
❙✉♣♣♦rt ❘❡❝♦✈❡r②✴❉❡t❡❝t✐♦♥ ❈❧❛ss✐❝❛❧ ❋r❛♠❡✇♦r❦ ▲❡t ① = ( x ✶ , . . . , x n ) T ∈ R n ❞❡♥♦t❡ t❤❡ ✉♥❦♥♦✇♥ s✐❣♥❛❧ ✇❤❡r❡ � µ ✱ ✐❢ i ∈ S x i = , ∈ S ✵ ✱ ✐❢ i / ✇✐t❤ µ > ✵ ✜①❡❞ ❛♥❞ S ∈ C ✇❤❡r❡ C ✐s ❛ ❝❧❛ss ♦❢ s❡ts✳
❙✉♣♣♦rt ❘❡❝♦✈❡r②✴❉❡t❡❝t✐♦♥ ❈❧❛ss✐❝❛❧ ❋r❛♠❡✇♦r❦ ❲❡ ♦❜s❡r✈❡ iid Y i = x i + W i , W i ∼ N ( ✵ , ✶ ) , i = ✶ , . . . , n . ❖✉r ❣♦❛❧ ✐s t♦ r❡❝♦✈❡r S ♦r t♦ ❞❡t❡❝t ✐ts ♣r❡s❡♥❝❡ ✭❞❡❝✐❞❡ ❜❡t✇❡❡♥ H ✵ : S = ∅ ❛♥❞ H ✶ : ∅ � = S ∈ C ✮✳ ❍♦✇ ❞♦❡s µ ♥❡❡❞ t♦ s❝❛❧❡ s♦ t❤❛t t❤❡ ❛❜♦✈❡ t❛s❦s ❛r❡ ♣♦ss✐❜❧❡❄
❙✉♣♣♦rt ❘❡❝♦✈❡r②✴❉❡t❡❝t✐♦♥ ❈❧❛ss✐❝❛❧ ❋r❛♠❡✇♦r❦ ❉❡♣❡♥❞s ♦♥ t❤❡ ❝❧❛ss C ✳ ❋r♦♠ ♥♦✇ ♦♥✱ ❛ss✉♠❡ | S | = s ≪ n ∀ S ∈ C ✭s♣❛rs❡ s✐❣♥❛❧s✮✳ ❲❡ ✇❛♥t ♠❛① S ∈C P ( Error ) t♦ ❜❡ s♠❛❧❧✳ � ♠❛① X i ❞♦❡s t❤❡ ❥♦❜ ✭✐♥ t❤❡ s♣❛rs❡ r❡❣✐♠❡✮✳ S ∈C i ∈ S ❙❡❡ ❡✳❣✳ ▲✉❣♦s✐ ❡t✳ ❛❧✳ ✭✷✵✶✵✮✿ ❖♥ ❝♦♠❜✐♥❛t♦r✐❛❧ t❡st✐♥❣ ♣r♦❜❧❡♠s❀ ❆r✐❛s✲❈❛str♦ ❡t✳ ❛❧✳ ✭✷✵✶✶✮✿ ❉❡t❡❝t✐♦♥ ♦❢ ❛♥ ❆♥♦♠❛❧♦✉s ❈❧✉st❡r ✐♥ ❛ ◆❡t✇♦r❦
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ▲❡❛r♥✐♥❣ t♦ ❧❡❛r♥ • ❍♦✇ ❝❛♥ ✇❡ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ❢❡❡❞❜❛❝❦❄ • ❍♦✇ ♠✉❝❤ ❝❛♥ ✇❡ ❣❛✐♥❄
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❋r❛♠❡✇♦r❦ ❚❤❡ ✉♥❦♥♦✇♥ s✐❣♥❛❧ ❛♥❞ t❤❡ ❣♦❛❧s ❛r❡ t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✳ ▼❡❛s✉r❡♠❡♥t ♠♦❞❡❧✿ iid Y t = x A t + W t , W t ∼ N ( ✵ , ✶ ) , t = ✶ , . . . , n , ✇❤❡r❡ A t ∈ { ✶ , . . . , n } ❝❛♥ ❞❡♣❡♥❞ ♦♥ ♣❛st ♦❜s❡r✈❛t✐♦♥s { A j , Y j } t − ✶ j = ✶ ✳
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❋r❛♠❡✇♦r❦ ❚❤❡ ✉♥❦♥♦✇♥ s✐❣♥❛❧ ❛♥❞ t❤❡ ❣♦❛❧s ❛r❡ t❤❡ s❛♠❡ ❛s ❜❡❢♦r❡✳ ▼❡❛s✉r❡♠❡♥t ♠♦❞❡❧✿ Y t = x A t + Γ − ✶ / ✷ iid W t , W t ∼ N ( ✵ , ✶ ) , t = ✶ , ✷ , . . . , t ✇❤❡r❡ A t ∈ { ✶ , . . . , n } , Γ t > ✵ ❝❛♥ ❞❡♣❡♥❞ ♦♥ ♣❛st ♦❜s❡r✈❛t✐♦♥s { A j , Γ j , Y j } t − ✶ j = ✶ ✱ ❛♥❞ ♠✉st s❛t✐s❢② �� � Γ t ≤ n , ∀ S ∈ C . E S t
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❙✐♠♣❧❡ ♣r♦❝❡❞✉r❡ ❢♦r r❡❝♦✈❡r② ▲❡t C ❜❡ t❤❡ ❝❧❛ss ♦❢ ❛❧❧ s ✲s♣❛rs❡ s❡ts ❛♥❞ s✉♣♣♦s❡ ✇❡ ✇✐s❤ t♦ r❡❝♦✈❡r t❤❡ s✉♣♣♦rt ✭✇❡ ✇❛♥t � S ∈C P S ( � S s✳t✳ ♠❛① S � = S ) ≤ ε ✮✳ ❆❧❣♦r✐t❤♠ • ❋✐① Γ t = Γ = ✶ / ✸ ∀ t ∈ N • ❋♦r ❡❛❝❤ ❡♥tr② x i , i = ✶ , . . . , n ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿ • ▼❡❛s✉r❡ Y i , j = x i + Γ − ✶ / ✷ W i , j = ✶ , . . . , τ i ✱ ✇❤❡r❡ τ i = ♠✐♥ { j : Y i , j ≤ ✵ } ∧ ❧♦❣ ✷ ( n /ε ) ✳ • i ∈ � S ⇐ ⇒ Y i , j > ✵ ∀ j = ✶ , . . . , ❧♦❣ ✷ ( n /ε ) ✳
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❙✐♠♣❧❡ ♣r♦❝❡❞✉r❡ ❢♦r r❡❝♦✈❡r② ✲ ❛♥❛❧②s✐s ∈ S ❋♦r i / S ) = P ( Y i , j > ✵ ∀ j ) ≤ ( ✶ / ✷ ) ❧♦❣ ✷ ( n /ε ) = ε/ n . P ( i ∈ � ❋♦r i ∈ S S ) ≤ P ( ∃ j : Y i , j ≤ ✵ ) ≤ ❧♦❣ ✷ ( n /ε ) e − µ ✷ / ✻ ≤ ε/ s ∈ � P ( i / ✷ � � � ε + ❧♦❣ ❧♦❣ ✷ ( n /ε ) ❧♦❣ s ✇❤❡♥❡✈❡r µ ≥ ✻ ✳ ✷ ❍❡♥❝❡ � � P S ( � P ( i ∈ � ∈ � S � = S ) ≤ S ) + P ( i / S ) ≤ ε . i / ∈ S i ∈ S
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❙✐♠♣❧❡ ♣r♦❝❡❞✉r❡ ❢♦r r❡❝♦✈❡r② ✲ ❛♥❛❧②s✐s ❍♦✇ ♠✉❝❤ ♣r❡❝✐s✐♦♥ ❞♦ ✇❡ ✉s❡ ✐♥ ❡①♣❡❝t❛t✐♦♥❄ �� � �� � � ≤ ✶ ≤ Γ ✸ ( ✷ n + s ❧♦❣ ( n /ε )) ≤ n E S Γ t ✷ + ❧♦❣ ✷ ( n /ε ) t i ∈ S i / ∈ S ✐❢ s ≪ n ✳ ❚♦ s✉♠♠❛r✐③❡✱ t❤✐s s✐♠♣❧❡ ♣r♦❝❡❞✉r❡ s✉❝❝❡❡❞s ✇❤❡♥ � ❧♦❣ s + ❧♦❣ ❧♦❣ n + ❧♦❣ ✶ µ � ε .
❘❡♠✐♥❞❡r ✲ ❙▲❘❚ ✭❲❛❧❞✮ ❲❡ ✇✐s❤ t♦ t❡st H ✵ : Y j ∼ N ( ✵ , Γ − ✶ ) ❛♥❞ H ✶ : Y j ∼ N ( µ, Γ − ✶ ) ✱ j ∈ N ✇✐t❤ ❛s ❢❡✇ ♦❜s❡r✈❛t✐♦♥s ❛s ♣♦ss✐❜❧❡ ✭✐♥ ❡①♣❡❝t❛t✐♦♥✮ ✇✐t❤ ♣r❡s❝r✐❜❡❞ ❡rr♦r ♣r♦❜❛❜✐❧✐t✐❡s α, β ✳ ❈♦♥s✐❞❡r t❤❡ ♣r♦❝❡ss t � ❧♦❣ f ✶ ( Y i ) Z ✵ = ✵ , Z t = f ✵ ( Y i ) , t = ✶ , ✷ , . . . j = ✶ ▲❡t l = ❧♦❣ β < ✵ < u = ❧♦❣ ( ✶ /α ) ❛♥❞ T = ✐♥❢ { t : Z t / ∈ ( l , u ) } ✳ ❲❡ t❤❡♥ ❤❛✈❡ P ✵ ( Z T ≥ u ) ≤ α ❛♥❞ P ✶ ( Z T ≤ l ) ≤ β ✳ ❆s Γ → ✵ ✇❡ ❛❧s♦ ❤❛✈❡ • P ✵ ( Z T ≥ u ) → α ❛♥❞ P ✶ ( Z T ≤ l ) → β Γ µ ✷ ❧♦❣ ✶ ✷ Γ µ ✷ ❧♦❣ ✶ ✷ • E ✵ ( T ) ≈ β ❛♥❞ E ✶ ( T ) ≈ α
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❘❡✜♥❡♠❡♥t ❘❡♣❧❛❝❡ t❤❡ ❝♦r❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s ♣r♦❝❡❞✉r❡ ✇✐t❤ ❛ ❙▲❘❚ t♦ t❡st ❜❡t✇❡❡♥ x i = ✵ ❛♥❞ x i = µ ✳ ❙❡t ❚②♣❡ ■ ❛♥❞ ■■ ❡rr♦r ♣r♦❜❛❜✐❧✐t✐❡s t♦ ❜❡ α = ε/ n ❛♥❞ β = ε/ s ✳ ❲❡ ❤❛✈❡ P S ( � S � = S ) ≤ ε ❛s ❜❡❢♦r❡✳ ❚❤❡ ♣r❡❝✐s✐♦♥ ✉s❡❞ ✭✐♥ ❡①♣❡❝t❛t✐♦♥✮ ✐s �� � � � ≤ ✷ n ❧♦❣ s ε + s ❧♦❣ n Γ t . E S µ ✷ ε t ■❢ n ✐s ❧❛r❣❡ ✭❛♥❞ s ≪ n ✮ t❤✐s ✐s ❛t ♠♦st n ✇❤❡♥❡✈❡r � ✷ ❧♦❣ s µ ≥ ε + o ( ✶ ) . ❚❤✐s ✐s ♦♣t✐♠❛❧✳
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❉❡t❡❝t✐♦♥ √ ε ✳ ❚❤✐s ❲❤❛t ❛❜♦✉t ❞❡t❡❝t✐♦♥❄ ❊❛s②✿ s❡t α ❛s ❜❡❢♦r❡ ❛♥❞ β = s ❡♥s✉r❡s t❤❛t ❛t ❧❡❛st ♦♥❡ s✐❣♥❛❧ ❝♦♠♣♦♥❡♥t ✐s ❢♦✉♥❞ ✇✳♣✳ ✶ − ε ✉♥❞❡r t❤❡ ❛❧t❡r♥❛t✐✈❡✳ ❙❝❛❧✐♥❣ ❧❛✇s ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ str✉❝t✉r❡ ❛♥②♠♦r❡ ✭❛s ❧♦♥❣ ❛s ✇❡ ❤❛✈❡ s②♠♠❡tr② ✐♥ t❤❡ ❝❧❛ss✮✦
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❙tr✉❝t✉r❡❞ ❘❡❝♦✈❡r② ❋♦r ❝❡rt❛✐♥ ❝❧❛ss❡s ✐t ✐s ❡♥♦✉❣❤ t♦ ✜♥❞ ♦♥❡ ❝♦♠♣♦♥❡♥t ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ✧❡❛s②✧✳ ❋♦r ✐♥st❛♥❝❡✱ ✐❢ t❤❡r❡ ✇❛s ♥♦ ♥♦✐s❡✿
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❙tr✉❝t✉r❡❞ ❘❡❝♦✈❡r② ❋♦r ❝❡rt❛✐♥ ❝❧❛ss❡s ✐t ✐s ❡♥♦✉❣❤ t♦ ✜♥❞ ♦♥❡ ❝♦♠♣♦♥❡♥t ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ✧❡❛s②✧✳ ❋♦r ✐♥st❛♥❝❡✱ ✐❢ t❤❡r❡ ✇❛s ♥♦ ♥♦✐s❡✿
❆❞❛♣t✐✈❡ s❡♥s✐♥❣ ❙tr✉❝t✉r❡❞ ❘❡❝♦✈❡r② ❋♦r ❝❡rt❛✐♥ ❝❧❛ss❡s ✐t ✐s ❡♥♦✉❣❤ t♦ ✜♥❞ ♦♥❡ ❝♦♠♣♦♥❡♥t ❛♥❞ t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s ✧❡❛s②✧✳ ❋♦r ✐♥st❛♥❝❡✱ ✐❢ t❤❡r❡ ✇❛s ♥♦ ♥♦✐s❡✿
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