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■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠② ❖♣❡r❛t✐♦♥❛❧ r✐s❦ ❆ ❢✉rt❤❡r r✐s❦ ❝❛t❡❣♦r② ✐s ♦♣❡r❛t✐♦♥❛❧ r✐s❦✿ t❤❡ r✐s❦ ♦❢ ❧♦ss❡s r❡s✉❧t✐♥❣ ❢r♦♠ ✐♥❛❞❡q✉❛t❡ ♦r ❢❛✐❧❡❞ ✐♥t❡r♥❛❧ ♣r♦❝❡ss❡s✱ ♣❡♦♣❧❡ ❛♥❞ s②st❡♠s✱ ♦r ❢r♦♠ ❡①t❡r♥❛❧ ❡✈❡♥ts✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ ❘❡st✐ ❛♥❞ ❙✐r♦♥✐ ✭✷✵✵✼✱ P❛rt ■❱✮ ❚❤❡ ❧♦ss r❡s✉❧t❡❞ ❢r♦♠ ✉♥❛✉t❤♦r✐s❡❞ s♣❡❝✉❧❛t✐✈❡ tr❛❞✐♥❣ ✐♥ ✈❛r✐♦✉s ❙✫P ✺✵✵✱ ❉❛① ❛♥❞ ❊✉r♦st♦①① ✐♥❞❡① ❢✉t✉r❡s ♦✈❡r t❤❡ ♣❛st t❤r❡❡ ♠♦♥t❤s✳ ❚❤❡ ♣♦s✐t✐♦♥s ❤❛❞ ❜❡❡♥ ♦✛s❡t ✐♥ ♦✉r s②st❡♠s ✇✐t❤ ✜❝t✐t✐♦✉s✱ ❢♦r✇❛r❞✲s❡tt❧✐♥❣✱ ❝❛s❤ ❊❚❋ ♣♦s✐t✐♦♥s✱ ❛❧❧❡❣❡❞❧② ❡①❡❝✉t❡❞ ❜② t❤❡ tr❛❞❡r✳ ❚❤❡s❡ ✜❝t✐t✐♦✉s tr❛❞❡s ❝♦♥❝❡❛❧❡❞ t❤❡ ❢❛❝t t❤❛t t❤❡ ✐♥❞❡① ❢✉t✉r❡s tr❛❞❡s ✈✐♦❧❛t❡❞ ❯❇❙✬s r✐s❦ ❧✐♠✐ts✳ ✭❯❇❙✱ ✶✽ ❙❡♣t❡♠❜❡r ✷✵✶✶✮ ✶✵ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠② ▼♦❞❡❧ r✐s❦ ▼♦❞❡❧ r✐s❦ ♠❛♥❛❣❡♠❡♥t ❤❛s ❜❡❝♦♠❡ ❛ ❜♦❛r❞✲❧❡✈❡❧ ♣r♦❝❡ss✳ ◆♦✇ t❤❡ ❝❤✐❡❢ r✐s❦ ♦✣❝❡r ❤❛s t♦ ❣♦ t♦ t❤❡ ❜♦❛r❞ ❛♥❞ ♥♦t ♦♥❧② t❛❧❦ ❛❜♦✉t ♠❛r❦❡t r✐s❦✱ ❝r❡❞✐t r✐s❦ ❛♥❞ ♦♣❡r❛t✐♦♥❛❧ r✐s❦✱ ❤❡ ❛❧s♦ ❤❛s t♦ t❛❧❦ ❛❜♦✉t ♠♦❞❡❧ r✐s❦✳ ■t ✐s ❛ ❤✉❣❡ ♦r❣❛♥✐s❛t✐♦♥❛❧ ❝❤❛♥❣❡✳ ✭◆❡✇ ❨♦r❦✲❜❛s❡❞ ♠♦❞❡❧ r✐s❦ ♠❛♥❛❣❡r❀ ❙❤❡r✐❢ ✭✷✵✶✻✮✮ ❛r✐s❡s ❢r♦♠ ❛ ♠✐ss♣❡❝✐✜❡❞ ♠♦❞❡❧ ❡✳❣✳ ✉s✐♥❣ ❇❧❛❝❦✲❙❝❤♦❧❡s ✇❤❡♥ ♠♦❞❡❧ ❛ss✉♠♣t✐♦♥s ❞♦♥✬t ❤♦❧❞ ✭❡✳❣✳ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ r❡t✉r♥s✮ ✏❛❧✇❛②s ♣r❡s❡♥t t♦ s♦♠❡ ❞❡❣r❡❡✑ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ q✳✈✳ ❘❡❜♦♥❛t♦ ✭✷✵✵✼✮✱ ❙✉♣❡r✈✐s♦r② ●✉✐❞❛♥❝❡ ♦♥ ♠♦❞❡❧ r✐s❦ ♠❛♥❛❣❡♠❡♥t ✭✷✵✶✶✮✱ ▼♦r✐♥✐ ✭✷✵✶✶✮ ✶✶ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❆ st❛♥❞❛r❞ r✐s❦ t❛①♦♥♦♠② ▲✐q✉✐❞✐t② r✐s❦ ❲❤❡♥ ✇❡ t❛❧❦ ❛❜♦✉t ❧✐q✉✐❞✐t② r✐s❦ ✇❡ ❛r❡ ❣❡♥❡r❛❧❧② r❡❢❡rr✐♥❣ t♦ ♣r✐❝❡ ♦r ♠❛r❦❡t ❧✐q✉✐❞✐t② r✐s❦✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❜r♦❛❞❧② ❞❡✜♥❡❞ ❛s t❤❡ r✐s❦ st❡♠♠✐♥❣ ❢r♦♠ t❤❡ ❧❛❝❦ ♦❢ ♠❛r❦❡t❛❜✐❧✐t② ♦❢ ❛♥ ✐♥✈❡st♠❡♥t t❤❛t ❝❛♥♥♦t ❜❡ ❜♦✉❣❤t ♦r s♦❧❞ q✉✐❝❦❧② ❡♥♦✉❣❤ t♦ ♣r❡✈❡♥t ♦r ♠✐♥✐♠✐③❡ ❛ ❧♦ss✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ ■♥ ❜❛♥❦✐♥❣✱ t❤❡r❡ ✐s ❛❧s♦ t❤❡ ❝♦♥❝❡♣t ♦❢ ❢✉♥❞✐♥❣ ❧✐q✉✐❞✐t② r✐s❦✱ ✇❤✐❝❤ r❡❢❡rs t♦ t❤❡ ❡❛s❡r ✇✐t❤ ✇❤✐❝❤ ✐♥st✐t✉t✐♦♥s ❝❛♥ r❛✐s❡ ❢✉♥❞✐♥❣ t♦ ♠❛❦❡ ♣❛②♠❡♥ts ❛♥❞ ♠❡❡t ✇✐t❤❞r❛✇❛❧s ❛s t❤❡② ❛r✐s❡✳ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✺✮ ✶✷ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛ ❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛ ✶ ❢♦r ❡❛❝❤ s❡❝✉r✐t②✱ ✐❞❡♥t✐❢② t❤❡ ✐✐❞ st♦❝❤❛st✐❝ t❡r♠s ✭➓✸✳✶✮ ✷ ❡st✐♠❛t❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✭➓✹✮ ✸ ♣r♦❥❡❝t t❤❡ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ✭➓✸✳✷✮ ✹ ❞✐♠❡♥s✐♦♥ r❡❞✉❝❡ t♦ ♠❛❦❡ t❤❡ ♣r♦❜❧❡♠ ♠♦r❡ tr❛❝t❛❜❧❡ ✭➓✸✳✹✮ ✺ ❡✈❛❧✉❛t❡ t❤❡ ♣♦rt❢♦❧✐♦ ♣❡r❢♦r♠❛♥❝❡ ❛t t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ✭➓✺✮ ✇❤❛t ✐s ②♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥❄ ✻ ♣✐❝❦ t❤❡ ♣♦rt❢♦❧✐♦ t❤❛t ♦♣t✐♠✐s❡s ②♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✭➓✻✮ ✼ ❛❝❝♦✉♥t ❢♦r ❡st✐♠❛t✐♦♥ r✐s❦ r❡♣❧❛❝❡ ♣♦✐♥t ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✇✐t❤ ❇❛②❡s✐❛♥ ❞✐str✐❜✉t✐♦♥s ✭➓✼✮ ✶ r❡✲❡✈❛❧✉❛t❡ t❤❡ ♣♦rt❢♦❧✐♦ ❞✐str✐❜✉t✐♦♥s ✐♥ t❤✐s ❧✐❣❤t ✭➓✽✮ ✷ r♦❜✉st❧② r❡✲♦♣t✐♠✐s❡ ✭➓✾✮ ✸ ❖❜s❡r✈❛t✐♦♥ s❤♦✇s t❤❛t s♦♠❡ st❛t✐st✐❝❛❧ ❢r❡q✉❡♥❝✐❡s ❛r❡✱ ✇✐t❤✐♥ ♥❛rr♦✇❡r ♦r ✇✐❞❡r ❧✐♠✐ts✱ st❛❜❧❡✳ ❇✉t st❛❜❧❡ ❢r❡q✉❡♥❝✐❡s ❛r❡ ♥♦t ✈❡r② ❝♦♠♠♦♥✱ ❛♥❞ ❝❛♥♥♦t ❜❡ ❛ss✉♠❡❞ ❧✐❣❤t❧②✳ ❑❡②♥❡s ✭✶✾✷✶✱ ♣✳✸✽✶✮ ✶✹ ✴ ✷✷✹

■♥tr♦❞✉❝t✐♦♥ ❚❤❡ ▼❡✉❝❝✐ ♠❛♥tr❛ ◆♦t❛t✐♦♥❛❧ ❝♦♥✈❡♥t✐♦♥s τ ✱ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ T ✱ t✐♠❡ ❛t ✇❤✐❝❤ t❤❡ ❛❧❧♦❝❛t✐♦♥ ❞❡❝✐s✐♦♥ ✐s ♠❛❞❡ t❤✉s✱ T + τ ✐s ✇❤❡♥ t❤❡ ✐♥✈❡st♠❡♥ts ❛r❡ t♦ ❜❡ ❡✈❛❧✉❛t❡❞ P t ✱ t❤❡ ✈❡❝t♦r ♦❢ ♣r✐❝❡s ❛t t✐♠❡ t X t ✱ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ t❤❛t ✇✐❧❧ r❡❛❧✐s❡ ❛t t✐♠❡ t x t ✱ ❛ r❡❛❧✐s❛t✐♦♥ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ i T ≡ { x ✶ , . . . , x T } ✱ ❛ ❞❛t❛s❡t ♦❢ ♦❜s❡r✈❡❞ r❡❛❧✐s❛t✐♦♥s ✶✺ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛ ♥✉♠❜❡r ✇❤♦s❡ r❡❛❧✐s❛t✐♦♥ ✐s✱ ❛s ②❡t✱ ✉♥❦♥♦✇♥ ✐ts ❞✐str✐❜✉t✐♦♥ ♠❛② ❜❡ ❦♥♦✇♥ ❛ s♣❛❝❡ ♦❢ ❡✈❡♥ts✱ E ❛ ♣r♦❜❛❜✐❧✐t② ❞✐str✐❜✉t✐♦♥✱ P ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ t❤❡ s♣❛❝❡ ♦❢ ❡✈❡♥ts t♦ t❤❡ r❡❛❧ ❧✐♥❡ t❤✉s✱ x = X ( e ) ❢♦r s♦♠❡ ❡✈❡♥t e ✐♥ E t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ❛♥ ❡✈❡♥t ❣✐✈✐♥❣ r✐s❡ t♦ ❛ r❡❛❧✐s❡❞ x ∈ [ x , ¯ x ] ✿ x ] } ≡ P { e ∈ E s✳t✳ X ( e ) ∈ [ x , ¯ P { X ∈ [ x , ¯ x ] } . r❡❛❞s✿ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t r❛♥❞♦♠ ✈❛r✐❛❜❧❡ X t❛❦❡s ♦♥ ❛ ✈❛❧✉❡ ✐♥ [ x , ¯ x ] ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ t❤❡ s❡t ♦❢ ❡✈❡♥ts ②✐❡❧❞✐♥❣ ❛ r❡❛❧✐s❡❞ ✈❛❧✉❡ ♦❢ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✳ ✳ ✳ ❣♦✐♥❣ ❢♦r✇❛r❞✱ t②♣✐❝❛❧❧② s✉♣♣r❡ss ❞❡♣❡♥❞❡♥❝❡ ♦♥ e ✱ r❡❢❡r ❥✉st t♦ X ♥❛ï✈❡ st❛t❡♠❡♥t ✶✼ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ Pr♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✭P❉❋✮✱ f X ❊①❛♠♣❧❡ ✶ ✶ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ r✈ X √ π e − x ✷ f X ( x ) = t❛❦❡s ♦♥ ❛ ✈❛❧✉❡ ✇✐t❤✐♥ ❛ ❣✐✈❡♥ ✐♥t❡r✈❛❧ � ¯ x P { X ∈ [ x , ¯ x ] } ≡ f X ( x ) dx ✵✳✹ x ✵✳✸ ✷ ✇❤② ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧s♦ ❤♦❧❞❄ ✵✳✷ ② f X ( x ) ≥ ✵ � ∞ ✵✳✶ f X ( x ) dx = ✶ −∞ ✵✳✵ − ✸ − ✷ − ✶ ✵ ✶ ✷ ✸ ① ✶✽ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ❈✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❈❉❋✮✱ F X ✶ t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ r✈ X t❛❦❡s ♦♥ ❛ ✈❛❧✉❡ ❧❡ss t❤❛♥ x ❊①❛♠♣❧❡ F X ( x ) ≡ P { X ≤ x } � x F X ( x ) = ✶ ✷ [ ✶ + ❡r❢ ( x )] = f X ( u ) du −∞ ✇❤❡r❡ ❡r❢ ✐s t❤❡ ❡rr♦r ❢✉♥❝t✐♦♥ ✷ ✇❤② ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧s♦ ❤♦❧❞❄ � x ✷ e − t ✷ dt ❡r❢ ( x ) ≡ √ π F X ( −∞ ) = ✵ ✵ F X ( ∞ ) = ✶ F X ♥♦♥✲❞❡❝r❡❛s✐♥❣ ✶✾ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ❈❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✭❝❢✮✱ φ X ❊①❛♠♣❧❡ φ X ( ω ) = e − ✶ ✷ ω ✷ ✶ � e i ω X � φ X ( ω ) ≡ E , ω ∈ R ✶✳✵ ✷ ✐ts ♣r♦♣❡rt✐❡s ❛r❡ ❧❡ss ✐♥t✉✐t✐✈❡ ✵✳✽ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ q✳✈✳ ♣♣✳✻✲✼✮ ✵✳✻ ✸ ♣❛rt✐❝✉❧❛r❧② ✉s❡❢✉❧ ❢♦r ❤❛♥❞❧✐♥❣ ♣❤✐ ✵✳✹ ✭✇❡✐❣❤t❡❞✮ s✉♠s ♦❢ ✐♥❞❡♣❡♥❞❡♥t ✵✳✷ r✈s ✵✳✵ − ✸ − ✷ − ✶ ✵ ✶ ✷ ✸ ♦♠❡❣❛ ✷✵ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ◗✉❛♥t✐❧❡✱ Q X ✶ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❈❉❋ ❊①❛♠♣❧❡ Q X ( p ) ≡ F − ✶ Q X ( p ) = ❡r❢ − ✶ ( ✷ p − ✶ ) ( p ) X ✷ t❤❡ ♥✉♠❜❡r x s✉❝❤ t❤❛t t❤❡ ❊①❛♠♣❧❡ ♣r♦❜❛❜✐❧✐t② t❤❛t X ❜❡ ❧❡ss t❤❛♥ x ✐s p ✿ ❋♦r t❤❡ ♠❡❞✐❛♥✱ p = ✶ ✷ P { X ≤ Q X ( p ) } = p ❱❛❘ ✷✶ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡ q✉❛♥t✐❧❡ ❛♥❞ t❤❡ ❈❉❋ p = F X ( x ) p ✐♥✈❡rt✐❜✐❧✐t② r❡q✉✐r❡s f X > ✵ p ′ ♦t❤❡r✇✐s❡✱ ❝❛♥ r❡❣✉❧❛r✐s❡ f X ✇✐t❤ x ′ = Q X ( p ′ ) f X ; ε ▼❡✉❝❝✐ ✭✷✵✵✺✱ ❆♣♣✳ ❇✳✹✮ x x ′ ✷✷ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ▼♦✈✐♥❣ ❜❡t✇❡❡♥ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ r✈ X I ✐s t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦♣❡r❛t♦r D ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦♣❡r❛t♦r ✐♥✈❡rs❡s Q X F X F ✐s t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭❋❚✮ F − ✶ ✐s t❤❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ✭■❋❚✮ ❛❧❧ ♦❢ t❤❡s❡ ❛r❡ ❡①❛♠♣❧❡s ♦❢ ❧✐♥❡❛r I ◦ F − ✶ F ◦ D ♦♣❡r❛t♦rs✱ A [ v ] ( ① ) I D A ✱ t❤❡ ❧✐♥❡❛r ♦♣❡r❛t♦r v ✱ t❤❡ ❢✉♥❝t✐♦♥ t♦ ✇❤✐❝❤ ✐t ✐s ❛♣♣❧✐❡❞ φ X ① ✱ t❤❡ ❢✉♥❝t✐♦♥✬s ❛r❣✉♠❡♥t F − ✶ q✳✈✳ ▼❡✉❝❝✐✱ ❆♣♣❡♥❞✐① ❇✳✸ f X F ✭♥✳❜✳ f X ❡①✐sts ✐✛ F X ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s❀ φ X ❛❧✇❛②s ❡①✐sts✮ ✷✸ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❘❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡✐r r❡♣r❡s❡♥t❛t✐♦♥ ▲❡❝t✉r❡ ✶ ❡①❡r❝✐s❡s ▼❡✉❝❝✐ ❡①❡r❝✐s❡s ♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✶✳✶✳✶✱ ✶✳✶✳✷✱ ✶✳✶✳✸✱ ✶✳✶✳✺✱ ✶✳✶✳✻✱ ✶✳✷✳✷ ▼❆❚▲❆❇✿ ✶✳✶✳✹✱ ✶✳✶✳✼✱ ✶✳✶✳✽ ♣r♦❥❡❝t ♣✐❝❦ ❛ ❝♦✉♥tr② ❢♦r ②♦✉r ♣r♦❥❡❝t ✭❡♥s✉r✐♥❣ t❤❛t ♥♦ ♦♥❡ ❡❧s❡ ✐s ✉s✐♥❣ ✐t✮✱ ❛♥❞ ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ❡q✉✐t② ✐♥❞❡① ❛♥❞ ❝♦♠♠♦❞✐t② ✭✐❞❡❛❧❧② ❣♦❧❞✮✳ ❇❡❣✐♥ t♦ ❡①♣❡r✐♠❡♥t ✇✐t❤ ②♦✉r ■♥t❡r❛❝t✐✈❡ ❇r♦❦❡rs tr❛❞✐♥❣ ♣❧❛t❢♦r♠✳ ✷✹ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙✉♠♠❛r② st❛t✐st✐❝s ❑❡② s✉♠♠❛r② ♣❛r❛♠❡t❡rs ❢✉❧❧ ❞✐str✐❜✉t✐♦♥s ❝❛♥ ❜❡ ❡①♣❡♥s✐✈❡ t♦ r❡♣r❡s❡♥t ✇❤❛t s✉♠♠❛r② ✐♥❢♦r♠❛t✐♦♥ ❤❡❧♣s ❝❛♣t✉r❡ ❦❡② ❢❡❛t✉r❡s❄ ✶ ❧♦❝❛t✐♦♥✱ Loc { X } ✐❢ ❤❛❞ ♦♥❡ ❣✉❡ss ❛s t♦ ✇❤❡r❡ X ✇♦✉❧❞ t❛❦❡ ✐ts ✈❛❧✉❡ s❤♦✉❧❞ s❛t✐s❢② Loc { a } = a ❛♥❞ ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ Loc { a + bX } = a + bLoc { X } t♦ ❡♥s✉r❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ♠❡❛s✉r❡♠❡♥t s❝❛❧❡✴❝♦♦r❞✐♥❛t❡ s②st❡♠ ✷ ❞✐s♣❡rs✐♦♥✱ Dis { X } ❤♦✇ ❛❝❝✉r❛t❡ t❤❡ ❧♦❝❛t✐♦♥ ❣✉❡ss✱ ❛❜♦✈❡✱ ✐s ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ✐s ♥♦✇ Dis { a + bX } = | b | Dis { X } ✇❤❡r❡ |·| ❞❡♥♦t❡s ❛❜s♦❧✉t❡ ✈❛❧✉❡ ✸ ③✲s❝♦r❡ ♥♦r♠❛❧✐s❡s ❛ r✈✱ Z X ≡ X − Loc { X } Dis { X } ✿ ✵ ❧♦❝❛t✐♦♥❀ ✶ ❞✐s♣❡rs✐♦♥ ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ♦❢ ❧♦❝❛t✐♦♥ ✫ ❞✐s♣❡rs✐♦♥ ⇔ ( Z a + bX ) ✷ = ( Z X ) ✷ ✷✻ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙✉♠♠❛r② st❛t✐st✐❝s ▼♦st ❝♦♠♠♦♥ ❧♦❝❛t✐♦♥ ❛♥❞ ❞✐s♣❡rs✐♦♥ ♠❡❛s✉r❡s ❵❧♦❝❛❧✬ ❵s❡♠✐✲❧♦❝❛❧✬ ❵❣❧♦❜❛❧✬ ❧♦❝❛t✐♦♥ ♠♦❞❡✱ Mod { X } ♠❡❞✐❛♥✱ Med { X } ♠❡❛♥ ✴ ❡①♣✬❞ ✈❛❧✱ E { X } � Med { X } � ∞ f X ( x ) dx = ✶ ❛r❣♠❛① x ∈ R f X ( x ) −∞ xf X ( x ) dx −∞ ✷ ❞✐s♣❡rs✐♦♥ ♠♦❞❛❧ ❞✐s♣❡rs✐♦♥ ✐♥t❡rq✉❛♥t✐❧❡ r❛♥❣❡ ✈❛r✐❛♥❝❡ � ∞ −∞ ( x − E { X } ) ✷ f X ( x ) dx ❵❣❧♦❜❛❧✬ ♠❡❛s✉r❡s ❛r❡ ❢♦r♠❡❞ ❢r♦♠ t❤❡ ✇❤♦❧❡ ❞✐str✐❜✉t✐♦♥ ❵s❡♠✐✲❧♦❝❛❧✬ ♠❡❛s✉r❡s ❛r❡ ❢♦r♠❡❞ ❢r♦♠ ❤❛❧❢ ✭♦r s♦✮ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ❵❧♦❝❛❧✬ ♠❡❛s✉r❡s ❛r❡ ❞r✐✈❡♥ ❜② ✐♥❞✐✈✐❞✉❛❧ ♦❜s❡r✈❛t✐♦♥s ❣❡♥❡r❛❧❧②✱ ✇❡ ❞❡✜♥❡ Dis { X } ≡ � X − Loc { X } � X ; p ✶ p ✐s t❤❡ ♥♦r♠ ♦♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ L p ✇❤❡r❡ � g � X ; p ≡ ( E {| g ( X ) | p } ) X p = ✶ ✐s t❤❡ ♠❡❛♥ ❛❜s♦❧✉t❡ ❞❡✈✐❛t✐♦♥✱ MAD { X } ≡ E {| X − E { X }|} | X − E { X }| ✷ �� ✶ � � ✷ p = ✷ ✐s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥✱ Sd { X } ≡ E ✷✼ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙✉♠♠❛r② st❛t✐st✐❝s ❍✐❣❤❡r ♦r❞❡r ♠♦♠❡♥ts ✶ k t❤ ✲r❛✇ ♠♦♠❡♥t � X k � RM X k ≡ E ✐s t❤❡ ❡①♣❡❝t❛t✐♦♥ ♦❢ t❤❡ k t❤ ♣♦✇❡r ♦❢ X ✷ k t❤ ✲❝❡♥tr❛❧ ♠♦♠❡♥t ✐s ♠♦r❡ ❝♦♠♠♦♥❧② ✉s❡❞ � ( X − E { X } ) k � CM X k ≡ E ❞❡✲♠❡❛♥s t❤❡ r❛✇ ♠♦♠❡♥t✱ ♠❛❦✐♥❣ ✐t ❧♦❝❛t✐♦♥✲✐♥❞❡♣❡♥❞❡♥t s❦❡✇♥❡ss✱ ❛ ♠❡❛s✉r❡ ♦❢ s②♠♠❡tr②✱ ✐s t❤❡ ♥♦r♠❛❧✐s❡❞ ✸ r❞ ❝❡♥tr❛❧ ♠♦♠❡♥t CM X ✸ Sk { X } ≡ ( Sd { X } ) ✸ ❦✉rt♦s✐s ♠❡❛s✉r❡s t❤❡ ✇❡✐❣❤ts ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥✬s t❛✐❧ r❡❧❛t✐✈❡ t♦ ✐ts ❝❡♥tr❡ CM X ✹ Ku { X } ≡ ( Sd { X } ) ✹ ✷✽ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❯♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✿ X ∼ U ([ a , b ]) s✐♠♣❧❡st ❞✐str✐❜✉t✐♦♥❀ s❤❛❧❧ ❜❡ ✉s❡❢✉❧ ✇❤❡♥ ♠♦❞❡❧❧✐♥❣ ❝♦♣✉❧❛s ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ a ✭❧♦✇❡r ❜♦✉♥❞✮ ❛♥❞ b ✭✉♣♣❡r ❜♦✉♥❞✮ ❛♥② ♦✉t❝♦♠❡ ✐♥ t❤❡ [ a , b ] ✐s ❡q✉❛❧❧② ❧✐❦❡❧② ❝❧♦s❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r f U a , b ( x ) , F U a , b ( x ) , φ U a , b ( ω ) ❛♥❞ Q U a , b ( p ) st❛♥❞❛r❞ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ✐s U ([ ✵ , ✶ ]) ✸✵ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s µ, σ ✷ � � ◆♦r♠❛❧ ✭●❛✉ss✐❛♥✮ ❞✐str✐❜✉t✐♦♥✿ X ∼ N ♠♦st ✇✐❞❡❧② ✉s❡❞✱ st✉❞✐❡❞ ❞✐str✐❜✉t✐♦♥ ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ µ ✭♠❡❛♥✮ ❛♥❞ σ ✷ ✭✈❛r✐❛♥❝❡✮ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ σ ✷ = ✶ ❛s ❛ st❛❜❧❡ ❞✐str✐❜✉t✐♦♥✱ t❤❡ s✉♠s ♦❢ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ r✈✬s ❛r❡ ♥♦r♠❛❧ ❝❧♦s❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r f N µ,σ ✷ ( x ) , F N µ,σ ✷ ( x ) , φ N µ,σ ✷ ( ω ) ❛♥❞ Q N µ,σ ✷ ( p ) ✇❤② ❞♦ ✇❡ ❝❛r❡ t❤❛t Ku { X } = ✸❄ ✸✶ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s µ, σ ✷ � � ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥✿ X ∼ Ca ❵❢❛t✲t❛✐❧❡❞✬ ❞✐str✐❜✉t✐♦♥✿ ✇❤❡♥ ♠✐❣❤t t❤✐s ❜❡ ✉s❡❢✉❧❄ ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ µ ❛♥❞ σ ✷ � − ✶ � ✶ + ( x − µ ) ✷ ✶ f Ca µ,σ ✷ ( x ) ≡ √ σ ✷ σ ✷ π ✇❤❛t ❛r❡ E { X } , Var { X } , Sk { X } ❛♥❞ Ku { X } ❄ s❡❡ ❤❡r❡ ❢♦r ❛ ❞✐s❝✉ss✐♦♥ st❛♥❞❛r❞ ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ σ ✷ = ✶ ✭❋❨■✿ ✐❢ X , Y ∼ NID ( ✵ , ✶ ) t❤❡♥ X Y ∼ Ca ( ✵ , ✶ ) ✮ ✸✷ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ν, µ, σ ✷ � � ❙t✉❞❡♥t t ❞✐str✐❜✉t✐♦♥✿ X ∼ St ❊①❛♠♣❧❡ ✭ ν = ✸✮ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ♣❛r❛♠❡t❡r✱ → d N ν, µ, σ ✷ � µ, σ ✷ � ν ✱ ❞❡t❡r♠✐♥❡s ❢❛t♥❡ss ♦❢ t❛✐❧s � � ν → ∞ ⇒ St ❛♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥s ❢♦r → d Ca ν, µ, σ ✷ � µ, σ ✷ � ν → ✶ ⇒ St � � f St ν,µ,σ ✷ , F St ν,µ,σ ✷ ❛♥❞ φ St ν,µ,σ ✷ ✉s❡ t❤❡ ❣❛♠♠❛✱ ❜❡t❛ ❛♥❞ ❇❡ss❡❧ ❢✉♥❝t✐♦♥s❀ ♥♦♥❡ ❢♦r Q St ν,µ,σ ✷ ❧✐♠✐t ♦❢ ❛♥❛❧②t✐❝❛❧ ❡①♣r❡ss✐♦♥s ✵✳✹ q✉✐❝❦❧② r❡❛❝❤❡❞ ✵✳✸ st❛♥❞❛r❞ ❙t✉❞❡♥t ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ σ ✷ = ✶ ✵✳✷ ② ✇❤❡♥ ❛r❡ ✵✳✶ E { X } , Var { X } , Sk { X } ❛♥❞ Ku { X } ❞❡✜♥❡❞❄ ✵✳✵ − ✸ − ✷ − ✶ ✵ ✶ ✷ ✸ ✸✸ ✴ ✷✷✹ ①

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s µ, σ ✷ � � ▲♦❣✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥✿ X ∼ LogN ✐❢ Y ∼ N � µ, σ ✷ � t❤❡♥ X ≡ e Y ∼ LogN µ, σ ✷ � � ❊①❛♠♣❧❡ ✭❇❛✐❧❡②✿ s❤♦✉❧❞ ❜❡ ❝❛❧❧❡❞ ❵❡①♣✲♥♦r♠❛❧✬ ❞✐str✐❜✉t✐♦♥❄✮ ♥♦✇ φ LogN µ,σ ✷ ❤❛s ♥♦ ❦♥♦✇♥ ✵✳✻ ❛♥❛❧②t✐❝ ❢♦r♠ ✵✳✺ ♣r♦♣❡rt✐❡s ✵✳✹ ② X > ✵ ✵✳✸ (% ❝❤❛♥❣❡s ✐♥ X ) ∼ N ✵✳✷ ❛s②♠♠❡tr✐❝ ✭♣♦s✐t✐✈❡❧② ✵✳✶ s❦❡✇❡❞✮ ✵✳✵ ❝♦♠♠♦♥❧② ❛♣♣❧✐❡❞ t♦ st♦❝❦ − ✸ − ✷ − ✶ ✵ ✶ ✷ ✸ ① ♣r✐❝❡s ✭❍✉❧❧ ✭✷✵✵✾✱ ➓✶✷✳✻✱ ➓✶✸✳✶✮✱ ❙t❡❢❛♥✐❝❛ ✭✷✵✶✶✱ ➓✹✳✻✮✮ ✸✹ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ν, µ, σ ✷ � � ●❛♠♠❛ ❞✐str✐❜✉t✐♦♥✿ X ∼ Ga ❧❡t Y ✶ , . . . , Y ν ∼ IID s✳t✳ Y t ∼ N � µ, σ ✷ � ∀ t ∈ { ✶ , . . . ν } ♥♦♥✲❝❡♥tr❛❧ ❣❛♠♠❛ ❞✐str✐❜✉t✐♦♥✱ ❊①❛♠♣❧❡ ✭ µ = ✵ , σ ✷ = ✶✮ X ≡ � ν t = ✶ Y ✷ � ν, µ, σ ✷ � t ∼ Ga ν ✿ ❞❡❣r❡❡s✲♦❢✲❢r❡❡❞♦♠ ✭s❤❛♣❡✮❀ µ ✿ ♥♦♥✲❝❡♥tr❛❧✐t②❀ σ ✷ ✿ s❝❛❧❡ ✵✳✷✺ ❇❛②❡s✐❛♥s✿ ❡❛❝❤ ♦❜s❡r✈❛t✐♦♥ ✐s ❛♥ r✈ ⇒ t❤❡✐r ✈❛r✐❛♥❝❡ ∼ Ga ✵✳✷✵ ✵✳✶✺ ② ✶ µ = ✵ ⇒ ❝❡♥tr❛❧ ❣❛♠♠❛ ✵✳✶✵ ❞✐str✐❜✉t✐♦♥✱ X ∼ Ga ν, σ ✷ � � ✵✳✵✺ ✭♠♦st ❝♦♠♠♦♥✮ ✵✳✵✵ ✷ σ ✷ = ✶ ⇒ ♥♦♥✲❝❡♥tr❛❧ ❝❤✐✲sq✉❛r❡ − ✸ − ✷ − ✶ ✵ ✶ ✷ ✸ ❞✐str✐❜✉t✐♦♥ ① ✸ µ = ✵ , σ ✷ = ✶ ⇒ ❝❤✐✲sq✉❛r❡ ❳ ∼ W ❞✐str✐❜✉t✐♦♥✱ X ∼ χ ✷ ν ✸✺ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥✿ X ∼ Em ( i T ) ❞❛t❛ ❞❡✜♥❡s ❞✐str✐❜✉t✐♦♥✿ ❢✉t✉r❡ ♦❝❝✉rs ✇✐t❤ s❛♠❡ ♣r♦❜❛❜✐❧✐t② ❛s ♣❛st T f i T ( x ) ≡ ✶ δ ( x t ) ( x ) � T t = ✶ T F i T ( x ) ≡ ✶ H ( x t ) ( x ) � T t = ✶ δ ( x t ) ( · ) ✐s ❉✐r❛❝✬s ❞❡❧t❛ ❢✉♥❝t✐♦♥ ❝❡♥tr❡❞ ❛t x t ✱ ❛ ❣❡♥❡r❛❧✐s❡❞ ❢✉♥❝t✐♦♥ ✭✐❢ ✇✐s❤ t♦ tr❡❛t X ❛s ❞✐s❝r❡t❡✱ ❑r♦♥❡❝❦❡r✬s ❞❡❧t❛ ❢✉♥❝t✐♦♥ ❞❡✜♥❡s ♣r♦❜❛❜✐❧✐t② ♠❛ss ❢✉♥❝t✐♦♥✮ H ( x t ) ( · ) ✐s ❍❡❛✈✐s✐❞❡✬s st❡♣ ❢✉♥❝t✐♦♥✱ ✇✐t❤ ✐ts st❡♣ ❛t x t ✇❤❛t ❞♦ t❤❡s❡ ❧♦♦❦ ❧✐❦❡❄ ❲❤❛t ❞♦ r❡❣✉❧❛r✐s❡❞ ✈❡rs✐♦♥s ❧♦♦❦ ❧✐❦❡❄ ❞❡✜♥✐♥❣ Q i T ( p ) ♦❜t❛✐♥❡❞ ❜② ❜❛♥❞✇✐❞t❤ t❡❝❤♥✐q✉❡s ♦❢ ❆♣♣❡♥❞✐① ❇✿ ♦r❞❡r ♦❜s❡r✈❛t✐♦♥s✱ t❤❡♥ ❝♦✉♥t ❢r♦♠ ❧♦✇❡st ❳ ∼ Em ✸✻ ✴ ✷✷✹

❯♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ▲❡❝t✉r❡ ✷ ❡①❡r❝✐s❡s ▼❡✉❝❝✐ ❡①❡r❝✐s❡s ♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✶✳✷✳✺ ✭♥♦t ▼❆❚▲❆❇✮✱ ✶✳✷✳✻✱ ✶✳✷✳✼ ▼❆❚▲❆❇✿ ✶✳✷✳✸✱ ✶✳✷✳✺ ✭▼❆❚▲❆❇✮ ♣r♦❥❡❝t ❡①♣❧♦r❡ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ r❡t✉r♥s ❢♦r ②♦✉r ❡q✉✐t② ✐♥❞❡①✱ ②♦✉r ❝♦♠♠♦❞✐t② ❛♥❞ ②♦✉r ❝✉rr❡♥❝✐❡s✬ ❡①❝❤❛♥❣❡ r❛t❡ ✈ t❤❡ ❯❙❉ ✸✼ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ❉✐r❡❝t ❡①t❡♥s✐♦♥s ♦❢ ✉♥✐✈❛r✐❛t❡ st❛t✐st✐❝s ✐❢ ✐♥t❡r❡st❡❞ ✐♥ ♣♦rt❢♦❧✐♦s ✭♦r ❡✈❡♥ ❛r❜✐tr❛❣❡✮✱ ♠✉st ❜❡ ❛❜❧❡ t♦ ❝♦♥s✐❞❡r ❤♦✇ ❛♥ ❛ss❡t✬s ♠♦✈❡♠❡♥ts ❞❡♣❡♥❞ ♦♥ ♦t❤❡rs✬ N ✲❞✐♠❡♥s✐♦♥❛❧ r✈✱ ❳ ≡ ( X ✶ , . . . , X N ) ′ ✱ s♦ t❤❛t ① ∈ R N ♣r♦❜❛❜✐❧✐t② ❞❡♥s✐t② ❢✉♥❝t✐♦♥ � � f ❳ ( ① ) d ① , st f ❳ ( ① ) ≥ ✵ , R N f ❳ ( ① ) d ① = ✶ P { ❳ ∈ R} ≡ R ❝✉♠✉❧❛t✐✈❡ ♦r ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❞❢✱ ❉❋✱ ❈❉❋✱ ❏❉❋ ✳ ✳ ✳ ✮ � x ✶ � x N F ❳ ( ① ) ≡ P { ❳ ≤ ① } = · · · f ❳ ( u ✶ , . . . , u N ) du N · · · du ✶ −∞ −∞ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ � e i ω ′ ❳ � , ω ∈ R N φ ❳ ( ω ) ≡ E ✇❤❛t ❛❜♦✉t t❤❡ q✉❛♥t✐❧❡❄ ✭❤✐♥t✿ F ❳ : R N → R ✶ ✮ ✸✾ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ▼❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥✴❞❡♥s✐t② ♦❢ ❳ B ♣❛rt✐t✐♦♥ ❳ ✐♥t♦ K ✲❞✐♠❡♥s✐♦♥❛❧ ❳ A ❛♥❞ ( N − K ) ✲❞✐♠❡♥s✐♦♥❛❧ ❳ B ❞✐str✐❜✉t✐♦♥ ♦❢ ❳ B ✇❤❛t❡✈❡r ❳ A ✬s ✭t❡❝❤♥✐❝❛❧❧②✿ ✐♥t❡❣r❛t❡s ♦✉t ❳ A ✮ F ❳ B ( ① B ) ≡ P { ❳ B ≤ ① B } ❊①❛♠♣❧❡ = P { ❳ A ≤ ∞ , ❳ B ≤ ① B } = F ❳ ( ∞ , ① B ) � f ❳ B ( ① B ) ≡ R K f ❳ ( ① A , ① B ) d ① A ❢ � � e i ω ′ ❳ B φ ❳ B ( ω ) ≡ E � � e i ψ ′ ❳ A + i ω ′ ❳ B = E | ψ = ✵ ① ❴ ❜ ①❴❛ = φ ❳ ( ✵ , ω ) ✹✵ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ❲❤❛t✱ r♦✉❣❤❧②✱ ❞♦ t❤❡ z ♠❛r❣✐♥❛❧s ♦❢ t❤✐s ♣❞❢ ❧♦♦❦ ❧✐❦❡❄ ❝♦♣✉❧❛ ❞❡✜♥❡❞ y x ✹✶ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ❈♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥✴❞❡♥s✐t② ♦❢ ❳ A ❣✐✈❡♥ ① B ❡✳❣✳ ✜① ❛ss❡ts B ✬s r❡t✉r♥s ❛t ① B ❀ ✇❤❛t ✐s t❤❛t ♦❢ ❛ss❡ts A ❄ f ❳ A | ① B ( ① A ) ≡ f ❳ ( ① A , ① B ) ❊①❛♠♣❧❡ f ❳ B ( ① B ) ❝❛♥ ❞❡❝♦♠♣♦s❡ ❏❉❋ ✐♥t♦ ♣r♦❞✉❝t ♦❢ ♠❛r❣✐♥❛❧ ❛♥❞ ❢ ❝♦♥❞✐t✐♦♥❛❧ ❇❛②❡s✬ r✉❧❡ ❢♦r ✉♣❞❛t✐♥❣ ❜❡❧✐❡❢s ✐s ❛♥ ✐♠♠❡❞✐❛t❡ ❝♦♥s❡q✉❡♥❝❡ ①❴❜ ①❴❛ f ❳ A | ① B ( ① A ) = f ❳ B | ① A ( ① B ) f ❳ A ( ① A ) f ❳ B ( ① B ) ✹✷ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ▲♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r✱ Loc { ❳ } ❞❡s✐❞❡r❛t❛ ♦❢ ❧♦❝❛t✐♦♥ ❡①t❡♥❞ ❞✐r❡❝t❧② ❢r♦♠ ✉♥✐✈❛r✐❛t❡ ❝❛s❡ ❢♦r ❝♦♥st❛♥t ♠ ✱ Loc { ♠ } = ♠ ✶ ❢♦r ✐♥✈❡rt✐❜❧❡ ❇ ✱ ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ♥♦✇ ✷ Loc { ❛ + ❇❳ } = ❛ + ❇ Loc { ❳ } ❡①♣❡❝t❡❞ ✈❛❧✉❡ E { ❳ } = ( E { X ✶ } , . . . , E { X N } ) ′ ✶ ❛✣♥❡ ❡q✉✐✈❛r✐❛♥❝❡ ♣r♦♣❡rt② ❤♦❧❞s ❢♦r ❛♥② ❝♦♥❢♦r♠❛❜❧❡ ❇ ✱ ♥♦t ❥✉st ✷ ✐♥✈❡rt✐❜❧❡ ✭ Med { ❳ } , Mod { ❳ } r❡q✉✐r❡ ✐♥✈❡rt✐❜❧❡✮ r❡❧❛t✐✈❡❧② ❡❛s② t♦ ❝❛❧❝✉❧❛t❡ ✇❤❡♥ φ ❳ ❦♥♦✇♥✱ ❛♥❛❧②t✐❝❛❧ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ✸ ➓❚✷✳✶✵✮ ✹✸ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❇✉✐❧❞✐♥❣ ❜❧♦❝❦s ❉✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡r✱ Dis { ❳ } r❡❝❛❧❧✿ ✐♥ t❤❡ ✉♥✐✈❛r✐❛t❡ ❝❛s❡✱ t❤❡ z ✲s❝♦r❡ ♥♦r♠❛❧✐s❡s ❛ ❞✐str✐❜✉t✐♦♥ s♦ t❤❛t ✐t ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥s � ( X − Loc { X } ) ( X − Loc { X } ) | Z a + bX | = | Z X | ≡ Dis { X } ❧❡t Σ ❜❡ ❛ s②♠♠❡tr✐❝ P❉ ♦r P❙❉ ♠❛tr✐①❀ t❤❡♥ ▼❛❤❛❧❛♥♦❜✐s ❞✐st❛♥❝❡ ❢r♦♠ ① t♦ µ ✱ ♥♦r♠❛❧✐s❡❞ ❜② t❤❡ ♠❡tr✐❝ Σ ✱ ✐s � ( ① − µ ) ′ Σ − ✶ ( ① − µ ) Ma ( ① , µ , Σ ) ≡ ❣✐✈❡♥ ❛♥ ❡❧❧✐♣s♦✐❞ ❝❡♥tr❡❞ ❛t µ ✇❤♦s❡ ♣r✐♥❝✐♣❛❧ ❛①❡s✬ ❧❡♥❣t❤s ❡q✉❛❧ t❤❡ sq✉❛r❡ r♦♦ts ♦❢ t❤❡ ❡✐❣❡♥✈❛❧✉❡s ♦❢ Σ ✱ ❛❧❧ ① ♦♥ ✐ts s✉r❢❛❝❡ ❤❛✈❡ t❤❡ s❛♠❡ ▼❛❤❛❧❛♥♦❜✐s ❞✐st❛♥❝❡ ❢r♦♠ µ ■■❉ ❤❡✉r✐st✐❝ t❡st ✷ ♠✉❧t✐✈❛r✐❛t❡ z ✲s❝♦r❡ ✐s t❤❡♥ Ma ❳ ≡ Ma ( ❳ , Loc { ❳ } , DisSq { ❳ } ) ❜❡♥❝❤♠❛r❦ ✭sq✉❛r❡❞✮ ❞✐s♣❡rs✐♦♥ ♦r s❝❛tt❡r ♣❛r❛♠❡t❡r✿ ❝♦✈❛r✐❛♥❝❡ ✹✹ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❉❡♣❡♥❞❡♥❝❡ ❈♦rr❡❧❛t✐♦♥ ♥♦r♠❛❧✐s❡❞ ❝♦✈❛r✐❛♥❝❡ Cov { X m , X n } Sd { X m } Sd { X n } ∈ [ − ✶ , ✶ ] ρ ( X m , X n ) = Cor { X m , X n } ≡ ✇❤❡r❡ Cov { X m , X n } ≡ E { ( X m − E { X m } ) ( X n − E { X n } ) } ✇❤❡♥ ✐s t❤✐s ♥♦t ❞❡✜♥❡❞❄ ❛ ♠❡❛s✉r❡ ♦❢ ❧✐♥❡❛r ❞❡♣❡♥❞❡♥❝❡✱ ✐♥✈❛r✐❛♥t ✉♥❞❡r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ρ ( α m + β m X m , α n + β n X n ) = ρ ( X m , X n ) ❢❛❧❧❛❝② ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✷✹✶✮✿ ❣✐✈❡♥ ♠❛r❣✐♥❛❧ ❞❢s F ✶ ❛♥❞ F ✷ ❛♥❞ ❛♥② ρ ∈ [ − ✶ , ✶ ] ✱ ❝❛♥ ❛❧✇❛②s ✜♥❞ ❛ ❏❉❋ F ❜✐♥❞✐♥❣ t❤❡♠ t♦❣❡t❤❡r tr✉❡ ❢♦r ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s❀ ❣❡♥❡r❛❧❧②✱ ❛tt❛✐♥❛❜❧❡ ❝♦rr❡❧❛t✐♦♥s ❛r❡ ❛ str✐❝t s✉❜s❡t ♦❢ [ − ✶ , ✶ ] ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ❊①❛♠♣❧❡ ✼✳✷✾✮ ✹✻ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❉❡♣❡♥❞❡♥❝❡ ❙t❛♥❞❛r❞ ♥♦r♠❛❧ ♠❛r❣✐♥❛❧s✱ ρ ≈ . ✼ ❢❛❧❧❛❝② ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✷✸✾✮✿ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛♥❞ ♣❛✐r✇✐s❡ ❝♦rr❡❧❛t✐♦♥s ♦❢ ❛ r✈ ❞❡t❡r♠✐♥❡ ✐ts ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ✹✼ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❉❡♣❡♥❞❡♥❝❡ ■♥❞❡♣❡♥❞❡♥❝❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ♦♥❡ ✈❛r✐❛❜❧❡ ❞♦❡s ♥♦t ❛✛❡❝t ❞✐str✐❜✉t✐♦♥ ♦❢ ♦t❤❡rs f ❳ B ( ① B ) = f ❳ B | ① A ( ① B ) ♣r♦❜❛❜✐❧✐t② ♦❢ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts✿ P { e ∩ f } = P { e } P { f } F ❳ ( ① A , ① B ) = F ❳ A ( ① A ) F ❳ B ( ① B ) ❢r♦♠ ❞❡✜♥✐t✐♦♥s ♦❢ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥❝❡ ✭tr② ✐t✦✮ f ❳ ( ① A , ① B ) = f ❳ A ( ① A ) f ❳ B ( ① B ) ❛❜♦✈❡ tr✉❡ ✐❢ ❳ A , ❳ B tr❛♥s❢♦r♠❡❞ ❜② ❛r❜✐tr❛r② g ( · ) ❛♥❞ h ( · ) ✿ ✐❢ ① A ❞♦❡s♥✬t ❡①♣❧❛✐♥ ❳ B ✱ tr❛♥s❢♦r♠❡❞ ✈❡rs✐♦♥s ✇♦♥✬t ❡✐t❤❡r ❧✐♥❡❛r r❡t✉r♥s ♣❧♦t t❤❡r❡❢♦r❡ ❛❧❧♦✇s ♥♦♥✲❧✐♥❡❛r r❡❧❛t✐♦♥s ✐♥❞❡♣❡♥❞❡♥t ✐♠♣❧✐❡s ✉♥❝♦rr❡❧❛t❡❞✱ ❜✉t ♥♦t t❤❡ ❝♦♥✈❡rs❡ ❊①❛♠♣❧❡ ●✐✈❡♥ X ✷ + Y ✷ = ✶✱ ❛r❡ t❤❡ r✈s X ❛♥❞ Y ✭✉♥✮❝♦rr❡❧❛t❡❞✱ ✭✐♥✮❞❡♣❡♥❞❡♥t❄ ❍✐♥t✿ ✐❢ ✜tt✐♥❣ y i = mx i + b + ε i ✱ ✇❤❛t ❛r❡ m , ˆ m ❄ ✹✽ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❯♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ✐❞❡❛ ✐s ❛s ✐♥ ✉♥✐✈❛r✐❛t❡ ❝❛s❡✱ ❜✉t ❞♦♠❛✐♥ ♠❛② ❜❡ ❛♥②t❤✐♥❣ ♦❢t❡♥ ❡❧❧✐♣t✐❝❛❧ ❞♦♠❛✐♥✱ E µ , Σ ✇❤❡r❡ µ ✐s ❝❡♥tr♦✐❞✱ Σ ✐s ♣♦s✐t✐✈❡ ♠❛tr✐① ❊①❛♠♣❧❡ f X ✶ , X ✷ ( x ✶ , x ✷ ) = ✶ ✷ ≤ ✶ } ( x ✶ , x ✷ ) π I { x ✷ ✶ + x ✷ ✇❤❡r❡ I S ✐s t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦♥ t❤❡ s❡t S � √ ✶ − x ✷ � π dx ✷ = ✷ ✶ ♠❛r❣✐♥❛❧ ❞❡♥s✐t②✿ f X ✶ ( x ✶ ) = − √ ✶ − x ✷ ✶ ✶ π ✶ − x ✷ ✶ f X ✶ , X ✷ ( x ✶ , x ✷ ) ✶ ❝♦♥❞✐t✐♦♥❛❧ ❞❡♥s✐t②✿ f X ✶ | x ✷ ( x ✶ ) = ✷ √ = f X ✷ ( x ✷ ) ✶ − x ✷ ✷ ❛r❡ X ✶ ❛♥❞ X ✷ ✭✉♥✮❝♦rr❡❧❛t❡❞✱ ✭✐♥✮❞❡♣❡♥❞❡♥t❄ ✺✵ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ◆♦r♠❛❧ ✭●❛✉ss✐❛♥✮ ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ N ( µ , Σ ) ♠♦st ✇✐❞❡❧② ✉s❡❞✱ st✉❞✐❡❞ ❞✐str✐❜✉t✐♦♥ ❢✉❧❧② ❞❡s❝r✐❜❡❞ ❜② t✇♦ ♣❛r❛♠❡t❡rs✱ µ ✭❧♦❝❛t✐♦♥✮ ❛♥❞ Σ ✭❞✐s♣❡rs✐♦♥✮ st❛♥❞❛r❞ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ Σ = ■ ✭✐❞❡♥t✐t② ♠❛tr✐①✮ ❝❧♦s❡❞ ❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥s ❢♦r f N µ , Σ ( ① ) , F N µ , Σ ( ① ) ✱ ❛♥❞ φ N µ , Σ ( ω ) ❛s s②♠♠❡tr✐❝ ❛♥❞ ✉♥✐♠♦❞❛❧ E { ❳ } = Mod { ❳ } = Med { ❳ } = µ Cov { ❳ } = Σ ♠❛r❣✐♥❛❧✱ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛❧s♦ ♥♦r♠❛❧ ✺✶ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❙t✉❞❡♥t t ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ St ( ν, µ , Σ ) ❛❣❛✐♥✱ s②♠♠❡tr✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ❛❜♦✉t ❛ ♣❡❛❦ ❛❣❛✐♥✱ t❤r❡❡ ♣❛r❛♠❡t❡rs ❛s s②♠♠❡tr✐❝ ❛♥❞ ✉♥✐♠♦❞❛❧✱ E { ❳ } = Mod { ❳ } = Med { ❳ } = µ ν s❝❛tt❡r ♣❛r❛♠❡t❡r � = ❝♦✈❛r✐❛♥❝❡✿ Cov { ❳ } = ν − ✷ Σ st❛♥❞❛r❞ ❙t✉❞❡♥t t ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ Σ = ■ ❢♦r♠ ♦❢ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ❞❡♣❡♥❞s ♦♥ ✇❤❡t❤❡r ν ❡✈❡♥ ♦r ♦❞❞❀ s❡❡ ❇❡r❣ ❛♥❞ ❱✐❣♥❛t ✭✷✵✵✽✮ ❢♦r ❛ ❞✐✛❡r❡♥t ✭❝♦♥tr❛r②❄✮ ❛♣♣r♦❛❝❤ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛r❡ ❛❧s♦ t ❀ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥s ❛r❡ ♥♦t❀ t❤✉s✱ ✐❢ ❳ ∼ St ✱ ❝❛♥✬t ❜❡ ✐♥❞❡♣❡♥❞❡♥t ✺✷ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ Ca ( µ , Σ ) ❛s ✐♥ t❤❡ ✉♥✐✈❛r✐❛t❡ ❝❛s❡✱ t❤❡ ❢❛t✲t❛✐❧❡❞ ❧✐♠✐t ♦❢ t❤❡ ❙t✉❞❡♥t t ✲❞✐str✐❜✉t✐♦♥✿ Ca ( µ , Σ ) = St ( ✶ , µ , Σ ) st❛♥❞❛r❞ ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥ ✇❤❡♥ µ = ✵ ❛♥❞ Σ = ■ ✭✐❞❡♥t✐t② ♠❛tr✐①✮ s❛♠❡ ♣r♦❜❧❡♠ ✇✐t❤ ♠♦♠❡♥ts ❛s ✉♥✐✈❛r✐❛t❡ ❝❛s❡ ✺✸ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ▲♦❣✲❞✐str✐❜✉t✐♦♥s ❡①♣♦♥❡♥t✐❛❧s ♦❢ ♦t❤❡r ❞✐str✐❜✉t✐♦♥s✱ ❛♣♣❧✐❡❞ ❝♦♠♣♦♥❡♥t✲✇✐s❡ t❤✉s✱ ✉s❡❢✉❧ ❢♦r ♠♦❞❡❧❧✐♥❣ ♣♦s✐t✐✈❡ ✈❛❧✉❡s ✐❢ ❨ ❤❛s ♣❞❢ f ❨ t❤❡♥ ❳ ≡ e ❨ ✐s ❧♦❣✲ ❨ ❞✐str✐❜✉t❡❞ ❊①❛♠♣❧❡ ✭▲♦❣✲♥♦r♠❛❧✮ ▲❡t ❨ ∼ N ( µ , Σ ) ✳ ❚❤❡♥✱ ✐❢ ❳ ≡ e ❨ ✱ s♦ t❤❛t X i ≡ e Y i ❢♦r ❛❧❧ i = ✶ , . . . , N ✱ ❳ ∼ LogN ( µ , Σ ) ✳ ✺✹ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❲✐s❤❛rt ❞✐str✐❜✉t✐♦♥✿ ❲ ∼ W ( ν, Σ ) ❝♦♥s✐❞❡r N ✲❞✐♠❡♥s✐♦♥❛❧s ■■❉ r✈s ❳ t ∼ N ( ✵ , Σ ) ❢♦r t = ✶ , . . . , ν ≥ N t❤❡♥ ❲✐s❤❛rt ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ν ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✐s t❤❡ r❛♥❞♦♠ ♠❛tr✐① ❲ ≡ ❳ ✶ ❳ ′ ✶ + · · · + ❳ ν ❳ ′ ν ❛s Σ ✐s s②♠♠❡tr✐❝ ❛♥❞ P❉✱ s♦ ✐s ❲ ♠✉❧t✐✈❛r✐❛t❡ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ ❣❛♠♠❛ ❞✐str✐❜✉t✐♦♥ X ∼ Ga ❢✉rt❤❡r♠♦r❡✱ ❣✐✈❡♥ ❣❡♥❡r✐❝ ❛ ✱ ❲ ∼ W ( ν, Σ ) ⇒ ❛ ′ ❲❛ ∼ Ga ν, ❛ ′ Σ ❛ � � ❛s ✐♥✈❡rs❡ ♦❢ s②♠♠❡tr✐❝✱ P❉ ♠❛tr✐① ✐s s②♠♠❡tr✐❝✱ P❉✱ ✐♥✈❡rs❡ ❲✐s❤❛rt ❩ − ✶ ∼ W ν, Ψ − ✶ � � ⇒ ❩ ∼ IW ( ν, Ψ) ❛s ❛ r❛♥❞♦♠ P❉ ♠❛tr✐①✱ ❲✐s❤❛rt ✉s❡❢✉❧ ✐♥ ❡st✐♠❛t✐♥❣ r❛♥❞♦♠ Σ ♣r✐♦rs ✺✺ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❚❛①♦♥♦♠② ♦❢ ❞✐str✐❜✉t✐♦♥s ❊♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥✿ ❳ ∼ Em ( i T ) ❞✐r❡❝t ❡①t❡♥s✐♦♥ ♦❢ ✉♥✐✈❛r✐❛t❡ ❝❛s❡ X ∼ Em T f i T ( ① ) ≡ ✶ δ ( ① t ) ( ① ) � T t = ✶ T F i T ( ① ) ≡ ✶ H ( ① t ) ( ① ) � T t = ✶ T φ i T ( ω ) ≡ ✶ � e i ω ′ ① t T t = ✶ ♠♦♠❡♥ts ✐♥❝❧✉❞❡ s❛♠♣❧❡ ♠❡❛♥✿ ˆ � T E i T ≡ ✶ t = ✶ ① t ✶ T � ′ � � � ˆ � T ① t − ˆ ① t − ˆ Cov i T ≡ ✶ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡✿ E i T E i T ✷ T t = ✶ ✺✻ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s ❊❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s✿ ❳ ∼ El ( µ , Σ , g N ) ❤✐❣❤❧② s②♠♠❡tr✐❝❛❧✱ ❛♥❛❧②t✐❝❛❧❧② tr❛❝t❛❜❧❡✱ ✢❡①✐❜❧❡ ❳ ✐s ❡❧❧✐♣t✐❝❛❧❧② ❞✐str✐❜✉t❡❞ ✇✐t❤ ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r µ ❛♥❞ s❝❛tt❡r ♠❛tr✐① Σ ✐❢ ✐ts ✐s♦✲♣r♦❜❛❜✐❧✐t② ❝♦♥t♦✉rs ❢♦r♠ ❡❧❧✐♣s♦✐❞s ❝❡♥tr❡❞ ❛t µ ✇❤♦s❡ ♣r✐♥❝✐♣❛❧ ❛①❡s✬ ❧❡♥❣t❤s ❛r❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ sq✉❛r❡ r♦♦ts ♦❢ Σ ✬s ❡✐❣❡♥✈❛❧✉❡s ❡❧❧✐♣t✐❝❛❧ ♣❞❢ ♠✉st ❜❡ f µ , Σ ( ① ) = | Σ | − ✶ Ma ✷ ( ① , µ , Σ ) ✷ g N � � ✇❤❡r❡ g N ( · ) ≥ ✵ ✐s ❛ ❣❡♥❡r❛t♦r ❢✉♥❝t✐♦♥ r♦t❛t❡❞ t♦ ❢♦r♠ t❤❡ ❞✐str✐❜✉t✐♦♥✳ ❡①❛♠♣❧❡s ✐♥❝❧✉❞❡✿ ✉♥✐❢♦r♠ ✭s♦♠❡t✐♠❡s✮✱ ♥♦r♠❛❧✱ ❙t✉❞❡♥t t ✱ ❈❛✉❝❤② ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥s✿ ❢♦r ❛♥② K ✲✈❡❝t♦r ❛ ✱ K × N ♠❛tr✐① ❇ ✱ ❛♥❞ t❤❡ r✐❣❤t ❣❡♥❡r❛t♦r g K ✱ ❳ ∼ El ( µ , Σ , g N ) ⇒ ❛ + ❇❳ ∼ El � ❛ + ❇ µ , ❇ Σ ❇ ′ , g K � ❝♦rr❡❧❛t✐♦♥ ❝❛♣t✉r❡s ❛❧❧ ❞❡♣❡♥❞❡♥❝❡ str✉❝t✉r❡ ✭❝♦♣✉❧❛ ❛❞❞s ♥♦t❤✐♥❣✮ ✺✽ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s ❙t❛❜❧❡ ❞✐str✐❜✉t✐♦♥s ❧❡t ❳ , ❨ ❛♥❞ ❩ ❜❡ ■■❉ r✈s❀ t❤❡✐r ❞✐str✐❜✉t✐♦♥ ✐s st❛❜❧❡ ✐❢ ❢♦r ❛♥② ❝♦♥st❛♥ts α, β > ✵ t❤❡r❡ ❡①✐st ❝♦♥st❛♥ts γ ❛♥❞ δ > ✵ s✉❝❤ t❤❛t α ❳ + β ❨ d = γ + δ ❩ ❡①❛♠♣❧❡s✿ ♥♦r♠❛❧✱ ❈❛✉❝❤② ✭❜✉t ♥♦t ❧♦❣♥♦r♠❛❧✱ ♦r ❣❡♥❡r✐❝ ❙t✉❞❡♥t t ✮ ❝❧♦s❡❞ ✉♥❞❡r ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥s✱ t❤✉s ❛❧❧♦✇s ❡❛s② ♣r♦❥❡❝t✐♦♥ t♦ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥s st❛❜✐❧✐t② ✐♠♣❧✐❡s ❛❞❞✐t✐✈✐t② ✭t❤❡ s✉♠ ♦❢ t✇♦ ■■❉ r✈s ❜❡❧♦♥❣s t♦ t❤❡ s❛♠❡ ❢❛♠✐❧② ♦❢ ❞✐str✐❜✉t✐♦♥s✮✱ ❜✉t ♥♦t t❤❡ r❡✈❡rs❡ ❊①❛♠♣❧❡ √ √ ⇒ X + Y d ✶ st❛❜❧❡ ⇒ ❛❞❞✐t✐✈❡✿ X , Y , Z ∼ NID � ✶ , σ ✷ � = ✷ − ✷ + ✷ Z ✷ ❛❞❞✐t✐✈❡ �⇒ st❛❜❧❡✿ d ❳ , ❨ , ❩ ∼ WID ( ν, Σ ) ⇒ ❳ + ❨ ∼ W ( ✷ ν, Σ ) � = γ + δ ❩ ✺✾ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s ■♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ❞✐str✐❜✉t✐♦♥s t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ r✈ ❳ ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✐❢✱ ❢♦r ❛♥② ✐♥t❡❣❡r T ❳ d = ❨ ✶ + · · · + ❨ T ❢♦r s♦♠❡ ■■❉ r✈s ❨ ✶ , . . . , ❨ T ❡①❛♠♣❧❡s ✐♥❝❧✉❞❡✿ ❛❧❧ ❡❧❧✐♣t✐❝❛❧✱ ❣❛♠♠❛✱ ▲♦❣◆ ✭❜✉t ♥♦t ❲✐s❤❛rt ❢♦r N > ✶✮ s❤❛❧❧ s❡❡✿ ❛ss✐sts ✐♥ ♣r♦❥❡❝t✐♦♥ t♦ ❛r❜✐tr❛r② ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥s ✭❡✳❣✳ ❛♥② T ✮ ✻✵ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❙♣❡❝✐❛❧ ❝❧❛ss❡s ♦❢ ❞✐str✐❜✉t✐♦♥s ▲❡❝t✉r❡ ✸ ❡①❡r❝✐s❡s ▼❡✉❝❝✐ ❡①❡r❝✐s❡s ♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✶✳✷✳✻✱ ✶✳✷✳✼✱ ✶✳✸✳✶✱ ✶✳✸✳✹✱ ✷✳✶✳✸ ▼❆❚▲❆❇✿ ✶✳✷✳✽✱ ✶✳✸✳✷✱ ✶✳✸✳✸✱ ✷✳✶✳✶✱ ✷✳✶✳✷ ♣r♦❥❡❝t s♣❡❝✐❢② ❛ ♣♦rt❢♦❧✐♦ ✉s✐♥❣ ②♦✉r t❤r❡❡ ❛ss❡ts✳ ■t ♥❡❡❞ ♥♦t ❜❡ ♦♣t✐♠❛❧✳ ❯♣❞❛t❡ ✐t ❛s t❤❡ t❡r♠ ❣♦❡s ♦♥✳ ✻✶ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ■♥tr♦❞✉❝t✐♦♥ t❤❡ ❝♦♣✉❧❛ ✐s ❛ st❛♥❞❛r❞✐③❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ♣✉r❡❧② ❥♦✐♥t ❢❡❛t✉r❡s ♦❢ ❛ ♠✉❧t✐✈❛r✐❛t❡ ❞✐str✐❜✉t✐♦♥✱ ✇❤✐❝❤ ✐s ♦❜t❛✐♥❡❞ ❜② ✜❧t❡r✐♥❣ ♦✉t ❛❧❧ t❤❡ ♣✉r❡❧② ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❢❡❛t✉r❡s✱ ♥❛♠❡❧② t❤❡ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ❡❛❝❤ ❡♥tr② X n ✳ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✹✵✮ ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ❈❤ ✼✮ ❣♦❡s ✐♥t♦ ♠♦r❡ ❞❡t❛✐❧ t❤❛♥ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ❈❤ ✷✮ ♦♥ ❝♦♣✉❧❛s ♠♦r❡ ♠❛t❡r✐❛❧ ❛❜♦✉t t❤❡ ❜♦♦❦ ✐s ❛✈❛✐❧❛❜❧❡ ❛t ✇✇✇✳qr♠t✉t♦r✐❛❧✳♦r❣ s❡❡ ❊♠❜r❡❝❤ts ✭✷✵✵✾✮ ❢♦r t❤♦✉❣❤ts ♦♥ t❤❡ ✏❝♦♣✉❧❛ ❝r❛③❡✑✱ ❢r♦♠ ♦♥❡ ♦❢ ✐ts ♣✐♦♥❡❡rs✱ ❛♥❞ ❛ ✏♠✉st✲r❡❛❞✑ ❢♦r ❝♦♥t❡①t t❤❡ ❝❧❛ss✐❝ t❡①t ✐s ◆❡❧s❡♥ ✭✷✵✵✻✮❀ ✐t ❝♦♥t❛✐♥s ✇♦r❦❡❞ ❡①❛♠♣❧❡s ❛♥❞ s❡t q✉❡st✐♦♥s✱ ❛♥❞ ❤❛s t❤❡ s♣❛❝❡ t♦ ♣r♦♣❡r❧② ❞❡✈❡❧♦♣ t❤❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ❛ ✷✵✵✾ ✇✐r❡❞✳❝♦♠ ❛rt✐❝❧❡ ❜❧❛♠❡❞ t❤❡ ●❛✉ss✐❛♥ ❝♦♣✉❧❛ ❢♦r♠✉❧❛ ❢♦r ✏❦✐❧❧✐♥❣✑ ❲❛❧❧ ❙tr❡❡t ✻✸ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ❈♦♣✉❧❛s ❞❡✜♥❡❞ ❉❡✜♥✐t✐♦♥ ❆♥ N ✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♣✉❧❛✱ ❯ ✱ ✐s ❞❡✜♥❡❞ ♦♥ [ ✵ , ✶ ] N ❀ ✐ts ❏❉❋✱ F ❯ ✱ ❤❛s st❛♥❞❛r❞ ✉♥✐❢♦r♠ ♠❛r❣✐♥❛❧ ❞✐str✐❜✉t✐♦♥s✳ ❝♦♣✉❧❛ ❡①❛♠♣❧❡ ❊♠❜r❡❝❤ts ✭✷✵✵✾✱ ♣✳✻✹✵✮ ♥♦t❡s t❤❛t ♦t❤❡r st❛♥❞❛r❞✐s❛t✐♦♥s t❤❛♥ t❤❡ ❝♦♣✉❧❛✬s t♦ U ([ ✵ , ✶ ]) ♠❛② ❜❡ ♠♦r❡ ✉s❡❢✉❧ ✉♥❞❡r ❝❡rt❛✐♥ ❝✐r❝✉♠st❛♥❝❡s ✻✹ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ❙❦❧❛r✬s t❤❡♦r❡♠ ❚❤❡♦r❡♠ ✭❙❦❧❛r✱ ✶✾✺✾✮ ▲❡t F ❳ ❜❡ ❛ ❏❉❋ ✇✐t❤ ♠❛r❣✐♥❛❧s✱ F X ✶ , . . . , F X N ✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ❝♦♣✉❧❛✱ ❯ ✱ ✇✐t❤ ❏❉❋ F ❯ : [ ✵ , ✶ ] N → [ ✵ , ✶ ] s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ x ✶ , . . . , x N ∈ R ✱ ✭✶✮ F ❳ ( ① ) = F ❯ ( F X ✶ ( x ✶ ) , . . . , F X N ( x N )) . ■❢ t❤❡ ♠❛r❣✐♥❛❧s ❛r❡ ❝♦♥t✐♥✉♦✉s✱ F ❯ ✐s ✉♥✐q✉❡✳ ❈♦♥✈❡rs❡❧②✱ ✐❢ ❯ ✐s ❛ ❝♦♣✉❧❛ ❛♥❞ F X ✶ , . . . , F X N ❛r❡ ✉♥✐✈❛r✐❛t❡ ❈❉❋s✱ t❤❡♥ F ❳ ✱ ❞❡✜♥❡❞ ✐♥ ❡q✉❛t✐♦♥ ✶ ✐s ❛ ❏❉❋ ✇✐t❤ ♠❛r❣✐♥❛❧s F X ✶ , . . . , F X N ✳ ❯s❡❢✉❧ t♦ ❞❡❝♦♠♣♦s❡ r✈ ✐♥t♦ ♠❛r❣✐♥❛❧s ❛♥❞ ❝♦♣✉❧❛✿ ✶ ♠❛② ❤❛✈❡ ♠♦r❡ ❝♦♥✜❞❡♥❝❡ ✐♥ ♠❛r❣✐♥❛❧s t❤❛♥ ❏❉❋ ❡✳❣✳ ♠✉❧t✐✈❛r✐❛t❡ t ✇✐t❤ ❞✐✛❡r✐♥❣ t❛✐❧✲t❤✐❝❦♥❡ss ♣❛r❛♠❡t❡rs ❝❛♥ ♠♦❞✐❢② ❥♦✐♥t ❞✐str✐❜✉t✐♦♥s ♦❢ ❡①tr❡♠❡ ✈❛❧✉❡s ✷ ❝❛♥ ❡①♣❡r✐♠❡♥t ✇✐t❤ s❤♦❝❦s✿ ✐❞✐♦s②♥❝r❛t✐❝ ✈✐❛ ♠❛r❣✐♥❛❧s✱ ❝♦♠♠♦♥ ✈✐❛ ❝♦♣✉❧❛ ▼❡✉❝❝✐ ✭✷✵✵✺✱ ✭✷✳✸✵✮✮ r❡❧❛t❡s f ❳ t♦ f ❯ ✿ s♦♠❡t✐♠❡s ♠♦r❡ ✉s❡❢✉❧ ✻✺ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s Pr♦❜❛❜✐❧✐t② ❛♥❞ q✉❛♥t✐❧❡ tr❛♥s❢♦r♠❛t✐♦♥s ■❢ ✇❛♥t t♦ st♦❝❤❛st✐❝❛❧❧② s✐♠✉❧❛t❡ Z ✱ ❜✉t X ✐s ❡❛s✐❡r t♦ ❣❡♥❡r❛t❡✱ ❝❛♥ tr❛♥s❢♦r♠ ❛♥② r✈ X ✇✐t❤ ❝♦♥t✐♥✉♦✉s ❈❉❋ ✐♥t♦ ❛♥② ♦t❤❡r Z ✈✐❛ ❛ ♥❡✇ r✈ U ❚❤❡♦r❡♠ ✭Pr♦♣♦s✐t✐♦♥ ✼✳✷ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✮❀ ▼❡✉❝❝✐ ✷✳✷✺ ✲ ✷✳✷✼✮ ▲❡t F X ❜❡ ❛ ❈❉❋ ❛♥❞ ❧❡t Q X ❞❡♥♦t❡ ✐ts ✐♥✈❡rs❡✳ ❚❤❡♥ ✶ ✐❢ X ❤❛s ❛ ❝♦♥t✐♥✉♦✉s ✉♥✐✈❛r✐❛t❡ ❈❉❋✱ F X ✱ t❤❡♥ F X ( X ) ∼ U ([ ✵ , ✶ ]) ♣r♦♦❢ ✷ ✐❢ U ≡ F X ( X ) d = F Z ( Z ) ∼ U ([ ✵ , ✶ ]) ✱ t❤❡♥ Z d = Q Z ( U ) t❤❡ ♥❡✇ r✈✱ U ✐s t❤❡ ❣r❛❞❡ ♦❢ X ♥♦✇ ❤❛✈❡ ✸ r❞ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ❝♦♣✉❧❛s✿ ❯ ✱ t❤❡ ❝♦♣✉❧❛ ♦❢ ❛ ♠✉❧t✐✈❛r✐❛t❡ r✈✱ ❳ ✱ ✐s t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ ✐ts ❣r❛❞❡s ( U ✶ , . . . , U N ) ′ ≡ ( F X ✶ ( X ✶ ) , . . . , F X N ( X N )) ′ ✻✻ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ■♥❞❡♣❡♥❞❡♥❝❡ ❝♦♣✉❧❛ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ r✈s ⇔ ❏❉❋ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡✐r ✉♥✐✈❛r✐❛t❡ ❈❉❋s ❛♣♣❧②✐♥❣ ❙❦❧❛r✬s t❤❡♦r❡♠ t♦ ✐♥❞❡♣❡♥❞❡♥t r✈s✱ X ✶ , . . . , X N N � F ❳ ( ① ) = F X n ( x n ) = F ❯ ( F X ✶ ( x ✶ ) , . . . , F X N ( x N )) n = ✶ t❤✉s✱ s✉❜st✐t✉t✐♥❣ F X n ( x n ) = u n ✱ ♣r♦✈✐❞❡s t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ❝♦♣✉❧❛ N � Π ( ✉ ) ≡ F ❯ ( u ✶ , . . . , u N ) = u n n = ✶ ✇❤✐❝❤ ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ t❤❡ ✉♥✐t ❤②♣❡r✲❝✉❜❡✱ ✇✐t❤ ❛ ❤♦r✐③♦♥t❛❧ ♣❞❢✱ π ( ✉ ) = ✶ ❙❝❤✇❡✐③❡r✲❲♦❧❢ ♠❡❛s✉r❡s ♦❢ ❞❡♣❡♥❞❡♥❝❡ ✭✐♥❞❡①❡❞ ❜② p ✐♥ L p ✲♥♦r♠✮✿ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛ ❝♦♣✉❧❛ ❛♥❞ t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ❝♦♣✉❧❛ ✻✼ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ❙tr✐❝t❧② ✐♥❝r❡❛s✐♥❣ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ♠❛r❣✐♥❛❧s r❡❝❛❧❧✿ ❝♦rr❡❧❛t✐♦♥ ♦♥❧② ✐♥✈❛r✐❛♥t ✉♥❞❡r ❧✐♥❡❛r tr❛♥s❢♦r♠❛t✐♦♥s ❚❤❡♦r❡♠ ✭Pr♦♣♦s✐t✐♦♥ ✼✳✼ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✮✮ ▲❡t ( X ✶ , . . . , X N ) ❜❡ ❛ r✈ ✇✐t❤ ❝♦♥t✐♥✉♦✉s ♠❛r❣✐♥❛❧s ❛♥❞ ❝♦♣✉❧❛ ❯ ✱ ❛♥❞ ❧❡t g ✶ , . . . , g N ❜❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥s✳ ❚❤❡♥ ( g ✶ ( X ✶ ) , . . . , g N ( X N )) ❛❧s♦ ❤❛s ❝♦♣✉❧❛ ❯ ✳ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤✐s ✐s t❤❡ ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛ ❧❡t t❤❡ r✈s X ✶ , . . . , X N ❤❛✈❡ ❝♦♥t✐♥✉♦✉s ❞❢s t❤❛t ❛r❡ ♣❡r❢❡❝t❧② ♣♦s✐t✐✈❡❧② ❞❡♣❡♥❞❡♥t✱ s♦ t❤❛t X n = g n ( X ✶ ) ❛❧♠♦st s✉r❡❧② ❢♦r ❛❧❧ n ∈ { ✷ , . . . , N } ❢♦r str✐❝t❧② ✐♥❝r❡❛s✐♥❣ g n ( · ) ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛ ✐s t❤❡♥ M ( ✉ ) ≡ ♠✐♥ { u ✶ , . . . , u N } ✇❤❡r❡ t❤❡ ❏❉❋ ♦❢ t❤❡ r✈ ( U , . . . , U ) ✐s s✳t✳ U ∼ U ([ ✵ , ✶ ]) ✭▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts✱ ✷✵✶✺✱ ♣✳✷✷✻✮ ✻✽ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ❋ré❝❤❡t✲❍♦❡✛❞✐♥❣ ❜♦✉♥❞s ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛✱ M ✱ ✐s ❋ré❝❤❡t✲❍♦❡✛❞✐♥❣ ✉♣♣❡r ❜♦✉♥❞ ❋ré❝❤❡t✲❍♦❡✛❞✐♥❣ ❧♦✇❡r ❜♦✉♥❞✱ W ✱ ✐s♥✬t ❝♦♣✉❧❛ ❢♦r N > ✷✿ � N � � W ( ✉ ) ≡ ♠❛① ✶ − N + u n , ✵ n = ✶ ❛♥② ❝♦♣✉❧❛✬s ❈❉❋ ✜ts ❜❡t✇❡❡♥ t❤❡s❡ W ( ✉ ) ≤ F ❯ ( ✉ ) ≤ M ( ✉ ) ✇❤✐❝❤ ❝♦♣✉❧❛ ✐s ✷ ♥❞ ✜❣✉r❡❄ ❘ ❝♦❞❡✿ ❍är❞❧❡ ❛♥❞ ❖❦❤r✐♥ ✭✷✵✶✵✮ ✻✾ ✴ ✷✷✹

▼✉❧t✐✈❛r✐❛t❡ st❛t✐st✐❝s ❈♦♣✉❧❛s ❆ ❝❛❧❧ ♦♣t✐♦♥ ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r t✇♦ st♦❝❦ ♣r✐❝❡s✱ t❤❡ r✈s ❳ = ( X ✶ , X ✷ ) ✱ ❛♥❞ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ♦♥ t❤❡ ✜rst ✇✐t❤ str✐❦❡ ♣r✐❝❡ K ✳ ❚❤❡ ♣❛②♦✛ ♦♥ t❤✐s ♦♣t✐♦♥ ✐s t❤❡r❡❢♦r❡ ❛❧s♦ ❛ r✈✱ C ✶ ≡ ♠❛① { X ✶ − K , ✵ } ✳ ❚❤✉s✱ C ✶ ❛♥❞ X ✶ ❛r❡ ❝♦✲♠♦♥♦t♦♥✐❝❀ t❤❡✐r ❝♦♣✉❧❛ ✐s M ✱ t❤❡ ❝♦✲♠♦♥♦t♦♥✐❝✐t② ❝♦♣✉❧❛✳ ❋✉rt❤❡r✱ ( X ✶ , X ✷ ) ❛♥❞ ( C ✶ , X ✷ ) ❛r❡ ❛❧s♦ ❝♦✲♠♦♥♦t♦♥✐❝❀ t❤❡ ❝♦♣✉❧❛ ♦❢ ( X ✶ , X ✷ ) ✐s t❤❡ s❛♠❡ ❛s t❤❛t ♦❢ ( C ✶ , X ✷ ) ✳ ❲❤❛t t❡❝❤♥✐❝❛❧ ❞❡t❛✐❧ ✐s t❤❡ ❛❜♦✈❡ ♠✐ss✐♥❣❄ ❍♦✇ ✐s t❤✐s ♦✈❡r❝♦♠❡❄ ❝♦✲♠♦♥♦t♦♥✐❝ ❛❞❞✐t✐✈✐t② ✼✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❈♦♥❝❡♣t✉❛❧ ♦✈❡r✈✐❡✇ ▼❡✉❝❝✐ ✭✷✵✵✺✮ ✐❞❡♥t✐✜❡s t❤❡ ❢♦❧❧♦✇✐♥❣ st❡♣s ❢♦r ❜✉✐❧❞✐♥❣ t❤❡ ❧✐♥❦ ❜❡t✇❡❡♥ ❤✐st♦r✐❝❛❧ ♣❡r❢♦r♠❛♥❝❡ ❛♥❞ ❢✉t✉r❡ ❞✐str✐❜✉t✐♦♥s ✶ ❞❡t❡❝t✐♥❣ t❤❡ ✐♥✈❛r✐❛♥ts ✇❤❛t ♠❛r❦❡t ✈❛r✐❛❜❧❡s ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ■■❉ r✈s❄ ✷ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥✈❛r✐❛♥ts ❤♦✇ ❢r❡q✉❡♥t❧② ❞♦ t❤❡s❡ ❝❤❛♥❣❡ ✭q✳✈✳ ❇❛✉❡r ❛♥❞ ❇r❛✉♥ ✭✷✵✶✵✮✮ ✸ ♣r♦❥❡❝t✐♥❣ t❤❡ ✐♥✈❛r✐❛♥ts ✐♥t♦ t❤❡ ❢✉t✉r❡ ✹ ♠❛♣♣✐♥❣ t❤❡ ✐♥✈❛r✐❛♥ts ✐♥t♦ t❤❡ ♠❛r❦❡t ♣r✐❝❡s ❆s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ ❵♠♦st✬ r❛♥❞♦♠♥❡ss ♠❛② ❜❡ ♠✉❝❤ ❧❡ss t❤❛♥ t❤❛t ♦❢ t❤❡ ♣♦rt❢♦❧✐♦ s♣❛❝❡✱ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ t❡❝❤♥✐q✉❡s ✇✐❧❧ ❡♥❤❛♥❝❡ tr❛❝t❛❜✐❧✐t② ✼✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❙t②❧✐s❡❞ ❢❛❝ts ❯♥✐✈❛r✐❛t❡ st②❧✐s❡❞ ❢❛❝ts ●✐✈❡♥ ❛♥ ❛ss❡t ♣r✐❝❡ P t ✱ ❧❡t ✐ts ❝♦♠♣♦✉♥❞ r❡t✉r♥ ❛t t✐♠❡ t ❢♦r ❤♦r✐③♦♥ τ ❜❡ C t ,τ ≡ ❧♥ P t P t − τ ❚❤❡♥✱ ❢♦❧❧♦✇✐♥❣ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✱ ➓✸✳✶✮✿ ✶ s❡r✐❡s ♦❢ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❛r❡ ♥♦t ■■❉✱ ❜✉t s❤♦✇ ❧✐tt❧❡ s❡r✐❛❧ ❝♦rr❡❧❛t✐♦♥ ❛❝r♦ss ❞✐✛❡r❡♥t ❧❛❣s ✐❢ ♥♦t ■■❉✱ t❤❡♥ ♣r✐❝❡s ❞♦♥✬t ❢♦❧❧♦✇ r❛♥❞♦♠ ✇❛❧❦ ✐❢ ♥❡✐t❤❡r ■■❉ ♥♦r ♥♦r♠❛❧✱ ❇❧❛❝❦✲❙❝❤♦❧❡s✲▼❡rt♦♥ ♣r✐❝✐♥❣ ✐s ✐♥ tr♦✉❜❧❡ ✷ ✈♦❧❛t✐❧✐t② ❝❧✉st❡r✐♥❣✿ s❡r✐❡s ♦❢ | C t ,τ | ♦r C ✷ t ,τ s❤♦✇ ♣r♦❢♦✉♥❞ s❡r✐❛❧ ❝♦rr❡❧❛t✐♦♥ ✸ ❝♦♥❞✐t✐♦♥❛❧ ✭♦♥ ❛♥② ❤✐st♦r②✮ ❡①♣❡❝t❡❞ r❡t✉r♥s ❛r❡ ❝❧♦s❡ t♦ ③❡r♦ ✹ ✈♦❧❛t✐❧✐t② ❛♣♣❡❛rs t♦ ✈❛r② ♦✈❡r t✐♠❡ ✺ ❡①tr❡♠❡ r❡t✉r♥s ❛♣♣❡❛r ✐♥ ❝❧✉st❡rs ✻ r❡t✉r♥s s❡r✐❡s ❛r❡ ❧❡♣t♦❦✉rt✐❝ ✭❤❡❛✈②✲t❛✐❧❡❞✮ ❛s ❤♦r✐③♦♥ ✐♥❝r❡❛s❡s✱ r❡t✉r♥s ♠♦r❡ ■■❉✱ ❧❡ss ❤❡❛✈②✲t❛✐❧❡❞ ✼✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❙t②❧✐s❡❞ ❢❛❝ts ▼✉❧t✐✈❛r✐❛t❡ st②❧✐s❡❞ ❢❛❝ts ●✐✈❡♥ ❛ ✈❡❝t♦r ♦❢ ❛ss❡t ♣r✐❝❡s P t ✱ ❧❡t ✐ts ❝♦♠♣♦✉♥❞ r❡t✉r♥ ❛t t✐♠❡ t ❢♦r ❤♦r✐③♦♥ τ ❜❡ ❞❡✜♥❡❞ ❝♦♠♣♦♥❡♥t✲✇✐s❡ ❛s ❈ t ,τ ≡ ❧♥ P t P t − τ ❋♦❧❧♦✇✐♥❣ ▼❝◆❡✐❧✱ ❋r❡②✱ ❛♥❞ ❊♠❜r❡❝❤ts ✭✷✵✶✺✱ ➓✸✳✷✮ ✶ ❈ t ,τ s❡r✐❡s s❤♦✇ ❧✐tt❧❡ ❡✈✐❞❡♥❝❡ ♦❢ ✭s❡r✐❛❧✮ ❝r♦ss✲❝♦rr❡❧❛t✐♦♥✱ ❡①❝❡♣t ❢♦r ❝♦♥t❡♠♣♦r❛♥❡♦✉s r❡t✉r♥s ✷ | ❈ t ,τ | s❡r✐❡s s❤♦✇ ♣r♦❢♦✉♥❞ ❡✈✐❞❡♥❝❡ ♦❢ ✭s❡r✐❛❧✮ ❝r♦ss✲❝♦rr❡❧❛t✐♦♥ ✸ ❝♦rr❡❧❛t✐♦♥s ❜❡t✇❡❡♥ ❝♦♥t❡♠♣♦r❛♥❡♦✉s r❡t✉r♥s ✈❛r② ♦✈❡r t✐♠❡ ✹ ❡①tr❡♠❡ r❡t✉r♥s ✐♥ ♦♥❡ s❡r✐❡s ♦❢t❡♥ ❝♦✐♥❝✐❞❡ ✇✐t❤ ❡①tr❡♠❡ r❡t✉r♥s ✐♥ s❡✈❡r❛❧ ♦t❤❡r s❡r✐❡s ✼✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ▼❛r❦❡t ✐♥✈❛r✐❛♥ts ♠❛r❦❡t ✐♥✈❛r✐❛♥ts✱ X t ✱ ❛r❡ ■■❉ r✈s ✶ t❛❦✐♥❣ ♦♥ r❡❛❧✐s❡❞ ✈❛❧✉❡s x t ❛t t✐♠❡ t ✷ t❤❡② ❛r❡ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s ✐❢ t❤❡ ■■❉ ❞✐str✐❜✉t✐♦♥ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ ❛ r❡❢❡r❡♥❝❡ ❞❛t❡✱ ˜ t ✐♥✈❛r✐❛♥ts ❧✐❦❡ t❤✐s ♠❛❦❡ ✐t ❵❡❛s②✬ t♦ ❢♦r❡❝❛st ❤♦✇ t❡st ❢♦r ■■❉ ✭❈❛♠♣❜❡❧❧✱ ▲♦✱ ❛♥❞ ▼❛❝❑✐♥❧❛②✱ ✶✾✾✼✱ ❈❤❛♣t❡r ✷✮❄ ✐♥ ♣❛rt✐❝✉❧❛r✱ ❤♦✇ ♣♦s✐t t❤❡ r✐❣❤t H ✶ ❄ t❡sts ❛❣❛✐♥st ♣❛rt✐❝✉❧❛r H ✶ ✬s ♦❢t❡♥ ♠✐ss❡❞ ♥♦♥✲❧✐♥❡❛r ❞❡t❡r♠✐♥✐st✐❝ r❡❧❛t✐♦♥s❤✐♣s ❡✳❣✳ ❧♦❣✐st✐❝ ♠❛♣✱ x t + ✶ = rx t ( ✶ − x t ) ❛♥❞ t❡♥t ♠❛♣✱ � ✐❢ x t < ✶ � µ x t x t + ✶ = ✷ µ ( ✶ − x t ) ♦t❤❡r✇✐s❡ ❇❉❙✭▲✮ t❡st ✭❇r♦❝❦ ❡t ❛❧✳✱ ✶✾✾✻✮ ❞❡s✐❣♥❡❞ t♦ ❝❛♣t✉r❡ t❤✐s✱ ❜✉t ❢❛✐❧s ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ r❡❛❧ ♥♦✐s❡❀ ♥♦t ♦❢t❡♥ ✉s❡❞ ❞✉❡ t♦ str♦♥❣ t❤❡♦r❡t✐❝❛❧ ♣r✐♦rs ♦♥ H ✶ ✇❡ t❤❡r❡❢♦r❡ ♣r❡s❡♥t t✇♦ ❤❡✉r✐st✐❝ t❡sts ✭q✳✈✳ ▼❡✉❝❝✐✱ ✷✵✵✾✱ ➓✷✮ ✼✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❍❡✉r✐st✐❝ t❡st ✶✿ ❝♦♠♣❛r❡ s♣❧✐t s❛♠♣❧❡ ❤✐st♦❣r❛♠s ❜② t❤❡ ●❧✐✈❡♥❦♦✲❈❛♥t❡❧❧✐ t❤❡♦r❡♠✱ ❡♠♣✐r✐❝❛❧ ♣❞❢ → tr✉❡ ♣❞❢ ❛s t❤❡ ♥✉♠❜❡r ♦❢ ■■❉ ♦❜s❡r✈❛t✐♦♥s ❣r♦✇s s♣❧✐t t❤❡ t✐♠❡ s❡r✐❡s ✐♥ ❤❛❧❢ ❛♥❞ ❝♦♠♣❛r❡ t❤❡ t✇♦ ❤✐st♦❣r❛♠s ✇❤❛t s❤♦✉❧❞ t❤❡ t✇♦ ❤✐st♦❣r❛♠s ❧♦♦❦ ❧✐❦❡ ✐❢ ■■❉❄ ✼✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉♦ st♦❝❦ ♣r✐❝❡s✱ P t ✱ ♣❛ss t❤❡ ❤✐st♦❣r❛♠ t❡st❄ ❈❛✈❡❛t✿ ❛♣♣❛r❡♥t s✐♠✐❧❛r✐t② ❝❤❛♥❣❡s ✇✐t❤ ❜✐♥ s✐③❡ ❝❤♦✐❝❡ ❆❧❧ ❞❛t❛✿ ❚❍❆❘●❊❙✿■❉ ✵✶✴✵✶✴✵✼ ✕ ✶✵✴✵✾✴✵✾ ✼✽ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉♦ ❧✐♥❡❛r st♦❝❦ r❡t✉r♥s✱ L t ,τ ✱ ♣❛ss t❤❡ ❤✐st♦❣r❛♠ t❡st❄ P t ▲✐♥❡❛r r❡t✉r♥s ❛r❡ L t ,τ ≡ P t − τ − ✶ ✼✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉♦ ❝♦♠♣♦✉♥❞ st♦❝❦ r❡t✉r♥s✱ C t ,τ ✱ ♣❛ss t❤❡ ❤✐st♦❣r❛♠ t❡st❄ P t ❈♦♠♣♦✉♥❞ r❡t✉r♥s ❛r❡ C t ,τ ≡ ❧♥ P t − τ ✽✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❍❡✉r✐st✐❝ t❡st ✷✿ ♣❧♦t x t ✈ x t − ˜ τ ♣❧♦t x t ✈ x t − ˜ τ ✱ ✇❤❡r❡ ˜ τ ✐s t❤❡ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧ ✇❤❛t s❤♦✉❧❞ t❤❡ ♣❧♦t ❧♦♦❦ ❧✐❦❡ ✐❢ ■■❉❄ s②♠♠❡tr✐❝ ❛❜♦✉t t❤❡ ❞✐❛❣♦♥❛❧✿ ✐❢ ■■❉✱ ❞♦❡s♥✬t ♠❛tt❡r ✐❢ ♣❧♦t x t ✈ x t − ˜ τ ♦r x t − ˜ τ ✈ x t ❝✐r❝✉❧❛r✿ ♠❡❛♥✲✈❛r✐❛♥❝❡ ❡❧❧✐♣s♦✐❞ ✇✐t❤ ♠❡❛♥ ( µ, µ ) ✱ ✈❛r✐❛♥❝❡ s❛♠❡ ✐♥ ❡❛❝❤ ❞✐r❡❝t✐♦♥✱ ❛❧✐❣♥❡❞ ✇✐t❤ ❝♦♦r❞✐♥❛t❡ ❛①❡s ❛s ❝♦✈❛r✐❛♥❝❡ ③❡r♦ ✭❞✉❡ t♦ ✐♥❞❡♣❡♥❞❡♥❝❡✮ ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✺✺✮ ❤✐♥t ✽✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉♦ st♦❝❦ ♣r✐❝❡s✱ P t ✱ ♣❛ss t❤❡ ❧❛❣❣❡❞ ♣❧♦t t❡st❄ ❲❤❛t ❞♦❡s t❤✐s t❡❧❧ ✉s ❛❜♦✉t st♦❝❦ ♣r✐❝❡s❄ ✽✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉♦ ❧✐♥❡❛r st♦❝❦ r❡t✉r♥s✱ L t ,τ ✱ ♣❛ss t❤❡ ❧❛❣❣❡❞ ♣❧♦t t❡st❄ ❲❤❛t ❞♦ ✇❡ ❡①♣❡❝t ❝♦♠♣♦✉♥❞ r❡t✉r♥s t♦ ❧♦♦❦ ❧✐❦❡✱ ❛s ❛ r❡s✉❧t❄ ✐♥❞❡♣❡♥❞❡♥❝❡ ✽✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉♦ ❝♦♠♣♦✉♥❞ st♦❝❦ r❡t✉r♥s✱ C t ,τ ✱ ♣❛ss t❤❡ ❧❛❣❣❡❞ ♣❧♦t t❡st❄ ❲❤❛t ❞♦ ✇❡ ❡①♣❡❝t t♦t❛❧ r❡t✉r♥s✱ P t − τ t♦ ❧♦♦❦ P t H t ,τ ≡ ❧✐❦❡❄ ✽✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ■♥✈❛r✐❛♥ts ❢♦r ❡q✉✐t✐❡s✱ ❝♦♠♠♦❞✐t✐❡s ❛♥❞ ❡①❝❤❛♥❣❡ r❛t❡s ❤❛✈❡ s❡❡♥ ❧✐♥❡❛r✱ ❝♦♠♣♦✉♥❞✱ t♦t❛❧ r❡t✉r♥s ❢♦r ❚❍❆❘●❊❙ ❡q✉✐t② ❢✉♥❞ ♣❛ss t❤❡ ❤❡✉r✐st✐❝ t❡sts ♣r❡❢❡r t♦ ✉s❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❛s s❤❛❧❧ s❡❡ t❤❛t ❝❛♥ ♠♦r❡ ❡❛s✐❧② ♣r♦❥❡❝t ❞✐str✐❜✉t✐♦♥s t♦ ✐♥✈❡st♠❡♥t ✶ ❤♦r✐③♦♥ ❣r❡❛t❡r s②♠♠❡tr② ❢❛❝✐❧✐t❛t❡s ♠♦❞❡❧❧✐♥❣ ❜② ❡❧❧✐♣t✐❝❛❧ ❞✐str✐❜✉t✐♦♥s ✷ ∆ ❨❚▼ ✐♥❞✐✈✐❞✉❛❧ ❡q✉✐t✐❡s✱ ❝♦♠♠♦❞✐t✐❡s✱ ❡①❝❤❛♥❣❡ r❛t❡s ❤❛✈❡ s✐♠✐❧❛r ♣r♦♣❡rt✐❡s✿ ♥♦ t✐♠❡ ❤♦r✐③♦♥s ❦❡② ❛ss✉♠♣t✐♦♥s ❡q✉✐t✐❡s✿ ❡✐t❤❡r ♥♦ ❞✐✈✐❞❡♥❞s✱ ♦r ❞✐✈✐❞❡♥❞s ♣❧♦✉❣❤❡❞ ❜❛❝❦ ✐♥ ✶ ❣❡♥❡r❛❧❧②✱ ♥♦♥✲♦✈❡r❧❛♣♣✐♥❣ ✲ s❡❡ W t ✐♥ ▼❡✉❝❝✐✬s ♦♥❧✐♥❡ ❡①❡r❝✐s❡ ✸✳✷✳✶ ✷ ✭❖❝t ✷✵✵✾✮ ❛s ❛ ❝♦✉♥t❡r✲❡①❛♠♣❧❡ ❛❝❝❡♣t ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❛s ■■❉ ❛s ❡①♣♦s✐t✐♦♥❛❧ ❞❡✈✐❝❡ ✭r❡❝❛❧❧ st②❧✐s❡❞ ❢❛❝ts✮❀ s❡❡ ▼❡✉❝❝✐ ✭✷✵✵✾✮ ❢♦r ♠♦r❡ ❞✐s❝✉ss✐♦♥ ✽✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ▲❡❝t✉r❡ ✹ ❡①❡r❝✐s❡s ◆❡❧s❡♥ ✭✷✵✵✻✱ ❊①❡r❝✐s❡ ✷✳✶✷✮ ▲❡t X ❛♥❞ Y ❜❡ r✈s ✇✐t❤ ❏❉❋ ✶ + e − x + e − y � − ✶ � H ( x , y ) = ❢♦r ❛❧❧ x , y ∈ ¯ R ✱ t❤❡ ❡①t❡♥❞❡❞ r❡❛❧s✳ s❤♦✇ t❤❛t X ❛♥❞ Y ❤❛✈❡ st❛♥❞❛r❞ ✭✉♥✐✈❛r✐❛t❡✮ ❧♦❣✐st✐❝ ❞✐str✐❜✉t✐♦♥s ✶ ✶ + e − x � − ✶ ❛♥❞ G ( y ) = ✶ + e − y � − ✶ . � � F ( x ) = uv s❤♦✇ t❤❛t t❤❡ ❝♦♣✉❧❛ ♦❢ X ❛♥❞ Y ✐s C ( u , v ) = u + v − uv ✳ ✷ ▼❡✉❝❝✐ ❡①❡r❝✐s❡s ♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✸✳✷✳✶❀ ✐♥ q✉❡st✐♦♥ ✼✳✶✳✶✱ ✇❤② ✐s ❡q✉❛t✐♦♥ ✹✹✵ ♥♦t ❛ t②♣♦❄❀ ✐♥ q✉❡st✐♦♥ ✼✳✸✳✷✱ ❤♦✇ ❞♦ ✇❡ ✏♥♦t✐❝❡ t❤❛t ♥♦r♠❛❧ ♠❛r❣✐♥❛❧s ❬❜♦✉♥❞ t♦❣❡t❤❡r ❜②❪ ❛ ♥♦r♠❛❧ ❝♦♣✉❧❛ ❣✐✈❡ r✐s❡ t♦ ❛ ♥♦r♠❛❧ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥✑❄ ▼❆❚▲❆❇✿ ✷✳✶✳✹✱ ✷✳✷✳✶✱ ✷✳✷✳✷✱ ✷✳✷✳✸✱ ✷✳✷✳✹✱ ✷✳✷✳✺✱ ✷✳✷✳✻✱ ✷✳✷✳✼✱ ✸✳✶✳✶✱ ✸✳✶✳✷✱ ✸✳✶✳✸ ♣r♦❥❡❝t ❞♦ ②♦✉r ❛ss❡ts ♣♦ss❡ss t❤❡ r❡❧❡✈❛♥t ✐♥✈❛r✐❛♥t ♣r♦♣❡rt✐❡s❄ ✽✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❋✐①❡❞ ✐♥❝♦♠❡✿ ③❡r♦✲❝♦✉♣♦♥ ❜♦♥❞s ♠❛❦❡ ♥♦ t❡r♠❧② ♣❛②♠❡♥ts ❛s s✐♠♣❧❡st ❢♦r♠ ♦❢ ❜♦♥❞✱ ❢♦r♠ ❜❛s✐s ❢♦r ❛♥❛❧②s✐s ♦❢ ❜♦♥❞s ✜①❡❞ ✐♥❝♦♠❡ ❛s ❝❡rt❛✐♥ ❬❄❪ ♣❛②♦✉t ❛t ❢❛❝❡ ♦r r❡❞❡♠♣t✐♦♥ ✈❛❧✉❡ ✭s❡❡ ❇r✐❣♦✱ ▼♦r✐♥✐✱ ❛♥❞ P❛❧❧❛✈✐❝✐♥✐ ✭✷✵✶✸✮ ❢♦r r✐❝❤❡r r✐s❦ ♠♦❞❡❧❧✐♥❣✮ ❜♦♥❞ ♣r✐❝❡ t❤❡♥ Z ( E ) ✱ ✇❤❡r❡ t ≤ E ✐s ❞❛t❡✱ ❛♥❞ E ✐s ♠❛t✉r✐t② ❞❛t❡ t ♥♦r♠❛❧✐s❡ Z ( E ) = ✶ E ❛r❡ ❜♦♥❞ ♣r✐❝❡s ✐♥✈❛r✐❛♥ts❄ st♦❝❦ ♣r✐❝❡s ✇❡r❡♥✬t ✶ t✐♠❡ ❤♦♠♦❣❡♥❡✐t② ✈✐♦❧❛t❡❞ ✷ ❛r❡ r❡t✉r♥s ✭t♦t❛❧✱ s✐♠♣❧❡✱ ❝♦♠♣♦✉♥❞✮ ✐♥✈❛r✐❛♥ts❄ ✽✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❋✐①❡❞ ✐♥❝♦♠❡✿ ❛ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s ❢r❛♠❡✇♦r❦ ❝♦♥str✉❝t ❛ s②♥t❤❡t✐❝ s❡r✐❡s ♦❢ ❜♦♥❞ ♣r✐❝❡s ✇✐t❤ t❤❡ s❛♠❡ t✐♠❡ t♦ ♠❛t✉r✐t②✱ v ✿ Z ( E ) ✭❡✳❣✳ ◆♦✈ ✷✵✶✻ ♣r✐❝❡ ♦❢ ❛ ❜♦♥❞ t❤❛t ♠❛t✉r❡s ✐♥ ❋❡❜ ✷✵✷✶✮ ✶ t Z ( E − ˜ τ ) ✭❡✳❣✳ ◆♦✈ ✷✵✶✺ ♣r✐❝❡ ♦❢ ❛ ❜♦♥❞ t❤❛t ♠❛t✉r❡s ✐♥ ❋❡❜ ✷✵✷✵✮ ✷ t − ˜ τ Z ( E − ✷ ˜ τ ) ✭❡✳❣✳ ◆♦✈ ✷✵✶✹ ♣r✐❝❡ ♦❢ ❛ ❜♦♥❞ t❤❛t ♠❛t✉r❡s ✐♥ ❋❡❜ ✷✵✶✾✮ ✸ t − ✷ ˜ τ ✳ ✳ ✳ ✹ t❛r❣❡t ❞✉r❛t✐♦♥ ❢✉♥❞s✿ ❛♥ ❡st❛❜❧✐s❤❡❞ ✜①❡❞ ✐♥❝♦♠❡ str❛t❡❣② ✭▲❛♥❣❡t✐❡❣✱ ▲❡✐❜♦✇✐t③✱ ❛♥❞ ❑♦❣❡❧♠❛♥✱ ✶✾✾✵✮ ❝❛♥ ♥♦✇ ❞❡✜♥❡ ♣s❡✉❞♦✲r❡t✉r♥s✱ ♦r r♦❧❧✐♥❣ ✭t♦t❛❧✮ r❡t✉r♥s t♦ ♠❛t✉r✐t② Z ( t + v ) R ( v ) t τ ≡ t , ˜ Z ( t − ˜ τ + v ) t − ˜ τ ✇❤❡r❡ ˜ τ ✐s t❤❡ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧ ✭❡✳❣✳ ❛ ②❡❛r✮ t❤❡s❡ ♣❛ss t❤❡ t✇♦ ❤❡✉r✐st✐❝ t❡sts ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ❋✐❣✉r❡ ✸✳✺✮ ✽✽ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❋✐①❡❞ ✐♥❝♦♠❡✿ ②✐❡❧❞ t♦ ♠❛t✉r✐t② ✇❤❛t ✐s t❤❡ ♠♦st ❝♦♥✈❡♥✐❡♥t ✜①❡❞ ✐♥❝♦♠❡ ✐♥✈❛r✐❛♥t t♦ ✇♦r❦ ✇✐t❤❄ ❞❡✜♥❡ Y ( v ) v ❧♥ Z ( t + v ) ≡ − ✶ ❛♥❞ ♠❛♥✐♣✉❧❛t❡ t♦ ♦❜t❛✐♥ ❛ ❝♦♠♣♦✉♥❞ t t r❡t✉r♥✿ = ❧♥ Z ( t + v ) ✶ vY ( v ) = − ❧♥ Z ( t + v ) = ❧♥ ✶ − ❧♥ Z ( t + v ) t + v = ❧♥ t t t Z ( t + v ) Z ( t + v ) t t Y ( v ) ✐s ②✐❡❧❞ t♦ ♠❛t✉r✐t② v ❀ ②✐❡❧❞ ❝✉r✈❡ ❣r❛♣❤s Y ( v ) ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ v t t ✐❢ ˜ τ ✐s ❛ ②❡❛r ✭st❛♥❞❛r❞✮✱ t❤❡♥ ❨❚▼ ✐s ❧✐❦❡ ❛♥ ❛♥♥✉❛❧✐s❡❞ ②✐❡❧❞ ❝❤❛♥❣❡s ✐♥ ②✐❡❧❞ t♦ ♠❛t✉r✐t② ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ r♦❧❧✐♥❣ r❡t✉r♥s t♦ ♠❛t✉r✐t②✱ v ❧♥ Z ( t + v ) τ = − ✶ = − ✶ X ( v ) τ ≡ Y ( v ) − Y ( v ) v ❧♥ R ( v ) t t , ˜ t t − ˜ t , ˜ τ Z ( t − ˜ τ + v ) t − ˜ τ ✉s✉❛❧❧② ♣❛ss t❤❡ ❤❡✉r✐st✐❝s✱ ❤❛✈❡ s✐♠✐❧❛r❧② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s t♦ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❢♦r ❡q✉✐t✐❡s ❝♦♠♣♦✉♥❞ r❡t✉r♥s ✽✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉❡r✐✈❛t✐✈❡s ❞❡r✐✈❡❞ ❢r♦♠ ✉♥❞❡r❧②✐♥❣ r❛✇ s❡❝✉r✐t✐❡s ✭❡✳❣✳ st♦❝❦s✱ ③❡r♦✲❝♦✉♣♦♥ ❜♦♥❞s✱ ✳ ✳ ✳ ✮ ✈❛♥✐❧❧❛ ❊✉r♦♣❡❛♥ ♦♣t✐♦♥s ❛r❡ t❤❡ ♠♦st ❧✐q✉✐❞ ❞❡r✐✈❛t✐✈❡s ✭✇❤②❄✮ t❤❡ r✐❣❤t✱ ❜✉t ♥♦t t❤❡ ♦❜❧✐❣❛t✐♦♥✱ t♦ ❜✉② ♦r s❡❧❧ ✳ ✳ ✳ ♦♥ ❡①♣✐r② ❞❛t❡ E ✳ ✳ ✳ ❛♥ ✉♥❞❡r❧②✐♥❣ s❡❝✉r✐t② tr❛❞✐♥❣ ❛t ♣r✐❝❡ U t ❛t t✐♠❡ t ✳ ✳ ✳ ❢♦r str✐❦❡ ♣r✐❝❡ K ❊①❛♠♣❧❡ ✭❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥✮ ❚❤❡ ♣r✐❝❡ ♦❢ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ❛t t✐♠❡ t ≤ E ✐s ♦❢t❡♥ ❡①♣r❡ss❡❞ ❛s ≡ C BSM � � C ( K , E ) E − t , K , U t , Z ( E ) , σ ( K , E ) s✳t✳ C ( K , E ) = ♠❛① { U E − K , ✵ } t t t E ✇❤❡r❡ E − t ✐s t❤❡ t✐♠❡ r❡♠❛✐♥✐♥❣✱ ❛♥❞ σ ( K , E ) ✐s t❤❡ ✈♦❧❛t✐❧✐t② ♦❢ U t ✳ t ❚❤❡ ♦♣t✐♦♥ ✐s ✐♥ t❤❡ ♠♦♥❡② ✇❤❡♥ U t > K ✱ ❛t t❤❡ ♠♦♥❡② ✇❤❡♥ U t = K ❛♥❞ ♦✉t ♦❢ t❤❡ ♠♦♥❡② ♦t❤❡r✇✐s❡✳ ✾✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉❡r✐✈❛t✐✈❡s✿ ✈♦❧❛t✐❧✐t② ♣r✐❝✐♥❣ ♦♣t✐♦♥s r❡q✉✐r❡s ❛ ♠❡❛s✉r❡ ♦❢ ✈♦❧❛t✐❧✐t② ❤✐st♦r✐❝❛❧ ♦r r❡❛❧✐s❡❞ ✈♦❧❛t✐❧✐t②✿ ❞❡t❡r♠✐♥❡❞ ❢r♦♠ ❤✐st♦r✐❝❛❧ ✈❛❧✉❡s ♦❢ U t ✶ ✭❡s♣✳ ❆❘❈❍ ♠♦❞❡❧s✮❀ ❜❛❝❦✇❛r❞ ❧♦♦❦✐♥❣ ❜✉t ♠♦❞❡❧✲❢r❡❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t②✿ ❛s t❤❡ ❝❛❧❧ ♦♣t✐♦♥✬s ♣r✐❝❡ ✐♥❝r❡❛s❡s ✐♥ σ t ✱ t❤❡ ❇❙▼ ✷ ♣r✐❝✐♥❣ ❢♦r♠✉❧❛ ❤❛s ❛♥ ✐♥✈❡rs❡✱ ❛❧❧♦✇✐♥❣ ✈♦❧❛t✐❧✐t② t♦ ❜❡ ✐♠♣❧✐❡❞ ❢r♦♠ ♦♣t✐♦♥ ♣r✐❝❡s❀ ❢♦r✇❛r❞ ❧♦♦❦✐♥❣✱ ❜✉t ♠♦❞❡❧✲❞❡♣❡♥❞❡♥t❀ ❡✳❣✳ ❱❳❖ ♠♦❞❡❧✲❢r❡❡ ✈♦❧❛t✐❧✐t② ❡①♣❡❝t❛t✐♦♥s✿ r✐s❦✲♥❡✉tr❛❧ ❡①♣❡❝t❛t✐♦♥ ♦❢ ❖❚▼ ✸ ♦♣t✐♦♥ ♣r✐❝❡s❀ ❢♦r✇❛r❞ ❧♦♦❦✐♥❣✱ ❧❡ss ♠♦❞❡❧✲❞❡♣❡♥❞❡♥t ✭❜✉t ❛ss✉♠❡s st♦❝❤❛st✐❝ ♣r♦❝❡ss ❞♦❡s♥✬t ❥✉♠♣✮❀ ❡✳❣✳ ❱■❳ ❚❛②❧♦r✱ ❨❛❞❛✈✱ ❛♥❞ ❩❤❛♥❣ ✭✷✵✶✵✮ ❝♦♠♣❛r❡ t❤❡ t❤r❡❡ ✈♦❧❛t✐❧✐t② ♠❡❛s✉r❡s ❊✉r♦♣❡✿ ❱❙❚❖❳❳✱ ❱❋❚❙❊✱ ❱❉❆❳✱ ❱❉❆❳✲◆❊❲✱ ❱❈❆❈✱ ❱❙▼■✱ ❱❆❊❳✱ ❱❇❊▲ ❛t✲t❤❡✲♠♦♥❡②✲❢♦r✇❛r❞ ✭❆❚▼❋✮ ✐♠♣❧✐❡❞ ♣❡r❝❡♥t❛❣❡ ✈♦❧❛t✐❧✐t② ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣✿ ✏✐♠♣❧✐❡❞ ♣❡r❝❡♥t❛❣❡ ✈♦❧❛t✐❧✐t② ♦❢ ❛♥ ♦♣t✐♦♥ ✇❤♦s❡ str✐❦❡ ✐s ❡q✉❛❧ t♦ t❤❡ ❢♦r✇❛r❞ ♣r✐❝❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ❛t ❡①♣✐r②✑ ✭▼❡✉❝❝✐✱ ✷✵✵✺✮ ❜② ♥♦✲❛r❜✐tr❛❣❡ ❢♦r✇❛r❞ ♣r✐❝❡ ❢♦r♠✉❧❛ ✭❙t❡❢❛♥✐❝❛✱ ✷✵✶✶✱ ➓✶✳✶✵✮✱ e r t ( E − t ) = ✶✱ s♦ t❤❛t K t = e − r t ( E − t ) U t = Z ( E ) U t t Z ( E ) t ✇❤② ❆❚▼❋❄ ✾✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❚❤❡ q✉❡st ❢♦r ✐♥✈❛r✐❛♥❝❡ ❉❡r✐✈❛t✐✈❡s✿ ❛ t✐♠❡ ❤♦♠♦❣❡♥❡♦✉s ❢r❛♠❡✇♦r❦ ❛s ✇✐t❤ Z ( E ) ❢♦r ✜①❡❞ ✐♥❝♦♠❡✱ σ ( K , E ) ❝♦♥✈❡r❣❡s ❛s t → E t t ❝♦♥s✐❞❡r s❡t ♦❢ r♦❧❧✐♥❣ ✐♠♣❧✐❡❞ ♣❡r❝❡♥t❛❣❡ ✈♦❧❛t✐❧✐t✐❡s ✇✐t❤ s❛♠❡ t✐♠❡ t♦ ♠❛t✉r✐t② v ✱ σ ( K t , t + v ) t s✉❜st✐t✉t❡ ❆❚▼❋ ❞❡✜♥✐t✐♦♥ ❢♦r K t ✐♥t♦ C BSM ♣r✐❝✐♥❣ ❢♦r♠✉❧❛ ❢♦r � C ( K t , E ) � C ( K t , t + v ) � � ✽ ✷ π σ ( K t , E ) E − t ❡r❢ − ✶ t t ≈ = t U t v U t ❜② ✜rst ♦r❞❡r ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ ❡r❢ − ✶ ✭q✳✈✳ ❚❡❝❤♥✐❝❛❧ ❆♣♣❡♥❞✐① ➓✸✳✶✮ ♥♦r♠❛❧✐s❛t✐♦♥ ❜② U t s❤♦✉❧❞ r❡♠♦✈❡ ♥♦♥✲st❛t✐♦♥❛r✐t② ♦❢ σ ( K t , E ) t ❛s C ( K t , t + v ) , U t ♥♦t ✐♥✈❛r✐❛♥t✱ r❛t✐♦ ✉s✉❛❧❧② ♥♦t ✭▼❡✉❝❝✐✱ ✷✵✵✺✱ ♣✳✶✶✽✮✱ t ❜✉t ❝❤❛♥❣❡s ✐♥ r♦❧❧✐♥❣ ❆❚▼❋ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ♣❛ss ❤❡✉r✐st✐❝ t❡sts ✭❧✐❦❡ ❞✐✛❡r❡♥❝✐♥❣ I ( ✶ ) s❡r✐❡s❄✮ ✾✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ❤❛✈❡ ✐❞❡♥t✐✜❡❞ ✐♥✈❛r✐❛♥ts✱ ❳ t , ˜ τ ❣✐✈❡♥ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧ ˜ τ ✇❛♥t t♦ ❦♥♦✇ ❞✐str✐❜✉t✐♦♥ ♦❢ ❳ T + τ,τ ✱ r✈ ❛t ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥✱ τ ♦✉r ♣r❡❢❡rr❡❞ ✐♥✈❛r✐❛♥ts ❛r❡ s♣❡❝✐✜❡❞ ✐♥ t❡r♠s ♦❢ ❞✐✛❡r❡♥❝❡s ❝♦♠♣♦✉♥❞s r❡t✉r♥s ❢♦r ❡q✉✐t✐❡s✱ ❝♦♠♠♦❞✐t✐❡s✱ ❋❳ ✶ ❳ T + τ,τ = ❧♥ P T + τ − ❧♥ P T ❝❤❛♥❣❡s ✐♥ ❨❚▼ ❢♦r ✜①❡❞ ✐♥❝♦♠❡ ✷ ❳ T + τ,τ = Y T + τ − Y T ❝❤❛♥❣❡s ✐♥ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❢♦r ❞❡r✐✈❛t✐✈❡s ✸ ❳ T + τ,τ = σ T + τ − σ T ❛❧❧ ♦❢ ✇❤✐❝❤ ❛r❡ ❛❞❞✐t✐✈❡✱ s♦ t❤❛t t❤❡② s❛t✐s❢② ❳ T + τ,τ = ❳ T + τ, ˜ τ + ❳ T + τ − ˜ τ + · · · + ❳ T +˜ τ, ˜ τ, ˜ τ ✾✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ❉✐str✐❜✉t✐♦♥s ❛t t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ❢♦r ❡①♣♦s✐t✐♦♥❛❧ s✐♠♣❧✐❝✐t②✱ ❛ss✉♠❡ t❤❛t τ = k ˜ τ ✱ ✇❤❡r❡ k ∈ Z ++ ♥♦ ♣r♦❜❧❡♠ ✐❢ ♥♦t ❛s ❧♦♥❣ ❛s ❞✐str✐❜✉t✐♦♥ ✐s ✐♥✜♥✐t❡❧② ❞✐✈✐s✐❜❧❡ ✭✇❤②❄✮ ❛s ❛❧❧ ♦❢ t❤❡ ✐♥✈❛r✐❛♥ts ✐♥ ❳ T + τ,τ = ❳ T + τ, ˜ τ + ❳ T + τ − ˜ τ + · · · + ❳ T +˜ τ, ˜ τ, ˜ τ ❛r❡ ■■❉✱ t❤❡ ♣r♦❥❡❝t✐♦♥ ❢♦r♠✉❧❛ ✐s � τ � φ ❳ T + τ,τ = φ ❳ t , ˜ τ ˜ τ ♣r♦♦❢ ❝❛♥ tr❛♥s❧❛t❡ ❜❛❝❦ ❛♥❞ ❢♦rt❤ ❜❡t✇❡❡♥ ❝❢ ❛♥❞ ♣❞❢ ✇✐t❤ ❋♦✉r✐❡r ❛♥❞ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠s φ ❳ = F [ f ❳ ] ❛♥❞ f ❳ = F − ✶ [ φ ❳ ] ❜② ❝♦♥tr❛st✱ ❧✐♥❡❛r r❡t✉r♥ ♣r♦❥❡❝t✐♦♥s ②✐❡❧❞ ▲ T + τ,τ = diag ( ✶ + ▲ T + τ, ˜ τ ) × · · · × diag ( ✶ + ▲ T +˜ τ ) − ✶ τ, ˜ ✇❤❡r❡ t❤❡ ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ✐♥ t❤❡ N × N diag ♠❛tr✐① ❛r❡ t❤♦s❡ ✐♥ ✐ts ✈❡❝t♦r✲✈❛❧✉❡❞ ❛r❣✉♠❡♥t❀ ✐ts ♦✛✲❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ❛r❡ ③❡r♦ ✾✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ ❏♦✐♥t ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥s ❊①❛♠♣❧❡ ▲❡t t❤❡ ✇❡❡❦❧② ❝♦♠♣♦✉♥❞ r❡t✉r♥s ♦♥ ❛ st♦❝❦ ❛♥❞ t❤❡ ✇❡❡❦❧② ②✐❡❧❞ ❝❤❛♥❣❡s ❢♦r t❤r❡❡✲②❡❛r ❜♦♥❞s ❜❡ ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞✳ ❚❤✉s✱ t❤❡ ✐♥✈❛r✐❛♥ts ❛r❡ � � � � ❧♥ P t − ❧♥ P t − ˜ C t , ˜ τ τ ❳ t , ˜ τ = ≡ . X ( v ) Y ( v ) − Y ( v ) t , ˜ τ t t − ˜ τ ❇✐♥❞ t❤❡s❡ ♠❛r❣✐♥❛❧s s♦ t❤❛t t❤❡✐r ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ✐s ❛❧s♦ ♥♦r♠❛❧✱ τ ( ω ) = e i ω ′ µ − ✶ ✷ ω ′ Σ ω ✳ τ ∼ N ( µ , Σ ) ✳ ❇② ❥♦✐♥t ♥♦r♠❛❧✐t②✱ t❤❡ ❝❢ ✐s φ ❳ t , ˜ ❳ t , ˜ ❋r♦♠ t❤❡ ♣r❡✈✐♦✉s s❧✐❞❡✱ ❳ T + τ,τ ❤❛s ❝❢ φ ❳ T + τ,τ ( ω ) = e i ω ′ τ τ µ − ✶ ✷ ω ′ τ τ Σ ω ✳ ˜ ˜ ❚❤✉s✱ � τ τ µ , τ � ❳ T + τ,τ ∼ N τ Σ . ˜ ˜ ✾✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t Pr♦❥❡❝t✐♥❣ ✐♥✈❛r✐❛♥ts t♦ t❤❡ ✐♥✈❡st♠❡♥t ❤♦r✐③♦♥ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❤♦r✐③♦♥ ❞✐str✐❜✉t✐♦♥ t❤❡ ♣r♦❥❡❝t✐♦♥ ❢♦r♠✉❧❛ ❛❧❧♦✇s ❞❡r✐✈❛t✐♦♥ ♦❢ ♠♦♠❡♥ts ✭✇❤❡♥ t❤❡② ❛r❡ ❞❡✜♥❡❞✮ ❡①♣❡❝t❡❞ ✈❛❧✉❡s s✉♠ ✶ E { ❳ T + τ,τ } = τ τ E { ❳ t , ˜ τ } ˜ sq✉❛r❡✲r♦♦t ♦❢ t✐♠❡ r✉❧❡ ♦❢ r✐s❦ ♣r♦♣❛❣❛t✐♦♥ ✷ � τ Cov { ❳ T + τ,τ } = τ τ Cov { ❳ t , ˜ τ } ⇔ Sd { ❳ T + τ,τ } = τ Sd { ❳ t , ˜ τ } ˜ ˜ ◆♦r♠❛❧✐s✐♥❣ ˜ τ = ✶ ②❡❛r✿ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❤♦r✐③♦♥ ✐♥✈❛r✐❛♥t ✐s t❤❡ sq✉❛r❡ r♦♦t ♦❢ t❤❡ ❤♦r✐③♦♥ t✐♠❡s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❛♥♥✉❛❧✐s❡❞ ✐♥✈❛r✐❛♥t ✐♥t✉✐t✐♦♥❄ P♦rt❢♦❧✐♦ ❞✐✈❡rs✐✜❡s ✐ts❡❧❢ ❜② r❡❝❡✐✈✐♥❣ ■■❉ s❤♦❝❦s ♦✈❡r t✐♠❡ s❡❡ ❉❛♥✐❡❧ss♦♥ ❛♥❞ ❩✐❣r❛♥❞ ✭✷✵✵✻✮ ❢♦r ✇❛r♥✐♥❣s ❛❜♦✉t ♥♦♥✲r♦❜✉st♥❡ss ✾✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s ❘❛✇ s❡❝✉r✐t✐❡s✿ ❤♦r✐③♦♥ ♣r✐❝❡s ♣r✐❝❡s ❞❡♣❡♥❞ ♦♥ ✐♥✈❛r✐❛♥ts t❤r♦✉❣❤ s♦♠❡ ♣r✐❝✐♥❣ ❢✉♥❝t✐♦♥✱ P T + τ = ❣ ( ❳ T + τ,τ ) ✶ ❢♦r ❡q✉✐t✐❡s✱ ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥s ❢♦r♠✉❧❛ ②✐❡❧❞s P T + τ = P T e ❳ T + τ,τ ✷ ❢♦r ③❡r♦ ❝♦✉♣♦♥ ❜♦✉♥❞s✱ ♠❛♥✐♣✉❧❛t✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❘ ( E − T − τ ) ❛♥❞ T + τ,τ ❳ ( E − T − τ ) ②✐❡❧❞s T + τ,τ e − ( E − T − τ ) ❳ ( E − T − τ ) ❩ ( E ) T + τ = ❩ ( E − τ ) T + τ,τ T ♥✳❜✳ ❝♦✉❧❞ ✉s❡ v ≡ E − ( T + τ ) ✾✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s ❘❛✇ s❡❝✉r✐t✐❡s✿ ❤♦r✐③♦♥ ♣r✐❝❡ ❞✐str✐❜✉t✐♦♥ ❢♦r ❜♦t❤ ❡q✉✐t✐❡s ❛♥❞ ✜①❡❞ ✐♥❝♦♠❡✱ P T + τ = e ❨ T + τ,τ ✱ ✇❤❡r❡ ❨ T + τ,τ ≡ γ + diag ( ε ) ❳ T + τ,τ ❛♥ ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥ t❤✉s✱ t❤❡② ❤❛✈❡ ❛ ❧♦❣ − ❨ ❞✐str✐❜✉t✐♦♥ t❤✐s ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s φ ❨ T + τ,τ ( ω ) = e i ω ′ γ φ ❳ T + τ,τ ( diag ( ε ) ω ) ✉s✉❛❧❧② ✐♠♣♦ss✐❜❧❡ t♦ ❝♦♠♣✉t❡ ❝❧♦s❡❞ ❢♦r♠ ❢♦r ❢✉❧❧ ❞✐str✐❜✉t✐♦♥ ♠❛② s✉✣❝❡ ❥✉st t♦ ❝♦♠♣✉t❡ ✜rst ❢❡✇ ♠♦♠❡♥ts ❡✳❣✳ ❝❛♥ ❝♦♠♣✉t❡ E { P n } ❛♥❞ Cov { P m , P n } ❢r♦♠ ❝❢ ✶✵✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s ❉❡r✐✈❛t✐✈❡s✿ ❤♦r✐③♦♥ ♣r✐❝❡s ♣r✐❝❡s ❛r❡ st✐❧❧ ❢✉♥❝t✐♦♥s ♦❢ ✐♥✈❛r✐❛♥ts✱ P T + τ = ❣ ( ❳ T + τ,τ ) ❛s ♣r✐❝❡s r❡✢❡❝t ♠✉❧t✐♣❧❡ ✐♥✈❛r✐❛♥ts✱ ♥♦ ❧♦♥❣❡r s✐♠♣❧❡ ❧♦❣ − ❨ str✉❝t✉r❡ ❊①❛♠♣❧❡ ❆❣❛✐♥✿ ♣r✐❝❡ ♦❢ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ❛t ❤♦r✐③♦♥ T + τ ≤ E ✐s ≡ C BSM � � C ( K , E ) E − T − τ, K , U T + τ , Z ( E ) T + τ , σ ( K , E ) . T + τ T + τ ❚❤❡ ❤♦r✐③♦♥ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡ t❤r❡❡ ✐♥✈❛r✐❛♥ts ❛r❡ t❤❡♥ U T + τ = U T e X ✶ Z ( E ) T + τ = Z ( E − τ ) e − X ✷ v T σ ( K , E ) T + τ = σ ( K T , E − τ ) + X ✸ T ❢♦r v ≡ E − T − τ ❛♥❞ s✉✐t❛❜❧② ❞❡✜♥❡❞ K T ❛♥❞ ✐♥✈❛r✐❛♥ts✱ X ✶ t♦ X ✸ ✳ ✶✵✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s ❉❡r✐✈❛t✐✈❡s✿ ❛♣♣r♦①✐♠❛t✐♥❣ ❤♦r✐③♦♥ ♣r✐❝❡s ♦♣t✐♦♥s ♣r✐❝✐♥❣ ❢♦r♠✉❧❛ ✐s ❛❧r❡❛❞② ❝♦♠♣❧✐❝❛t❡❞✱ ♥♦♥✲❧✐♥❡❛r ❛❞❞✐♥❣ ✐♥ ♣♦ss✐❜❧② ❝♦♠♣❧✐❝❛t❡❞ ❤♦r✐③♦♥ ♣r♦❥❡❝t✐♦♥s ❛❧♠♦st ❝❡rt❛✐♥❧② ♣r❡✈❡♥ts ❡①❛❝t s♦❧✉t✐♦♥s ❜✉t ❝❛♥ ❛♣♣r♦①✐♠❛t❡ P T + τ = g ( ❳ T + τ,τ ) ✇✐t❤ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ P T + τ ≈ g ( ♠ ) + ( ❳ − ♠ ) ∇ g ( ♠ ) + ✶ ✷ ( ❳ − ♠ ) ′ H ( g ( ♠ )) ( ❳ − ♠ ) ✇❤❡r❡ ∇ g ( ♠ ) ✐s ❣r❛❞✐❡♥t✱ H ( g ( ♠ )) ❍❡ss✐❛♥ ❛♥❞ ♠ s♦♠❡ s✐❣♥✐✜❝❛♥t ✈❛❧✉❡ ♦❢ t❤❡ ✐♥✈❛r✐❛♥ts ❳ T + τ,τ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥ ♣r♦❞✉❝❡s t❤❡ ●r❡❡❦s ❊①❛♠♣❧❡ ✭❇❡t❖♥▼❛r❦❡ts✮ ❇❡t❖♥▼❛r❦❡ts ❤❛s t♦ ♣r✐❝❡ ❝✉st♦♠ ♦♣t✐♦♥s ✐♥ ❧❡ss t❤❛♥ ✶✺ s❡❝♦♥❞s✳ ▼♦♥t❡ ❈❛r❧♦ ✐s ❢❛r t♦♦ s❧♦✇❀ ❡✈❡♥ ❇❧❛❝❦✲❙❝❤♦❧❡s ♠❛② ❜❡✳ ❚❤❡② ✉s❡ ❱❛♥♥❛✲❱♦❧❣❛✳ ✶✵✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ▼❛♣♣✐♥❣ ✐♥✈❛r✐❛♥ts ✐♥t♦ ♠❛r❦❡t ♣r✐❝❡s ▲❡❝t✉r❡ ✺ ❡①❡r❝✐s❡s ▼❡✉❝❝✐ ❡①❡r❝✐s❡s ♣❡♥❝✐❧✲❛♥❞✲♣❛♣❡r✿ ✺✳✸ ▼❆❚▲❆❇✿ ✸✳✷✳✷✱ ✸✳✷✳✸✱ ✺✳✶ ✭♠♦❞✐❢② ❝♦❞❡ t♦ ❞✐s♣❧❛② ♦♥❡✲♣❡r✐♦❞ ❛♥❞ ❤♦r✐③♦♥ ❞✐str✐❜✉t✐♦♥s❀ ❝♦♥tr❛st t♦ ▼❡✉❝❝✐ ✭✷✵✵✺✮ ❡q✉❛t✐♦♥s ✸✳✾✺✱ ✸✳✾✻✮✱ ✺✳✺✳✶✱ ✺✳✺✳✷✱ ✺✳✻ ♣r♦❥❡❝t ♣r♦❞✉❝❡ ❤♦r✐③♦♥ ♣r✐❝❡ ❞✐str✐❜✉t✐♦♥s ❢♦r ②♦✉r ❛ss❡ts ✶✵✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❲❤② ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥❄ ✶ ❛❝t✉❛❧ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ♠❛r❦❡t ✐s ❧❡ss t❤❛♥ t❤❡ ♥✉♠❜❡r ♦❢ s❡❝✉r✐t✐❡s ❊①❛♠♣❧❡ ❈♦♥s✐❞❡r ❛ st♦❝❦ ✇❤♦s❡ ♣r✐❝❡ ✐s U t ❛♥❞ ❛ ❊✉r♦♣❡❛♥ ❝❛❧❧ ♦♣t✐♦♥ ♦♥ ✐t ✇✐t❤ str✐❦❡ K ❛♥❞ ❡①♣✐r② ❞❛t❡ T + τ ✳ ❚❤❡✐r ❤♦r✐③♦♥ ♣r✐❝❡s ❛r❡ � U T + τ � P T + τ = . ♠❛① { U T + τ − K , ✵ } ❚❤❡s❡ ❛r❡ ♣❡r❢❡❝t❧② ♣♦s✐t✐✈❡❧② ❞❡♣❡♥❞❡♥t✳ ✷ ❛❝t✉❛❧ r❛♥❞♦♠♥❡ss ✐♥ t❤❡ ♠❛r❦❡t ❝❛♥ ❜❡ ✇❡❧❧ ❛♣♣r♦①✐♠❛t❡❞ ✇✐t❤ ❢❡✇❡r t❤❛♥ N ❞✐♠❡♥s✐♦♥s ✭t❤❛t ♦❢ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts✱ ❳ ✮ t❤✐s ✐s t❤❡ ♣♦ss✐❜✐❧✐t② ❝♦♥s✐❞❡r❡❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s ❝❛♥ ❝♦♥s✐❞❡r❛❜❧② r❡❞✉❝❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦♠♣❧❡①✐t② ✶✵✺ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❈♦♠♠♦♥ ❢❛❝t♦rs ✇♦✉❧❞ ❧✐❦❡ t♦ ❡①♣r❡ss N ✲✈❡❝t♦r ❳ t , ˜ τ ✐♥ t❡r♠s ♦❢ ❛ K ✲✈❡❝t♦r ♦❢ ❝♦♠♠♦♥ ❢❛❝t♦rs✱ ❋ t , ˜ τ ❀ ✶ ❡①♣❧✐❝✐t ❢❛❝t♦rs ❛r❡ ♠❡❛s✉r❛❜❧❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ✶ ❤✐❞❞❡♥ ❢❛❝t♦rs ❛r❡ s②♥t❤❡t✐❝ ✐♥✈❛r✐❛♥ts ❡①tr❛❝t❡❞ ❢r♦♠ t❤❡ ♠❛r❦❡t ✷ ✐♥✈❛r✐❛♥ts ❛♥ N ✲✈❡❝t♦r ♦❢ r❡s✐❞✉❛❧ ♣❡rt✉r❜❛t✐♦♥s✱ ❯ t , ˜ ✷ τ ❛s ❢♦❧❧♦✇s ❳ t , ˜ τ = ❤ ( ❋ t , ˜ τ ) + ❯ t , ˜ τ ❢♦r tr❛❝t❛❜✐❧✐t②✱ ✉s✉❛❧❧② ✉s❡ ❧✐♥❡❛r ❢❛❝t♦r ♠♦❞❡❧ ✭✜rst ♦r❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥✮✱ ❳ t , ˜ τ = ❇❋ t , ˜ τ + ❯ t , ˜ τ ✇✐t❤ ❛♥ N × K ❢❛❝t♦r ❧♦❛❞✐♥❣ ♠❛tr✐①✱ ❇ ✶✵✻ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❈♦♠♠♦♥ ❢❛❝t♦rs✿ ❞❡s✐❞❡r❛t❛ ✶ s✉❜st❛♥t✐❛❧ ❞✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥✱ K ≪ N ✷ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ ❋ t , ˜ τ ❛♥❞ ❯ t , ˜ τ ✭✇❤②❄✮ ❤❛r❞ t♦ ❛tt❛✐♥✱ s♦ ♦❢t❡♥ r❡❧❛① t♦ Cor { ❋ t , ˜ τ } = ✵ K × N τ , ❯ t , ˜ ✸ ❣♦♦❞♥❡ss ♦❢ ✜t ✇❛♥t r❡❝♦✈❡r❡❞ ✐♥✈❛r✐❛♥ts t♦ ❜❡ ❝❧♦s❡✱ ˜ ❳ ≡ ❤ ( ❋ ) ≈ ❳ ✉s❡ ❣❡♥❡r❛❧✐s❡❞ R ✷ �� � ′ � �� ❳ − ˜ ❳ − ˜ E ❳ ❳ R ✷ � � ❳ , ˜ ❳ ≡ ✶ − tr { Cov { ❳ }} ✇❤❡r❡ t❤❡ tr❛❝❡ ♦❢ ❨ ✱ tr { ❨ } ✱ ✐s t❤❡ s✉♠ ♦❢ ✐ts ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s ✇❤❛t ✐s ✐♥ t❤❡ ♥✉♠❡r❛t♦r❄ ✶ ✇❤❛t ✐s ✐♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r❄ ✷ ❤♦✇ ❞♦❡s t❤✐s ❞✐✛❡r ❢r♦♠ t❤❡ ✉s✉❛❧ ❝♦❡✣❝✐❡♥t ♦❢ ❞❡t❡r♠✐♥❛t✐♦♥✱ R ✷ ❄ ✸ ✶✵✼ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❊①♣❧✐❝✐t ❢❛❝t♦rs s✉♣♣♦s❡ t❤❛t t❤❡♦r② ♣r♦✈✐❞❡s ❛ ❧✐st ♦❢ ❡①♣❧✐❝✐t ♠❛r❦❡t ✈❛r✐❛❜❧❡s ❛s ❢❛❝t♦rs✱ ❋ ❤♦✇ ❞♦❡s ♦♥❡ ❞❡t❡r♠✐♥❡ t❤❡ ❧♦❛❞✐♥❣s ♠❛tr✐①✱ ❇ ❄ ✇✐t❤ ❧✐♥❡❛r ❢❛❝t♦r ♠♦❞❡❧✱ ❳ = ❇❋ + ❯ ✱ ♣✐❝❦ ❇ t♦ ♠❛①✐♠✐s❡ ❣❡♥❡r❛❧✐s❡❞ R ✷ R ✷ { ❳ , ❇❋ } ❇ r ≡ ❛r❣♠❛① ❇ ✇❤❡r❡ t❤❡ s✉❜s❝r✐♣t ✐♥❞✐❝❛t❡s t❤❛t t❤❡s❡ ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② r❡❣r❡ss✐♦♥ t❤✐s ✐s s♦❧✈❡❞ ❜② ❋❋ ′ � − ✶ ❳❋ ′ � � � ❇ r = E E ❤♦✇ ❞♦❡s t❤✐s ❞✐✛❡r ❢r♦♠ ❖▲❙❄ ❡✈❡♥ ✇❡❛❦ ✈❡rs✐♦♥ ♦❢ s❡❝♦♥❞ ❞❡s✐❞❡r❛t✉♠✱ Cor { ❋ , ❯ } = ✵ K × N ♥♦t ❣❡♥❡r❛❧❧② s❛t✐s✜❡❞❀ ❜✉t✿ E { ❋ } = ✵ ⇒ Cor { ❋ , ❯ } = ✵ K × N ✶ ❛❞❞✐♥❣ ❝♦♥st❛♥t ❢❛❝t♦r t♦ ❋ ⇒ E { ❯ r } = ✵ , Cor { ❋ , ❯ r } = ✵ K × N ✷ ❝❢✳ ✐♥❝❧✉❞✐♥❣ ❝♦♥st❛♥t t❡r♠ ✐♥ ❖▲❙ r❡❣r❡ss✐♦♥ ✶✵✽ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❊①♣❧✐❝✐t ❢❛❝t♦rs✿ ♣✐❝❦✐♥❣ ❢❛❝t♦rs ✶ ✇❛♥t t❤❡ s❡t ♦❢ ❢❛❝t♦rs t♦ ❜❡ ❛s ❤✐❣❤❧② ❝♦rr❡❧❛t❡❞ ❛s ♣♦ss✐❜❧❡ ✇✐t❤ t❤❡ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts ♠❛①✐♠✐s❡s ❡①♣❧❛♥❛t♦r② ♣♦✇❡r ♦❢ t❤❡ ❢❛❝t♦rs ✐❢ ❞♦ ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥ts ❞❡❝♦♠♣♦s✐t✐♦♥ ♦♥ ❋ ✱ s♦ t❤❛t Cov { ❋ } = ❊ Λ ❊ ′ ❛♥❞ ❈ ❳❋ ≡ Cor { ❳ , ❊ ′ ❋ } ✭ ❊ ′ ❋ ❛r❡ r♦t❛t❡❞ ❢❛❝t♦rs✮ t❤❡♥ = tr ( ❈ ❳❋ ❈ ′ ❳❋ ) R ✷ � � ❳ , ˜ ❳ r N ✷ ✇❛♥t t❤❡ s❡t ♦❢ ❢❛❝t♦rs t♦ ❜❡ ❛s ✉♥❝♦rr❡❧❛t❡❞ ✇✐t❤ ❡❛❝❤ ♦t❤❡r ❛s ♣♦ss✐❜❧❡ ❡①tr❡♠❡ ✈❡rs✐♦♥ ♦❢ ❝♦rr❡❧❛t✐♦♥ ✐s ♠✉❧t✐❝♦❧❧✐♥❡❛r✐t② ✐♥ t❤✐s ❝❛s❡✱ ❛❞❞✐♥❣ ❛❞❞✐t✐♦♥❛❧ ❢❛❝t♦rs ❞♦❡s♥✬t ❛❞❞ ❡①♣❧❛♥❛t♦r② ♣♦✇❡r✱ ❛♥❞ ❧❡❛✈❡s r❡❣r❡ss✐♦♥ ♣❧❛♥❡ ✐❧❧ ❝♦♥❞✐t✐♦♥❡❞ ✸ ♠♦r❡ ❣❡♥❡r❛❧❧②✱ tr❛❞❡✲♦✛ ❜❡t✇❡❡♥ ♠♦r❡ ❛❝❝✉r❛❝② ❛♥❞ ♠♦r❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ✐♥t❡♥s✐✈✐t② ✇❤❡♥ ❛❞❞✐♥❣ ❢❛❝t♦rs ✶✵✾ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❊①❛♠♣❧❡ ✭❈❛♣✐t❛❧ ❛ss❡ts ♣r✐❝✐♥❣ ♠♦❞❡❧ ✭❈❆P▼✮✮ τ ≡ P ( n ) ❚❤❡ ❧✐♥❡❛r r❡t✉r♥s ✭✐♥✈❛r✐❛♥ts✮ ♦❢ N st♦❝❦s ❛r❡ L ( n ) − ✶✳ ■❢ t❤❡ ♣r✐❝❡ t t , ˜ P ( n ) t − ˜ τ ♦❢ t❤❡ ♠❛r❦❡t ✐♥❞❡① ✐s M t ✱ t❤❡ ❧✐♥❡❛r r❡t✉r♥ ♦♥ t❤❡ ♠❛r❦❡t ✐♥❞❡①✱ τ − ✶✱ ✐s ❛ ❧✐♥❡❛r ❢❛❝t♦r✳ ❘❡❣r❡ss✐♦♥ t❤❡♥ ❡st✐♠❛t❡s β ( n ) M t F M τ ✱ ❛ τ ≡ t , ˜ ˜ M t − ˜ ❤♦r✐③♦♥✲❞❡♣❡♥❞❡♥t ❝♦rr❡❧❛t✐♦♥✿ � � � � �� L ( n ) L ( n ) + β ( n ) ˜ F M F M τ = E τ − E . t , ˜ t , ˜ ˜ t , ˜ t , ˜ τ τ τ ■❢ ❡①♣❡❝t❡❞ r❡t✉r♥s ❝♦♥✈❡①❧② ❝♦♠❜✐♥❡ ♠❛r❦❡t ❛♥❞ r✐s❦✲❢r❡❡ r❡t✉r♥s✱ � � � � � � L ( n ) = β ( n ) ✶ − β ( n ) F M R f + E τ E t , ˜ t , ˜ t , ˜ τ ˜ τ ˜ τ τ t❤❡♥ t❤❡ ❈❆P▼ ❢♦❧❧♦✇s L ( n ) τ + β ( n ) � � ˜ τ = R f F M τ − R f . t , ˜ t , ˜ t , ˜ τ t , ˜ τ ˜ ✶✶✵ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❊①❛♠♣❧❡ ✭❋❛♠❛ ❛♥❞ ❋r❡♥❝❤ ✭✶✾✾✸✮ t❤r❡❡ ❢❛❝t♦r ♠♦❞❡❧✮ ❚❤❡ ❋❛♠❛ ❛♥❞ ❋r❡♥❝❤ ✭✶✾✾✸✮ t❤r❡❡ ❢❛❝t♦r ♠♦❞❡❧ r❡❞✉❝❡s t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥s✱ C ( n ) τ ♦❢ N st♦❝❦s t♦ t❤r❡❡ ❡①♣❧✐❝✐t ❧✐♥❡❛r ❢❛❝t♦rs ❛♥❞ ❛ ❝♦♥st❛♥t✿ t , ˜ ✶ C M ✱ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥ t♦ ❛ ❜r♦❛❞ st♦❝❦ ✐♥❞❡① ✷ SmB ✱ s♠❛❧❧ ♠✐♥✉s ❜✐❣✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥ t♦ ❛ s♠❛❧❧✲❝❛♣ st♦❝❦ ✐♥❞❡① ❛♥❞ ❛ ❧❛r❣❡✲❝❛♣ st♦❝❦ ✐♥❞❡① ✸ HmL ✱ ❤✐❣❤ ♠✐♥✉s ❧♦✇✱ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣♦✉♥❞ r❡t✉r♥ t♦ ❛ ❤✐❣❤ ❜♦♦❦✲t♦✲♠❛r❦❡t st♦❝❦ ✐♥❞❡① ❛♥❞ ❛ ❧♦✇ ❜♦♦❦✲t♦✲♠❛r❦❡t st♦❝❦ ✐♥❞❡① ❯♥❧✐❦❡ ♠♦❞❡❧s ♦❢ ❧✐♥❡❛r r❡t✉r♥s✱ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts ❤❡r❡ ❞❡♣❡♥❞ ♦♥ ❝♦✈❛r✐❛♥❝❡s❀ ❜② ❵sq✉❛r❡ r♦♦t✬ ♣r♦♣❡rt②✱ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❡st✐♠❛t✐♦♥ ✐♥t❡r✈❛❧✱ ˜ τ ✶✶✶ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❍✐❞❞❡♥ ❢❛❝t♦rs ♥♦✇ ❧❡t ❢❛❝t♦rs✱ ❋ ( ❳ t , ˜ τ ) ❜❡ s②♥t❤❡t✐❝ ✐♥✈❛r✐❛♥ts ❡①tr❛❝t❡❞ ❢r♦♠ ♠❛r❦❡t ✐♥✈❛r✐❛♥ts t❤✉s✱ t❤❡ ❛✣♥❡ ♠♦❞❡❧ ✐s τ = q + ❇❋ ( ❳ t , ˜ τ ) + ❯ t , ˜ ❳ t , ˜ τ ✇❤❛t ✐s t❤❡ tr✐✈✐❛❧ ✇❛② ♦❢ ♠❛①✐♠✐s✐♥❣ ❣❡♥❡r❛❧✐s❡❞ R ✷ ❄ ✇❤❛t ✐s t❤❡ ✇❡❛❦♥❡ss ♦❢ ❞♦✐♥❣ s♦❄ ♦t❤❡r✇✐s❡✱ ♠❛✐♥ ❛♣♣r♦❛❝❤ t❛❦❡♥ ✐s ♣r✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ✭P❈❆✮ ✶✶✷ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ Pr✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥t ❛♥❛❧②s✐s ✭P❈❆✮ ❛ss✉♠❡ t❤❡ ❤✐❞❞❡♥ ❢❛❝t♦rs ❛r❡ ❛✣♥❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❳ t , ˜ τ τ ) = ❞ p + ❆ ′ ❋ p ( ❳ t , ˜ p ❳ t , ˜ τ ❣✐✈❡♥ t❤❡s❡ ❛✣♥❡ ❛ss✉♠♣t✐♦♥s✱ t❤❡ ♦♣t✐♠❛❧❧② r❡❝♦✈❡r❡❞ ✐♥✈❛r✐❛♥ts ❛r❡ ❳ p = ♠ p + ❇ p ❆ ′ ˜ p ❳ t , ˜ τ ✇❤❡r❡ ❳ , ♠ + ❇❆ ′ ❳ t , ˜ ( ❇ p , ❆ p , ♠ p ) ≡ ❛r❣♠❛① R ✷ � � τ ❇ , ❆ , ♠ ❤❡✉r✐st✐❝❛❧❧② ✇❛♥t ♦rt❤♦❣♦♥❛❧ ❢❛❝t♦rs ❝♦♥s✐❞❡r ❧♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞ ❣❡♥❡r❛t❡❞ ❜② ❳ t , ˜ τ ❛s❦✐♥❣ ✇❤❛t ✐ts ❧♦♥❣❡st ♣r✐♥❝✐♣❛❧ ❛①❡s ❛r❡ ✶✶✸ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ▲♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞ ❝♦♥s✐❞❡r r✈ ❳ ✐♥ R ✸ ❣✐✈❡♥ ❧♦❝❛t✐♦♥ ❛♥❞ ❞✐s♣❡rs✐♦♥ ♣❛r❛♠❡t❡rs✱ µ ❛♥❞ Σ ✱ ❝❛♥ ❢♦r♠ ❧♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞ ✐❢ K = ✶✱ ✇❤✐❝❤ ❢❛❝t♦r ✇♦✉❧❞ ②♦✉ ❝❤♦♦s❡❄ ❲❤❛t ✇♦✉❧❞ ˜ ❳ p ❧♦♦❦ ❧✐❦❡❄ ✇❤❛t ✐❢ K = ✷❄ ✇❤❛t ✐❢ K = ✸❄ ✶✶✹ ✴ ✷✷✹

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❖♣t✐♠❛❧ ❢❛❝t♦rs ✐♥ P❈❆ ❝❤♦♦s✐♥❣ ♦♣t✐♠❛❧ ❢❛❝t♦rs t❤❡r❡❢♦r❡ r♦t❛t❡s✱ tr❛♥s❧❛t❡s ❛♥❞ ❝♦❧❧❛♣s❡s t❤❡ ❧♦❝❛t✐♦♥✲❞✐s♣❡rs✐♦♥ ❡❧❧✐♣s♦✐❞✬s ❝♦✲♦r❞✐♥❛t❡s ✭q✳✈✳ ▼❡✉❝❝✐✱ ✷✵✵✺✱ ❆♣♣ ❆✳✺✮ t❤✉s ■ N − ❊ K ❊ ′ � � � � ( ❇ p , ❆ p , ♠ p ) = ❊ K , ❊ K , E { ❳ t , ˜ τ } K ✇❤❡r❡ � ❡ ( ✶ ) , . . . , ❡ ( K ) � ❊ K ≡ ✇✐t❤ ❡ ( k ) ❜❡✐♥❣ t❤❡ ❡✐❣❡♥✈❡❝t♦r ♦❢ Cov { ❳ t , ˜ τ } ❝♦rr❡s♣♦♥❞✐♥❣ t♦ λ k ✱ t❤❡ k t❤ ❧❛r❣❡st ❡✐❣❡♥✈❛❧✉❡✳ ♠ p tr❛♥s❧❛t❡s✱ ❛♥❞ ❇ p ❆ ′ p r♦t❛t❡s ❛♥❞ ❝♦❧❧❛♣s❡s✱ ❢♦r ❳ p = ♠ p + ❇ p ❆ ′ ˜ ■ N − ❊ K ❊ ′ τ } + ❊ K ❊ ′ � � E { ❳ t , ˜ p ❳ t , ˜ τ = K ❳ t , ˜ τ K τ } + ❊ K ❊ ′ = E { ❳ t , ˜ K ( ❳ t , ˜ τ − E { ❳ t , ˜ τ } ) ✇❤② ❛r❡ E { ❯ p } = ✵ ❛♥❞ Cor { ❋ p , ❯ p } = ✵ K × N ❄ � K ❛s R ✷ � � k = ✶ λ k τ , ˜ n = ✶ λ n ✱ ❝❛♥ s❡❡ ❡✛❡❝t ♦❢ ❡❛❝❤ ❢✉rt❤❡r ❢❛❝t♦r ✶✶✺ ✴ ✷✷✹ ❳ t , ˜ ❳ p = � N

▼♦❞❡❧❧✐♥❣ t❤❡ ♠❛r❦❡t ❉✐♠❡♥s✐♦♥ r❡❞✉❝t✐♦♥ ❊①♣❧✐❝✐t ❢❛❝t♦rs ✈ P❈❆❄ ❛s P❈❆ ♣r♦❥❡❝ts ♦♥t♦ t❤❡ ♠♦st ✐♥❢♦r♠❛t✐✈❡ K ❞✐♠❡♥s✐♦♥s✱ ✐t ②✐❡❧❞s ❛ ❤✐❣❤❡r R ✷ t❤❛♥ ❛♥② K ✲❢❛❝t♦r ❡①♣❧✐❝✐t ❢❛❝t♦r ♠♦❞❡❧ ❤♦✇❡✈❡r✱ t❤❡ s②♥t❤❡t✐❝ ❞✐♠❡♥s✐♦♥s ♦❢ P❈❆ ❛r❡ ❤❛r❞❡r t♦ ✐♥t❡r♣r❡t✱ ❛♥❞ t❤❡r❡❢♦r❡ ♣❡r❤❛♣s t♦ ✉♥❞❡rst❛♥❞ ❜✉t s❡❡ ▼❡✉❝❝✐ ✭✷✵✵✺✱ ♣♣✳✶✺✼✲✮ ❢♦r ❛ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ s✇❛♣ ♠❛r❦❡t ②✐❡❧❞ ❝✉r✈❡ ✐♥t♦ ❧❡✈❡❧✱ s❧♦♣❡ ❛♥❞ ❝✉r✈❛t✉r❡ ❢❛❝t♦rs s❡❡ ♣♣✳✻✼✲ ♦❢ ❙♠✐t❤ ❛♥❞ ❋✉❡rt❡s✬ P❛♥❡❧ ❚✐♠❡ ❙❡r✐❡s ♥♦t❡s ❢♦r ❛ ❞✐s❝✉ss✐♦♥ ♦❢ ❤♦✇ t♦ ✉s❡ ❛♥❞ ✐♥t❡r♣r❡t P❈❆ ♠♦❞❡❧s ✶✶✻ ✴ ✷✷✹