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stss tstr r r s - - PowerPoint PPT Presentation

Dec 19, 2023 234 likes •641 views

stss tstr r r s P rt rss + rst

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SLIDE 1

❘♦❜✉st♥❡ss ❛♥❞ ❇♦♦tstr❛♣ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

P✳ ❇❡rt❛✐❧✯ ❼❏✳ ❚r❡ss♦✉ ❅ ❼❙✳ ❈❧é♠❡♥ç♦♥+

✯▼♦❞❛❧✬❳✱ ❯♥✐✈❡rs✐té P❛r✐s ◆❛♥t❡rr❡ +❚é❧é❝♦♠ P❛r✐s❚❡❝❤✱ ❯▼❘ ❈◆❘❙ ✺✶✹✶ ▲❚❈■✱ ●r♦✉♣❡ ❚❙■ ❅ ■◆❘❆ P❛r✐s

❙t❛t✐st✐q✉❡s ♣♦✉r ❧❡s P❉▼P ✱ ◆❛♥❝②✱ ✷✲✸ ❢é✈r✐❡r ✷✵✶✼ ✶❡r ❢é✈r✐❡r ✷✵✶✼

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶ ✴ ✸✽

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SLIDE 2

❖✉t❧✐♥❡s

✶ Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s

❋r❛♠❡✇♦r❦ ❍❛rr✐s r❡❝✉rr❡♥❝❡ ❚❤❡ ❛t♦♠✐❝ ❝❛s❡ ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✷ ❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s ▼❛✐♥ ❍②♣♦t❤❡s❡s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✸ ❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s

❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s ▲✐♥❡❛r ❘❛♥❦ s✐❣♥❡❞ st❛t✐st✐❝s ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

✹ ❆ ❢❡✇ r❡❢❡r❡♥❝❡s

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷ ✴ ✸✽

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SLIDE 3

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s

❖✉t❧✐♥❡s

✶ Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s

❋r❛♠❡✇♦r❦ ❍❛rr✐s r❡❝✉rr❡♥❝❡ ❚❤❡ ❛t♦♠✐❝ ❝❛s❡ ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✷ ❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s ▼❛✐♥ ❍②♣♦t❤❡s❡s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✸ ❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s

❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s ▲✐♥❡❛r ❘❛♥❦ s✐❣♥❡❞ st❛t✐st✐❝s ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

✹ ❆ ❢❡✇ r❡❢❡r❡♥❝❡s

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸ ✴ ✸✽

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SLIDE 4

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❋r❛♠❡✇♦r❦ ❍❛rr✐s r❡❝✉rr❡♥❝❡

  • ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ✿ ❍❛rr✐s r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥s

❳ = (❳♥)♥∈N✱ ❛ ψ✲✐rr❡❞✉❝✐❜❧❡ ❛♣❡r✐♦❞✐❝ t✐♠❡✲❤♦♠♦❣❡♥❡♦✉s ▼❛r❦♦✈ ❝❤❛✐♥✱ ✈❛❧✉❡❞ ✐♥ ❛ ✭❝♦✉♥t❛❜❧❡ ❣❡♥❡r❛t❡❞✮ ♠❡❛s✉r❛❜❧❡ s♣❛❝❡ (❊, E) ✇✐t❤ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② Π(①, ❞②) ❛♥❞ ✐♥✐t✐❛❧ ❞✐str✐❜✉t✐♦♥ ν. ◆♦t❛t✐♦♥s ✿ Pν ✭r❡s♣❡❝t✐✈❡❧②✱ P① ❢♦r ① ✐♥ ❊✮ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ s✉❝❤ t❤❛t ❳✵ ∼ ν ✭r❡s♣✳✱ ❝♦♥❞✐t✐♦♥❡❞ ✉♣♦♥ ❳✵ = ①✮✱ Eν[.] t❤❡ Pν✲❡①♣❡❝t❛t✐♦♥ ✭r❡s♣✳ E①[.] t❤❡ P① (.)✲❡①♣❡❝t❛t✐♦♥✮✳ E❆[.] ❞❡♥♦t❡s t❤❡ ❡①♣❡❝t❛t✐♦♥ ❝♦♥❞✐t✐♦♥❡❞ ♦♥ t❤❡ ❡✈❡♥t {❳✵ ∈ ❆}✳ ▼❛✐♥ t♦♦❧ ✿ ❘❡❣❡♥❡r❛t✐♦♥ ♣r♦♣❡rt✐❡s ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥s✳ ❘❡❢❡r t♦ t❤❡ ❜♦♦❦s ❜② ❘❡✈✉③ ✭✶✾✽✹✮✱ ▼❡②♥ ❛♥❞ ❚✇❡❡❞✐❡ ✭✶✾✾✹✮✱ ❚❤♦r✐ss♦♥✭✷✵✵✵✮✳ ❙❡r✐❡s ♦❢ ✇♦r❦s ❜② ❇❡rt❛✐❧ ❛♥❞ ❈❧é♠❡♥ç♦♥✱ P❘❚❋ ✭✷✵✵✹✮✱ ❇❡r♥♦✉❧❧✐✭✷✵✵✻✮✱ ❚❡st✭✷✵✵✽✮✱ ▼❛t✳ ▼❡t❤♦❞ ✐♥ ❙t❛t✐st✳ ✭✷✵✶✵✮ ❛♥❞ ❇❡rt❛✐❧✱ ❈❧é♠❡♥ç♦♥✱ ❚r❡ss♦✉ ❊①tr❡♠❡s✭✷✵✵✽✮ ❛♥❞ ❊❧❡❝tr♦♥✐❝ ❏♦✉r♥❛❧ ❙t❛t✳ ✭✷✵✶✸✮✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✹ ✴ ✸✽

slide-5
SLIDE 5

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❚❤❡ ❛t♦♠✐❝ ❝❛s❡

  • ❡♥❡r❛❧ ❢r❛♠❡✇♦r❦ ✿ ❍❛rr✐s r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥s

▼❛r❦♦✈ ❝❤❛✐♥ ❳ ✐s s❛✐❞ r❡❣❡♥❡r❛t✐✈❡ ✇❤❡♥ ✐t ♣♦ss❡ss❡s ❛♥ ❛❝❝❡ss✐❜❧❡ ❛t♦♠✱ ✐✳❡✳ ❛ ♠❡❛s✉r❛❜❧❡ s❡t ❆ s✉❝❤ t❤❛t ψ(❆) > ✵ ❛♥❞ Π(①, .) = Π(②, .) ❢♦r ❛❧❧ ①, ② ✐♥ ❆✱ ❉❡♥♦t❡ ❜② τ❆ = τ❆(✶) = ✐♥❢ {♥ ≥ ✶, ❳♥ ∈ ❆} t❤❡ ❤✐tt✐♥❣ t✐♠❡ ♦♥ ❆✳ P✉t ❛❧s♦ τ❆(❥ ) = ✐♥❢ {♥ > τ❆(❥ − ✶), ❳♥ ∈ ❆} , ❥ ≥ ✷ ❢♦r t❤❡ s✉❝❝❡ss✐✈❡ r❡t✉r♥ t✐♠❡s t♦ ❆✱ ❧♥ = ♥

✐=✶ I{❳✐ ∈ ❆} ♥✉♠❜❡r ♦❢ ✈✐s✐ts ♦❢ ❳ t♦ t❤❡ r❡❣❡♥❡r❛t✐♦♥ s❡t ❆

✉♥t✐❧ t✐♠❡ ♥✱ ✐❢ τ❆ < ∞ ♦r ✐❢ t❤❡r❡ ❡①✐sts ✵ < β s✉❝❤ t❤❛t ❊τβ

❆ < ∞✱ t❤❡ s❛♠♣❧❡

♣❛t❤s ♦❢ t❤❡ ❝❤❛✐♥ ♠❛② ❜❡ ❞✐✈✐❞❡❞ ✐♥t♦ ✐✳✐✳❞✳ ❜❧♦❝❦s ♦❢ r❛♥❞♦♠ ❧❡♥❣t❤ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❝♦♥s❡❝✉t✐✈❡ ✈✐s✐ts t♦ ❆✱ ❣❡♥❡r❛❧❧② ❝❛❧❧❡❞ r❡❣❡♥❡r❛t✐♦♥ ❝②❝❧❡s ✿ B✶ = (❳τ❆(✶)+✶, ..., ❳τ❆(✷)), ..., B❥ = (❳τ❆(❥ )+✶, ..., ❳τ❆(❥ +✶)), ...

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✺ ✴ ✸✽

slide-6
SLIDE 6

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❚❤❡ ❛t♦♠✐❝ ❝❛s❡

❊①❛♠♣❧❡ ✶ ✿ ❈r❛♠❡r✲▲✉♥❞❜❡r❣ ✇✐t❤ ❛ ❞✐✈✐❞❡♥❞ ❜❛rr✐❡r

s❡❡ ❢♦r ✐♥st❛♥❝❡ t❤❡ ❜♦♦❦s ❜② ❆s♠✉ss❡♥✭✷✵✵✸✮✱ ❊♠❜r❡❝❤ts✱ ❑❧ü♣♣❡❧❜❡r❣✱ ❛♥❞ ▼✐❦♦s❝❤✭✶✾✾✼✮ ◆✉♠❜❡r ♦❢ ❝❧❛✐♠s ❛♥ ✐♥t❡r✈❛❧ [✵, t] ✿ {◆(t), t ≥ ✵, ◆(✵) = ✵} ✿ ❛♥ ❤♦♠♦❣❡♥❡♦✉s P♦✐ss♦♥ ♣r♦❝❡ss ✇✐t❤ r❛t❡ λ,✳ ✐♥♣✉t t✐♠❡s (❚♥)♥∈N t✐♠❡s ♦❢ t❤❡ ❝❧❛✐♠s ❈❧❛✐♠s s✐③❡s ❯✐, ✐ = ✶,✳✳✳✳∞ ✱ ✐✳✐✳❞ r✈✬s ✇✐t❤ ❝❞❢ ❋. ❙(t) =

◆(t)

  • ✐=✶

❯✐ ❈♦♥st❛♥t ♣r❡♠✐✉♠ r❛t❡ ✭♣r✐❝❡ ♣❡r ✉♥✐t ♦❢ t✐♠❡✮ ❝. ❘❡s❡r✈❡ ♦❢ ❝♦♠♣❛♥② ✇✐t❤ ❛ ❝♦♥st❛♥t ❜❛rr✐❡r ❜, ♦✈❡r ✇❤✐❝❤ ♣r♦✜t ✐s r❡❞✐str✐❜✉t❡❞ ❡✈♦❧✈❡s ❧✐❦❡ ❳ (t) = (✉ + ❝t − ❙(t)) ∧ ❜, ❯♥❞❡r t❤❡ ♥❡t ♣r♦✜t ❝♦♥❞✐t✐♦♥✱ t❤❡ ❡♠❜❡❞❞❡❞ ❝❤❛✐♥ ❳♥ = ❳ (❚♥) ✐s ❛♥ ❛t♦♠✐❝ ▼❛r❦♦✈ ❝❤❛✐♥s ✇✐t❤ ❛♥ ❛t♦♠ ❛t ❜✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✻ ✴ ✸✽

slide-7
SLIDE 7

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❚❤❡ ❛t♦♠✐❝ ❝❛s❡

❚❤❡ ❜❧♦❝❦s ❝♦rr❡s♣♦♥❞✐♥❣ t♦ s✉❝❝❡ss✐✈❡ ✈✐s✐ts t♦ ❜ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ t❤❡ r✉✐♥ ✐s ❛tt❛✐♥❡❞ ✇❤❡♥ t❤❡ ♠✐♥ = t♦ t❤❡ ♠✐♥ ♦✈❡r r❡❣❡♥❡r❛t✐♦♥ ❜❧♦❝❦s ♦❢ t❤❡ ❡♠❜❡❞❞❡❞ ❝❤❛✐♥ ✐s ♥❡❣❛t✐✈❡✱ ❙♠✐t❤✭✶✾✺✺✮✳

20 40 60 80 100 2 4 6 8 time Compagny reserves

X(t) Cramer−Lundberg model with a barrier

❋✐❣✉r❡✿ ❈r❛♠ér✲▲✉♥❞❜❡r❣ ♠♦❞❡❧ ✇✐t❤ ❛ ❞✐✈✐❞❡♥❞ ❜❛rr✐❡r ❛t ❜✱ r✉✐♥ ❛t ✵✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✼ ✴ ✸✽

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SLIDE 8

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❚❤❡ ❛t♦♠✐❝ ❝❛s❡

❍❛rr✐s ❝❤❛✐♥s ✿ ◆✉♠♠❡❧✐♥ s♣❧✐tt✐♥❣ tr✐❝❦

❉❡✜♥✐t✐♦♥ ❆ s❡t ❙ ∈ E ✐s s❛✐❞ t♦ ❜❡ s♠❛❧❧ ❢♦r ❳ ✐❢ t❤❡r❡ ❡①✐st ♠ ∈ N∗✱ δ > ✵ ❛♥❞ ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ Φ s✉♣♣♦rt❡❞ ❜② ❙ s✉❝❤ t❤❛t✱ ❢♦r ❛❧❧ ① ∈ ❙, ❇ ∈ E✱ Π♠(①, ❇) ≥ δΦ(❇). ❙♦♠❡ s✐♠♣❧✐✜❝❛t✐♦♥s ✿ ♠❂✶ ✭❡✈❡♥ ✐❢ ✐t ❡♥t❛✐❧s r❡♣❧❛❝✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝❤❛✐♥ ❳ ❜② t❤❡ ❝❤❛✐♥ {(❳♥♠, ..., ❳♥(♠+✶)−✶)}♥∈N✮✳ Φ(❇) ♠❛② ❜❡ ❝❤♦s❡♥ t♦ ❜❡ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦✈❡r ❙

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✽ ✴ ✸✽

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SLIDE 9

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❚❤❡ ❛t♦♠✐❝ ❝❛s❡

❚❤❡ ◆✉♠♠❡❧✐♥ s♣❧✐tt✐♥❣ tr✐❝❦

◆✉♠♠❡❧✐♥ ✭✶✾✼✽✮✱ ❆t❤r❡②❛ ❛♥❞ ◆❡② ✭✶✾✼✽✮ ✿ ❆♥② ❍❛rr✐s r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥s ❝❛♥ ❜❡ ♠❛❞❡ ❛t♦♠✐❝ ✦ ❚❤❡ s❛♠♣❧❡ s♣❛❝❡ ✐s ❡①♣❛♥❞❡❞ s♦ ❛s t♦ ❞❡✜♥❡ ❛ s❡q✉❡♥❝❡ (❨♥)♥∈N ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❇❡r♥♦✉❧❧✐ r✳✈✳✬s ✇✐t❤ ♣❛r❛♠❡t❡r δ ❜② ❞❡✜♥✐♥❣ t❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ Pν,M✱ ✇❤♦s❡ ❝♦♥str✉❝t✐♦♥ r❡❧✐❡s ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ r❛♥❞♦♠✐③❛t✐♦♥ ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② Π ❡❛❝❤ t✐♠❡ t❤❡ ❝❤❛✐♥ ❤✐ts ❙✳ ■❢ ❳♥ ∈ ❙ ❛♥❞ ✐❢ ❨♥ = ✶ ✭✇✐t❤ ♣r♦❜❛❜✐❧✐t② δ ∈ ]✵, ✶[✮✱ t❤❡♥ ❳♥+✶ ∼ Φ✱ ✐❢ ❨♥ = ✵✱ t❤❡♥ ❳♥+✶ ∼ (✶ − δ)−✶(Π(❳♥+✶, .) − δΦ(.)). ❆❙ = ❙ × {✶} ✐s ❛♥ ❛t♦♠ ❢♦r t❤❡ ❜✐✈❛r✐❛t❡ ▼❛r❦♦✈ ❝❤❛✐♥ (❳ , ❨ )✱ ✇❤✐❝❤ ✐♥❤❡r✐ts ❛❧❧ ✐ts ❝♦♠♠✉♥✐❝❛t✐♦♥ ❛♥❞ st♦❝❤❛st✐❝ st❛❜✐❧✐t② ♣r♦♣❡rt✐❡s ❢r♦♠ ❳ ✭r❡❢❡r t♦ ❈❤❛♣t✳ ✶✹ ♦❢ ▼❡②♥ ❛♥❞ ❚✇❡❡❞✐❡ ✭✶✾✾✻✮✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✾ ✴ ✸✽

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SLIDE 10

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❚❤❡ ❛t♦♠✐❝ ❝❛s❡

❚❤❡ ◆✉♠♠❡❧✐♥ s♣❧✐tt✐♥❣ tr✐❝❦

❆ss✉♠♣t✐♦♥ ✿ {Π(①, ❞②)}①∈❊ ❛♥❞ t❤❡ ✐♥✐t✐❛❧ ❞✐str✐❜✉t✐♦♥ ν ❛r❡ ❞♦♠✐♥❛t❡❞ ❜② ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡ λ ♦❢ r❡❢❡r❡♥❝❡✱ s♦ t❤❛t ν(❞②) = ❢ν(②)λ(❞②) ❛♥❞ Π(①, ❞②) = π(①, ②)λ(❞②) ❢♦r ❛❧❧ ① ∈ ❊✳ Φ ✐s ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ✇✐t❤ r❡s♣❡❝t t♦ λ t♦♦✱ Φ(❞②) = φ(②).λ(❞②)✱ ∀① ∈ ❙, π(①, ②) ≥ δφ(②), λ(❞②)✲❛❧♠♦st s✉r❡❧②. ❚❤❡♦r❡t✐❝❛❧ ❙♣❧✐tt✐♥❣

  • ✐✈❡♥ t❤❡ s❛♠♣❧❡ ♣❛t❤ ❳ (♥+✶)✱ ❨✐✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s

❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ❨✐ ✐s t❤❡ ❇❡r♥♦✉❧❧✐ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❛r❛♠❡t❡r δφ(❳✐+✶) π(❳✐, ❳✐+✶) · I{❳✐ ∈ ❙} + δ · I{❳✐ / ∈ ❙}. ✭✶✮

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✵ ✴ ✸✽

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SLIDE 11

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥

❚❤❡ ❛♣♣r♦①✐♠❛t❡ s♣❧✐tt✐♥❣ tr✐❝❦

❯♥❢♦rt✉♥❛t❡❧② φ ✐s ✉♥❦♥♦✇♥ ✿ ✉s❡ ❛ ♣❧✉❣✲✐♥ r✉❧❡ t♦ ❝r❡❛t❡ ❛rt✐✜❝✐❛❧ ❇❡r♥♦✉❧❧✐ r✳✈✳✬s ✇✐t❤ ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs✳ ❆♣♣r♦①✐♠❛t❡ s♣❧✐tt✐♥❣ tr✐❝❦ ❛❧❣♦r✐t❤♠ ❈♦♠♣✉t❡ ✜rst ❛♥ ❡st✐♠❛t❡ π♥(①, ②) ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t② ♦✈❡r ❙ ✷✱ s✉❝❤ t❤❛t π♥(①, ②) ≥ δφ(②) ❛✳s✳ ❢♦r ❛❧❧ (①, ②) ∈ ❙ ✷

  • ✐✈❡♥ t❤❡ s❛♠♣❧❡ ♣❛t❤ ❳ (♥+✶)✱

❨✐✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ ❨✐ ✐s ❇❡r♥♦✉❧❧✐ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs

  • δφ(❳✐+✶)
  • π(❳✐, ❳✐+✶) · I{❳✐ ∈ ❙} +

δ · I{❳✐ / ∈ ❙}. ❙♣❧✐t t❤❡ ❝❤❛✐♥ ❡❛❝❤ t✐♠❡ ❳✐ ∈ ❙ ❛♥❞ ❨✐ = ✶✳ ❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❜❧♦❝❦s ❛r❡ ✧❛s②♠♣t♦t✐❝❛❧❧②✧ ✐♥❞❡♣❡♥❞❡♥t✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✶ ✴ ✸✽

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SLIDE 12

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥

❙♦♠❡ ❛❞❞✐t✐♦♥❛❧ ♥♦t❛t✐♦♥s ✐♥ t❤❡ ❛♣♣r♦①✐♠❛t❡ ✇♦r❧❞ ❆♣♣r♦①✐♠❛t❡ ♥✉♠❜❡r ♦❢ r❡❣❡♥❡r❛t✐♦♥s ❢♦r ❛ ❣✐✈❡♥ s♠❛❧❧ s❡t ^ ❧♥ =

  • ✶≤❦≤♥

I{(❳❦, ❨❦) ∈ ❙ × {✶}} ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ t✐♠❡s

  • τ❆❙ (❥ + ✶) = ✐♥❢{♥ ≥ ✶ +

τ❆❙ (❥ )/ (❳♥, ❨♥) ∈ ❙ × {✶}}✱ ❢♦r ✶ ≤ ❥ ≤ ❧♥ − ✶, Pr❛❝t✐❝❛❧ ❝❤♦✐❝❡ ♦❢ ❙ ✿ ♦♣t✐♠✐③❡ ♦✈❡r ❛ ❝❧❛ss ♦❢ s♠❛❧❧ s❡ts t♦ ♦❜t❛✐♥ t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❛♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐♦♥s

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✷ ✴ ✸✽

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SLIDE 13

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥

❊①❛♠♣❧❡ ✷ ✿ ❙♣❧✐tt✐♥❣ ❛ t✐♠❡ s❡r✐❡s

❆ t✐♠❡ s❡r✐❡s ♠♦❞❡❧ ❡①❤✐❜✐t✐♥❣ s♦♠❡ ♥♦♥❧✐♥❡❛r✐t✐❡s ❛♥❞ str✉❝t✉r❛❧ ❝❤❛♥❣❡s ❜♦t❤ ✐♥ ❧❡✈❡❧ ❛♥❞ ✈❛r✐❛♥❝❡ ✿ ❊❚❆❘✭✶✮✲❆❘❈❍✭✶✮ ♠♦❞❡❧ ✭❊①♣♦♥❡♥t✐❛❧ ❚❤r❡s❤♦❧❞ ❆✉t♦❘❡❣r❡ss✐✈❡ ▼♦❞❡❧ ✇✐t❤ ❆✉t♦❘❡❣r❡ss✐✈❡ ❈♦♥❞✐t✐♦♥❛❧ ❍❡t❡r♦s❝❡❞❛st✐❝✐t②✮ ❳t+✶ = (α✶ + α✷❡−❳ ✷

t )❳t + (✶ + β❳ ✷

t )✶/✷εt+✶ ,

✇❤❡r❡ t❤❡ ♥♦✐s❡ (εt)t=✶,...❚ ❛r❡ ✐✳✐✳❞ ✇✐t❤ ✈❛r✐❛♥❝❡ σ✷✳ ❯♥❞❡r st❛♥❞❛r❞ ❝♦♥❞✐t✐♦♥s✱ ❛❞♠✐ts ❛ st❛t✐♦♥❛r② s♦❧✉t✐♦♥s✱ ❜✉t ❡①❤✐❜✐ts s♦♠❡ t❤r❡s❤♦❧❞ ❛♥❞ ❤❡t❡r♦s❝❡❞❛t✐❝ ❜❡❤❛✈✐♦r✳ ❙❡❡ ❚❥øst❤❡✐♠✭✶✾✾✵✮✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✸ ✴ ✸✽

slide-14
SLIDE 14

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥

❋✐❣✉r❡✿ ❙♣❧✐tt✐♥❣ ❛ t✐♠❡✲s❡r✐❡s ❡①❤✐❜✐t✐♥❣ t❤r❡s❤♦❧❞s ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧ ❤❡t❡r♦s❝❡❞❛st✐❝✐t②✱ ♥❂✷✵✵✱α✶ = ✵.✻✵✱ α✷ = ✵.✹✺✱ β = ✵.✸✺ ❛♥❞ σ✷ = ✶✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✹ ✴ ✸✽

slide-15
SLIDE 15

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

■❞❡♥t✐❢② t❤❡ ✭♣s❡✉❞♦✲✮❜❧♦❝❦s B✶, . . . , B❧♥−✶ ❢r♦♠ t❤❡ ♦❜s❡r✈❡❞ tr❛❥❡❝t♦r② ❳✵, . . . , ❳♥ ❛♥❞ ❝♦♠♣✉t❡ t❤❡ st❛t✐st✐❝ ♦❢ ✐♥t❡r❡st ❛s

  • ❚♥ = ❚(B✶, . . . , B❧♥−✶) ❛♥❞ ✐ts st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥
  • σ♥ = σ(B✶, . . . , B❧♥−✶)✳

❉r❛✇ s❡q✉❡♥t✐❛❧❧② ❜♦♦tstr❛♣ ❞❛t❛ ❜❧♦❝❦s B∗

✶, . . . , B∗ ❦ ✐♥❞❡♣❡♥❞❡♥t❧② ❢r♦♠

t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❜❧♦❝❦s B✶, . . . , B❧♥−✶ ✉♥t✐❧ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❜♦♦tstr❛♣ s❡r✐❡s ❧∗(❦) = ❦

❥ =✶ ❧(B∗ ❥ ) ✐s ❧❛r❣❡r t❤❛♥ ♥✳ ▲❡t

❧∗

♥ = ✐♥❢{❦ ≥ ✶, ❧∗(❦) > ♥}✳

❋r♦♠ t❤❡ ❜♦♦tstr❛♣ ❞❛t❛ ❜❧♦❝❦s ❣❡♥❡r❛t❡❞ ❛t st❡♣ ✷✱ r❡❝♦♥str✉❝t ❛ ♣s❡✉❞♦✲tr❛❥❡❝t♦r② ❜② ❜✐♥❞✐♥❣ t❤❡ ❜❧♦❝❦s t♦❣❡t❤❡r✱ ❣❡tt✐♥❣ t❤❡ r❡❝♦♥str✉❝t❡❞ ❘❇❇ s❛♠♣❧❡ ♣❛t❤ ❳ ∗(♥) = (B∗

✶, . . . , B∗ ❧∗

♥ −✶) ♦❢ ❧❡♥❣t❤ ♥ ♦r

♥∗ ✭✇❤❡♥ ❣❡tt✐♥❣ r✐❞ ♦❢ t❤❡ ❧❛st ❜❧♦❝❦✳ ❈♦♠♣✉t❡ t❤❡ ❜♦♦tstr❛♣ ✈❡rs✐♦♥ ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❜❧♦❝❦s ❡st✐♠❛t♦r ✿ ❚ ∗

♥ = ❚(B∗ ✶, . . . , B∗ ❧∗

♥ −✶) ❛♥❞ ✐ts st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥

  • σ∗

♥ = σ(B∗ ✶, . . . , B∗ ❧∗

♥ −✶)✳ P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✺ ✴ ✸✽

slide-16
SLIDE 16

Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s ❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

❘❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s

❇♦♦tstr❛♣ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧s ✭❈■✮ ❛t ❧❡✈❡❧ ✶ − α ∈ (✶/✷, ✶) ❢♦r t❤❡ ♣❛r❛♠❡t❡r ♦❢ ✐♥t❡r❡st ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❝♦♠♣✉t✐♥❣ t❤❡ ❜♦♦tstr❛♣ r♦♦t✬s q✉❛♥t✐❧❡s q∗

α/✷ ❛♥❞ q∗ ✶−α/✷✱ ♦❢ ♦r❞❡rs α/✷ ❛♥❞ ✶ − α/✷

❇❛s✐❝ ♣❡r❝❡♥t✐❧❡ ❜♦♦tstr❛♣ ❈■ ✿ I❇P∗

✶−α = [q∗ α/✷, q∗ ✶−α/✷]✳

P❡r❝❡♥t✐❧❡ ❜♦♦str❛♣ ❈■ ✿ IP∗

✶−α =

  • ✷

❚♥ − q∗

✶−α/✷, ✷

❚♥ − q∗

α/✷

  • t✲P❡r❝❡♥t✐❧❡ ❜♦♦str❛♣ ❈■ ✿

ItP∗

✶−α =

  • ❚♥ − t∗

✶−α/✷

σ♥/√♥, ❚♥ − t∗

α/✷

σ♥/√♥

  • ✱ ✇❤❡r❡ t∗

♣ ✐s t❤❡ ♣t❤

q✉❛♥t✐❧❡ ♦❢ t❤❡ st✉❞❡♥t✐③❡❞ ❜♦♦tstr❛♣ r♦♦t

  • ❚ ∗

♥ −

❚♥

  • /
  • σ∗

♥/√♥

  • .

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✻ ✴ ✸✽

slide-17
SLIDE 17

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❖✉t❧✐♥❡s

✶ Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s

❋r❛♠❡✇♦r❦ ❍❛rr✐s r❡❝✉rr❡♥❝❡ ❚❤❡ ❛t♦♠✐❝ ❝❛s❡ ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✷ ❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s ▼❛✐♥ ❍②♣♦t❤❡s❡s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✸ ❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s

❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s ▲✐♥❡❛r ❘❛♥❦ s✐❣♥❡❞ st❛t✐st✐❝s ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

✹ ❆ ❢❡✇ r❡❢❡r❡♥❝❡s

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✼ ✴ ✸✽

slide-18
SLIDE 18

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

■♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

■♥ ❣❡♥❡r❛❧ ✐t ✐s ❞✐✣❝✉❧t t♦ ❞❡✜♥❡ t❤❡ ♥♦t✐♦♥ ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ❣❡♥❡r❛❧ t✐♠❡ s❡r✐❡s ✭ s❡❡ ▼❛rt✐♥ ❛♥❞ ❨♦❤❛✐✱ ✮ ✳ P❜ ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❝♦♥t❛♠✐♥❛t❡❞ ♠♦❞❡❧ ✭■♥♥♦✈❛t✐✈❡ ♦✉t❧✐❡rs✱ ❛❞❞✐t✐✈❡ ♦✉t❧✐❡rs✱ str✉❝t✉r❛❧ ❜r❡❛❦ ❡t❝✳✳✳✮ ❇❯❚ ❛♥ ❡❛s✐❡r ❛♥❞ ♥❛t✉r❛❧ ✈❡rs✐♦♥ ❢♦r ❍❛rr✐s ▼❛r❦♦✈ ❈❤❛✐♥✳ ❆❧❧ ❢✉♥❝t✐♦♥❛❧s ♦❢ t❤❡ st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ♠❛② ❜❡ s❡❡♥ ❛s ❢✉♥❝t✐♦♥❛❧s ♦❢ t❤❡ ❜❧♦❝❦ ❞✐str✐❜✉t✐♦♥ ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ t♦r✉s ♦♥ t❤❡ t♦r✉s T✳ ❉❡✜♥✐t✐♦♥ ✭■♥❢❧✉❡♥❝❡ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ t♦r✉s✮ ▲❡t (V, ||.||) ❜❡ ❛ s❡♣❛r❛❜❧❡ ❇❛♥❛❝❤ s♣❛❝❡✳ ▲❡t ❚ : PT → V ❜❡ ❛ ❢✉♥❝t✐♦♥❛❧ ♦♥ PT✳ ■❢✱ ❢♦r ❛❧❧ L ✐♥ PT✱ t−✶(❚((✶ − t)L + tδ❜) − ❚(L)) ❤❛s ❛ ✜♥✐t❡ ❧✐♠✐t ❛s t → ✵ ❢♦r ❛♥② ❜ ∈ T✱ t❤❡ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❞❡✜♥❡❞ ❜② ❚ (✶)(❜, L) = ❧✐♠

t→✵

❚((✶ − t)L + tδ❜) − ❚(L) t . ✭✷✮

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✽ ✴ ✸✽

slide-19
SLIDE 19

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❉❡✜♥✐t✐♦♥ ✭●r♦ss✲❡rr♦r s❡♥s✐✈✐t②✮ ❆ ❢✉♥❝t✐♦♥❛❧ ❚ ✐s s❛✐❞ t♦ ❜❡ ▼❛r❦♦✈✲r♦❜✉st ✐✛ ✐ts ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❚ (✶)(❜, L) ✐s ❜♦✉♥❞❡❞ ♦♥ t❤❡ t♦r✉s T✳ ❚❤❡ ❣r♦ss✲❡rr♦r s❡♥s✐t✐✈✐t② t♦ ❜❧♦❝❦ ❝♦♥t❛♠✐♥❛t✐♦♥ ✐s t❤❡♥ ❞❡✜♥❡❞ ❛s γ∗(❚, L) = s✉♣

❜∈T

||❚ (✶)(❜, L)||.

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✶✾ ✴ ✸✽

slide-20
SLIDE 20

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ♦♥ t❤❡ t♦r❛✉s

❉❡✜♥✐t✐♦♥ ✭❋ré❝❤❡t ❞✐❢❢❡r❡♥t✐❛❜✐❧✐t② ♦♥ t❤❡ t♦r✉s✮ ■t ✐s s❛✐❞ t❤❛t ❚ ✐s ❋ré❝❤❡t✲❞✐✛❡r❡♥t✐❛❜❧❡ ❛t L✵ ∈ PT ❢♦r ❛ ♠❡tr✐❝ ❞ ♦♥ t❤❡ ❚♦r✉s✱ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❝♦♥t✐♥✉♦✉s ❧✐♥❡❛r ♦♣❡r❛t♦r ❉❚L✵ ✭❢r♦♠ ♦♥ t❤❡ s❡t ♦❢ t❤❡ s✐❣♥❡❞ ♠❡❛s✉r❡s ♦❢ t❤❡ ❢♦r♠ L − L✵ ✐♥ (V, ||.||)) ❛♥❞ ❛ ❢✉♥❝t✐♦♥ ǫ(✶)(., L✵) : R → (V, ||.||)✱ ❝♦♥t✐♥✉♦✉s ❛t ✵ ✇✐t❤ ǫ(✶)(✵, L✵) = ✵ s✉❝❤ t❤❛t ✿ ∀L ∈ PT✱ ❚(L) − ❚(L✵) = ❉❚L✵(L − L✵) + ❘(✶)(L, L✵), ✇✐t❤ ❘(✶)(L, L✵) = ❞(L, L✵)ǫ(✶)(❞(L, L✵), L✵)✳ ■♥ ❛❞❞✐t✐♦♥✱ ❚ ✐s s❛✐❞ t♦ ❤❛✈❡ ❛ ❝❛♥♦♥✐❝❛❧ ❣r❛❞✐❡♥t ✭♦r ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥✮ ❚ (✶)(., L✵) ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡s❡♥t❛t✐♦♥ ❢♦r ❉❚L✵ ❤♦❧❞s ✿ ∀L ∈ PT, ❉❚L✵(L − L✵) =

  • T

❚ (✶)(❜, L✵)L(❞❜).

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✵ ✴ ✸✽

slide-21
SLIDE 21

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s

❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s

µ(❢ )

❞❡❢

= Eµ[❢ (❳ )]✳ ❉❡♥♦t❡ ❜② B ❛ r✳✈✳ ✈❛❧✉❡❞ ✐♥ T ✇✐t❤ ❞✐str✐❜✉t✐♦♥ L t❤❡♥ µ(❢ ) = EL [f(B)] /EL [▲(B)] = ❚(L), t❤❡♥ ❚ (✶)(❜, L) = ❞ ❞t (❚((✶ − t)L + t❜)|t=✵ = f(❜) − µ(❢ )▲(❜) EL [▲(B)] . ❊✈❡♥ ✐❢ ❢ ✐s ❜♦✉♥❞❡❞ t❤✐s ❢✉♥❝t✐♦♥❛❧ ✐s ♥♦t r♦❜✉st✳ ▼✲♣❛r❛♠❡t❡r✴ ❡st✐♠❛t♦rs ▲❡t θ ❜❡ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ✿ Eµ [❣(❳ , θ)] = ✵, ✭✸✮ ✇❤❡r❡ ❣ : R✷ → R ✐s ♦❢ ❝❧❛ss C✷ ✇❤✐❝❤ ✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ EL[g(B, θ)] = ✵ t❤❡♥ t❤❡ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② ❚ (✶)

❣

(❜, L) = − g(❜, θ) EL

  • ∂g(B,θ)

∂θ

,

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✶ ✴ ✸✽

slide-22
SLIDE 22

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s

❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s

❆ss✉♠❡ t❤❛t t❤❡ st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ❤❛s ❛ ❝♦♥t✐♥✉♦✉s str✐❝t❧② ✐♥❝r❡❛s✐♥❣ ❝✉♠✉❧❛t✐✈❡ ❞✐str✐❜✉t✐♦♥ ❢✉♥❝t✐♦♥ ✭❝❞❢✮ ❋µ(①) = µ(] − ∞, ①]) ❛♥❞ ❞❡♥s✐t② ❢µ(①)✳ ❈♦♥s✐❞❡r t❤❡ α−q✉❛♥t✐❧❡ ❚α(µ) = ❋ −✶

µ (α)✳ ✐❢

❢µ(❚α(L)) = ✵✱ t❤❡ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ✭s♠❡ ❛r❣✉♠❡♥ts ❛s s❡❡ ❬✶✵❪✮ ✐s ❣✐✈❡♥ ❤❡r❡ ❜② ❚ (✶)

α (❜, L) =

▲(❜)

✐=✶ (α − I{❜✐ ≤ ❚α(L)})

EL[▲(B)]❢µ(❚α(L)) ◆♦t ❜♦✉♥❞❡❞ ✦ ◗✉❛♥t✐❧❡s ❛r❡ ♥♦t r♦❜✉st ✐♥ t✐♠❡ s❡r✐❡s ❛♥❞ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ✿ ❛ s✐♥❣❧❡ ❝♦♥t❛♠✐♥❛t✐♦♥ ✭✐♥♥♦✈❛t✐✈❡ ♦✉t❧✐❡r✮ ♠❛② ❝r❡❛t❡ ❛ ✈❡r② ❧♦♥❣ ✭♦✉t❧✐❡r✮ ❜❧♦❝❦ ✭❜❡❢♦r❡ r❡❛❝❤✐♥❣ ❛❣❛✐♥ st❛t✐♦♥❛r✐t②✮✳ ◆❡❡❞❡❞ t♦ tr✉♥❝❛t❡ ❜♦t❤ t❤❡ ❢✉♥❝t✐♦♥❛❧ t♦ ♠❛❦❡ ✐t r♦❜✉st t♦ ❛❞❞✐t✐✈❡ ♦✉t❧✐❡r ❜✉t ❛❧s♦ ❧♦♥❣ ❜❧♦❝❦s t♦ ♠❛❦❡ ✐t r♦❜✉st t♦ ✐♥♥♦✈❛t✐✈❡ ♦✉t❧✐❡r✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✷ ✴ ✸✽

slide-23
SLIDE 23

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ▼❛✐♥ ❍②♣♦t❤❡s❡s

❍②♣♦t❤❡s❡s ❆✵✳ ▼❡❛♥ sq✉❛r❡❞ ❡rr♦r ✭▼❙❊✮ ♦❢ t❤❡ ❡st✐♠❛t♦r ♦❢ t❤❡ tr❛♥s✐t✐♦♥ ❞❡♥s✐t② α♥ = R♥(^ π♥, π) = E[( s✉♣

(①,②)∈❙ ✷ |^

π♥(①, ②) − π(①, ②)|)✷] − → ✵ ✐♥ Pν ♣r♦❜❛❜✐❧✐t② ❛s ♥ → ∞ ❆✶✳ t❤❡ ♣❛r❛♠❡t❡rs ❙ ❛♥❞ φ ❛r❡ ❝❤♦s❡♥ s♦ t❤❛t ✐♥❢①∈❙ φ(①) > ✵ ❆✷✳ s✉♣(①,②)∈❙ ✷ π(①, ②) < ∞ ❛♥❞ Pν✲❛❧♠♦st s✉r❡❧② s✉♣

♥∈N

s✉♣

(①,②)∈❙ ✷ ^

π♥(①, ②) < ∞ ✳ ❆✸ ✭❘❡❣❡♥❡r❛t✐✈❡ ❝❛s❡✮ H(κ) : E❆[τκ

❆] < ∞,

H(ν, κ) : Eν[τκ

❆] < ∞.

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✸ ✴ ✸✽

slide-24
SLIDE 24

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ▼❛✐♥ ❍②♣♦t❤❡s❡s

❍②♣♦t❤❡s❡s ❆✹ ✭●❡♥❡r❛❧ ❍❛rr✐s r❡❝✉rr❡♥t ❝❛s❡✮ ⑦ H(κ) : s✉♣①∈❙ E①[τκ

❙] < ∞,

⑦ H(ν, κ) : Eν[τκ

❙] < ∞.

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✹ ✴ ✸✽

slide-25
SLIDE 25

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥

P❧✉❣✲✐♥ ❡st✐♠❛t♦rs ❜❛s❡❞ ♦♥ ❜❧♦❝❦s

  • ✐✈❡♥ ❛♥ ♦❜s❡r✈❡❞ ♣❛t❤ ♦❢ ❧❡♥❣t❤ ♥✱ ♥❛t✉r❛❧ ❡♠♣✐r✐❝❛❧ ❡st✐♠❛t❡s ♦❢

♣❛r❛♠❡t❡rs ❚(L) ❛r❡ ♦❢ ❝♦✉rs❡ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦rs ❚(L♥) ❜❛s❡❞ ♦♥ t❤❡ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦❜s❡r✈❡❞ r❡❣❡♥❡r❛t✐♦♥ ❜❧♦❝❦s L♥ = (❧♥ − ✶)−✶

❧♥−✶

  • ❥ =✶

δB❥ ∈ PT ♦r ✐♥ t❤❡ ❣❡♥❡r❛❧ ❍❛rr✐s r❡❝✉rr❡♥t ❝❛s❡✱ ❚( L♥)

  • L♥ = (

❧♥ − ✶)

  • ❧♥−✶
  • ❥ =✶

δ

B❥ ,

❉✐st❛♥❝❡ ♦♥ t❤❡ ❚♦r✉s ✿ ❇♦✉♥❞❡❞ ▲✐♣s❝❤✐t③ t②♣❡ ♠❡tr✐❝ ♦♥ PT ❞❇▲(L, L′) = s✉♣

❢ ∈▲✐♣✶

T

{

  • ❢ (❜)L(❞❜) −
  • ❢ (❜)L′(❞❜)},

✭✹✮

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✺ ✴ ✸✽

slide-26
SLIDE 26

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥

❈♦♥tr♦❧❧✐♥❣ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❡♠♣✐r✐❝❛❧ ❜❧♦❝❦✲❞✐str✐❜✉t✐♦♥s

❚❤❡♦r❡♠ ✭❘❛t❡ ❜♦✉♥❞s ❢♦r t❤❡ ▲✐♣s❝❤✐t③ ❞✐st❛♥❝❡✮ ❯♥❞❡r H✵(✹) ❛♥❞

  • H✵(✹, ν)✱ ✇❡ ❤❛✈❡

❞❇▲(L♥, L) = ❖P(♥−✶/✷) ❛s ♥ → ∞✱ ❛♥❞ ❞❇▲(L♥, L♥) = ❖P(α✶/✷

♥

♥−✶/✷) = ♦P(♥−✶/✷)✱ ❛s ♥ → ∞.

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✻ ✴ ✸✽

slide-27
SLIDE 27

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥

❈▲❚ ❢♦r ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥❛❧s ♦♥ t❤❡ t♦r✉s

❚❤❡♦r❡♠ ✭❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠✮ ■♥ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❝❛s❡✱ ✐❢ ❚ : PT → R ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t L ❛♥❞ ❞(L♥, L) = ❖Pν(♥−✶/✷) ✭♦r ❘(✶)(L♥, L) = ♦Pν(♥−✶/✷)✮ ❛s ♥ → ∞✱ ❛♥❞ ✐❢ E❆[τ❆] < ∞ ❛♥❞ ✵ < ❱❛r❆(❚ (✶)(B✶, L)) < ∞ t❤❡♥✱ ✉♥❞❡r Pν✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ♥✶/✷ (❚(L♥) − ❚(L)) ⇒ N(✵, E❆[τ❆]❱❛r❆(❚ (✶)(B✶, L)), ❛s ♥ → ∞. ■♥ t❤❡ ❣❡♥❡r❛❧ ❍❛rr✐s ❝❛s❡✱ ✐❢ t❤❡ s♣❧✐t ❝❤❛✐♥ s❛t✐s✜❡s t❤❡ ❛ss✉♠♣t✐♦♥s ❛❜♦✈❡ ✭✇✐t❤ ❆ r❡♣❧❛❝❡❞ ❜② ❆❙ ❛♥❞ ❞( L♥, L) = ❖Pν(♥−✶/✷)✮✱ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✹✱ ❛s ♥ → ∞ ✇❡ ❤❛✈❡✱ ✉♥❞❡r Pν✱ ♥✶/✷ ❚( L♥) − ❚(L)

  • ⇒ N(✵, E❆❙ [τ❆❙ ]❱❛r❆❙ (❚ (✶)(B✶, L)).

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✼ ✴ ✸✽

slide-28
SLIDE 28

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

❚❤❡♦r❡♠ ✭❘❡❣❡♥❡r❛t✐✈❡ ❜♦♦tstr❛♣ ✈❡rs✐♦♥✮ ❉❡♥♦t❡ ❜② L∗

♥ t❤❡ ❡♠♣✐r✐❝❛❧

❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❜♦♦tstr❛♣ t❛❦❡♥ ✇✐t❤ r❡♣❧❛❝❡♠❡♥t ❡✐t❤❡r ❢r♦♠ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❜❧♦❝❦s ♦r t❤❡ ❛♣♣r♦①✐♠❛t❡ ❜❧♦❝❦s✳ ■♥ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❝❛s❡✱ ❛ss✉♠❡ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✺ ❤♦❧❞ ❢♦r t❤❡ ♠❡tr✐❝ ❝❤♦s❡♥ ❞ t❤❡♥ ✇❡ ❛❧s♦ ❤❛✈❡ ♥✶/✷ (❚(L∗

♥) − ❚(L♥)) ⇒ N(✵, E❆[τ❆]❱❛r❆(❚ (✶)(B✶, L)), ❛s ♥ → ∞.

❙✐♠✐❧❛r❧②✱ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❍❛rr✐s ♣♦s✐t✐✈❡ r❡❝✉rr❡♥t ❝❛s❡ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✺✱ ✇❡ ❤❛✈❡ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ✿ ♥✶/✷ ❚(L∗

♥) − ❚(

L♥)

  • ⇒ N(✵, E❆❙ [τ❆❙ ]❱❛r❆❙ (❚ (✶)(B✶, L)) ❛s ♥ → ∞.

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✽ ✴ ✸✽

slide-29
SLIDE 29

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

◆♦♥ r♦❜✉st♥❡ss ♦❢ t❤❡ ❇♦♦tstr❛♣ ❢♦r ♥♦♥ r♦❜✉st ❢✉♥❝t✐♦♥❛❧

❚❤❡ ❜♦♦tstr❛♣ ❞✐str✐❜✉t✐♦♥ ✐s ✈❡r② s❡♥s✐t✐✈❡ t♦ ♦✉t❧✐❡r ✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥s✳ ❚♦ ✐❧❧✉str❛t❡ t❤✐s ❢❛❝t ❝♦♥s✐❞❡r ✇❛✐t✐♥❣ t✐♠❡s ♦❢ ❛ ▼✴▼✴✶ ♣r♦❝❡ss ✭❝❢ ❬✶❪✮ ✇✐t❤ ❡①♣♦♥❡♥t✐❛❧ ♣❛r❛♠❡t❡rs λ✶ = ✵.✻✱ λ✷ = ✵.✽ ✇✐t❤ ♥❂✺✵✵✳ ■♥ t❤✐s ❝❛s❡✱ ❆ = {✵} ✭✐✳❡✳ ✧❡♠♣t② ✜❧❡✧✮ ✐s ❛♥ ❛t♦♠✳ ❚❤❡ st❛t✐♦♥❛r② ❞✐str✐❜✉t✐♦♥ ✐s ❛ ♠✐①t✉r❡ ✵.✷✺δ✵ + ✵.✼✺µ✵✱ ✇❤❡r❡ µ✵ ✐s ❛ ❤✐❣❤❧② ❛s②♠♠❡tr✐❝ ❝♦♥t✐♥✉♦✉s ❞✐str✐❜✉t✐♦♥ ♦♥ t❤❡ ❤❛❧❢✲❧✐♥❡ R+✳ ❲❤❛t ❤❛♣♣❡♥s ✐❢ ❛t t❂✷✺✵ t❤❡ s❡r✈❡r ❞♦❡s ♥♦t ✇♦r❦ ❢♦r ✷✵ ♣❡r✐♦❞ ✭t❤❛t ✐s t❤❡ q✉❡✉❡ ✐♥❝r❡❛s❡ ✇✐t❤ ❡♥tr② ✇✐t❤ ✵✳✻ ❡①♣♦♥❡♥t✐❛❧ ❞✐str✐❜✉t✐♦♥✮✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✷✾ ✴ ✸✽

slide-30
SLIDE 30

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

−2 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 True and Bootstrap dist., Mean in a MM1 0.8 0.6

❋✐❣✉r❡✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ tr✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ r❡❝❡♥t❡r❡❞ ♥♦r♠❛❧✐③❡❞ ♠❡❛♥ ✐♥ ❛ ▼▼✶ ♠♦❞❡❧✱ ✇✐t❤ ♣❛r❛♠❡t❡r ✵✳✽ ✵✳✻ ♥♦ ♦✉t❧✐❡r✱ ♥❂✺✵✵

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✵ ✴ ✸✽

slide-31
SLIDE 31

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

−2 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 True and Bootstrap dist., Mean in a MM1 0.8 0.6

❋✐❣✉r❡✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ tr✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ r❡❝❡♥t❡r❡❞ ♥♦r♠❛❧✐③❡❞ ♠❡❛♥ ✐♥ ❛ ▼▼✶ ♠♦❞❡❧✱ ✇✐t❤ ♣❛r❛♠❡t❡r ✵✳✽ ✵✳✻ ■♥♥♦✈❛t✐✈❡ ♦✉t❧✐❡r ❢♦r ✷✵ ♣❡r✐♦❞s✱ ♥❂✺✵✵

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✶ ✴ ✸✽

slide-32
SLIDE 32

❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

−2 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 0.5 True and Bootstrap dist., Median in a MM1 0.8 0.6

❋✐❣✉r❡✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ tr✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ r❡❝❡♥t❡r❡❞ ♥♦r♠❛❧✐③❡❞ ♠❡❞✐❛♥ ✐♥ ❛ ▼▼✶ ♠♦❞❡❧✱ ✇✐t❤ ♣❛r❛♠❡t❡r ✵✳✽ ✵✳✻ ■♥♥♦✈❛t✐✈❡ ♦✉t❧✐❡r ❢♦r ✷✵ ♣❡r✐♦❞s✱ ♥❂✺✵✵

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✷ ✴ ✸✽

slide-33
SLIDE 33

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s

❖✉t❧✐♥❡s

✶ Ps❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❢♦r ▼❛r❦♦✈ ❈❤❛✐♥s

❋r❛♠❡✇♦r❦ ❍❛rr✐s r❡❝✉rr❡♥❝❡ ❚❤❡ ❛t♦♠✐❝ ❝❛s❡ ❆♣♣r♦①✐♠❛t❡ r❡♥❡✇❛❧ s❡q✉❡♥❝❡ ✿ ♣s❡✉❞♦✲r❡❣❡♥❡r❛t✐♦♥ ❘❡❣❡♥❡r❛t✐✈❡ ❛♥❞ ❆♣♣r♦①✐♠❛t❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✷ ❘♦❜✉st♥❡ss ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❘♦❜✉st♥❡ss ❛♥❞ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s ❊①❛♠♣❧❡s ♦❢ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥s ▼❛✐♥ ❍②♣♦t❤❡s❡s ❛ ❈❡♥tr❛❧ ▲✐♠✐t ❚❤❡♦r❡♠ ❛♥❞ ✐ts ❜♦♦tstr❛♣ ✈❡rs✐♦♥ ❆s②♠♣t♦t✐❝ ✈❛❧✐❞✐t② ♦❢ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❇♦♦tstr❛♣

✸ ❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s

❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s ▲✐♥❡❛r ❘❛♥❦ s✐❣♥❡❞ st❛t✐st✐❝s ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

✹ ❆ ❢❡✇ r❡❢❡r❡♥❝❡s

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✸ ✴ ✸✽

slide-34
SLIDE 34

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s ❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s

❘♦❜✉st✐✜❡❞ ❡♠♣✐r✐❝❛❧ ❞✐str✐❜✉t✐♦♥

▲❡t ▼ > ✵ ❛♥❞ ❝♦♥s✐❞❡r t❤❡ r♦❜✉st✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❝❞❢ ❋µ(②) = µ(] − ∞, ②]) ❣✐✈❡♥ ❜② ❋L,▼ (②) = E❆ τ❆

✐=✶ I{❳✐ ≤ ②}

  • I{τ❆ ≤ ▼}
  • E❆[τ❆I{τ❆ ≤ ▼}]

. ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t t❤❡ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ♦❢ ❋L,▼ (②) ✐s ❣✐✈❡♥ ❜② ❋ (✶)

▼ (❜; ②, L) =

▲(❜)

✐=✶ (I{❜✐ ≤ ②} − ❋µ(②)) I{▲(❜) ≤ ▼}

E❆[τ❆I{τ❆ ≤ ▼}] ❢♦r ❛❧❧ ❜ ∈ T,

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✹ ✴ ✸✽

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SLIDE 35

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s ❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s

❋♦r ▼ ≥ ✶✱ ❞❡♥♦t❡ ❜② ❋ −✶

L,▼ (α) t❤❡ α−q✉❛♥t✐❧❡ ♦❢ ❋L,▼ ❛♥❞ ❜② ❢L,▼ ✐ts

❞❡♥s✐t②✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ✐s ❣✐✈❡♥ ❜② ✿ ∀❜ ∈ T ✱ ◗(✶)

L,▼ (❜; α, L) =

I{▲(❜) ≤ ▼} ▲(❜)

❥ =✶

  • α − I{❜❥ ≤ ❋ −✶

L,▼ (α)}

  • E❆[τ❆I{τ❆ ≤ ▼}]❢L,▼ (❋ −✶

L,▼ (α))

. ❉❡✜♥❡ ❛ r♦❜✉st ▲✲st❛t✐st✐❝s ❜② ❚❏,α,▼ (L) = ✶

✵

❏(✉)✇(❋ −✶

L,▼ (✉))❞✉,

✭✺✮ ✇❤❡r❡ ❏ : (✵, ✶) → R ❛ ❝♦♥t✐♥✉♦✉s s❝♦r❡ ❢✉♥❝t✐♦♥ ❜♦✉♥❞❡❞ ❜② ▼❏ ❛♥❞ ✇ : R → R ✐s ❛ ❝♦♥t✐♥✉♦✉s❧② ❞✐✛❡r❡♥t✐❛❜❧❡ ❜♦✉♥❞❡❞ ❜② ▲✇✳ ❚❤❡♥ |❚❏,α,▼ (L) − ❚❏,α,∞(L)| ≤ ✷▼❏▲✇ E❆[τ❆I{τ❆ ≤ ▼}]P(τ❆ ≥ ▼)✶/✷(E[τ❆])✶/✷(E❆[|

τ❆

  • ✐=✶

❳✐|])✶/✷.

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✺ ✴ ✸✽

slide-36
SLIDE 36

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s ❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s

❇♦♦tstr❛♣ ❛♥❞ ❆s②♠♣t✳ ♥♦r♠❛❧✐t② ♦❢ r♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s

❚❤❡♦r❡♠ ❚❤❡ ❢✉♥❝t✐♦♥❛❧ ❚❏,α,▼ (L) ✐s ❋ré❝❤❡t ❞✐✛❡r❡♥t✐❛❜❧❡ ❛t L ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❞❇▲ ♥♦r♠ ❛♥❞ ✇❡ ❤❛✈❡ ♥✶/✷(❚❏,α,▼ (L♥)−❚❏,α,▼ (L)) ⇒ N(✵, E❆[τ❆]❱❛r❆(❚ (✶)

❏,α,▼ (B✶, L)) ❛s ♥ → +∞

✇✐t❤ ❱❛r❆(❚ (✶)

❏,α,▼ (B✶, L)) = ❱❛r❆

∞

−∞

❏(❋L,▼ (✈))❋ (✶)

▼ (❜; ✈, L)✇ (✶)(✈)❞✈

  • .

■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ✭❆✮❘❇❇ ❛♣♣❧✐❡❞ t♦ t❤❡ r♦❜✉st ▲✲st❛t✐st✐❝ ❚❏,α,▼ (L♥) ✐s ❛s②♠♣t♦t✐❝❛❧❧② ✈❛❧✐❞✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✻ ✴ ✸✽

slide-37
SLIDE 37

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s ❘♦❜✉st✐✜❡❞ ▲✲st❛t✐st✐❝s

−4 −2 2 4 0.0 0.1 0.2 0.3 0.4 True and Bootstrap dist., Robust Winsorized Mean in a MM1 0.8 0.6

❋✐❣✉r❡✿ ❈♦♠♣❛r✐s♦♥ ♦❢ t❤❡ tr✉❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ r♦❜✉st✐✜❡❞ ✇✐♥③♦r✐③❡❞ ♠❡❛♥ ✐♥ ❛ ▼▼✶ ♠♦❞❡❧✱ ✇✐t❤ ♣❛r❛♠❡t❡r ✵✳✽ ✵✳✻ ■♥♥♦✈❛t✐✈❡ ♦✉t❧✐❡r ❢♦r ✷✵ ♣❡r✐♦❞s✱ ♥❂✺✵✵

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✼ ✴ ✸✽

slide-38
SLIDE 38

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s ✿ r♦❜✉st✐✜❡❞ ▲✲st❛t✐t✐❝s ❛♥❞ ❘✲st❛t✐st✐❝s ▲✐♥❡❛r ❘❛♥❦ s✐❣♥❡❞ st❛t✐st✐❝s ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s

❘✲♣❛r❛♠❡t❡r ❛♥❞ t❤❡✐r ❡♠♣✐r✐❝❛❧ ❝♦✉♥t❡r♣❛rt

❙t❛t✐st✐❝s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❢✉♥❝t✐♦♥❛❧ ❚φ(L) = ✶ EL [▲(B)]EL  

▲(B)

  • ✐=✶

φ(❋ +

µ (|❳✐|))s✐❣♥(❳✐)

  ✇❤❡r❡ ❋ +

µ (①) = Pµ(|❳✶| ≤ ①) ❛♥❞ t❤❡ s❝♦r❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥ φ ✐s ♦❢

❝❧❛ss C✶✳ ❍♦rr✐❜❧❡ ❢♦r♠✉❧❛s ❜✉t ✐t ✇♦r❦s✳✳✳ ◆♦t ♣♦ss✐❜❧❡ t♦ ❝♦♥tr♦❧ t❤❡ r❡♠❛✐♥❞❡r ✇✐t❤ t❤❡ ❇♦✉♥❞❡❞ ▲✐♣s❝❤✐t③ ❞✐st❛♥❝❡✳ ❈♦♠♣✉t❡ t❤❡ ✐♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡♥ ❝♦♥tr♦❧ ❞✐r❡❝t❧② t❤❡ r❡♠❛✐♥❞❡r ✇✐t❤ ❯✲st❛t✐st✐❝s ❛r❣✉♠❡♥ts✳ ❚❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❣❛✉ss✐❛♥ ❜✉t ♥♦t ♣✐✈♦t❛❧ ❞✉❡ t♦ t❤❡ ❞❡♣❡♥❞❛♥❝❡ str✉❝t✉r❡✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ❘❇❇ ❛♣♣❧✐❡❞ t♦ t❤❡ r❡❣❡♥❡r❛t✐✈❡ ❧✐♥❡❛r ❘✲st❛t✐st✐❝ ✐s ❛s②♠♣t♦t✐❝❛❧❧② ✈❛❧✐❞✳

P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✽ ✴ ✸✽

slide-39
SLIDE 39

❆ ❢❡✇ r❡❢❡r❡♥❝❡s

❆ ❢❡✇ ❜✐❜❧✐♦❣r❛♣❤✐❝❛❧ r❡❢❡r❡♥❝❡s

❙✳ ❆s♠✉ss❡♥ ❆♣♣❧✐❡❞ ♣r♦❜❛❜✐❧✐t② ❛♥❞ q✉❡✉❡s✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✷✵✵✸✳ P✳ ❇❡rt❛✐❧ ❛♥❞ ❙✳ ❈❧é♠❡♥ç♦♥✱ ❊❞❣❡✇♦rt❤ ❡①♣❛♥s✐♦♥s ❢♦r s✉✐t❛❜❧② ♥♦r♠❛❧✐③❡❞ s❛♠♣❧❡ ♠❡❛♥ st❛t✐st✐❝s ♦❢ ❛t♦♠✐❝ ▼❛r❦♦✈ ❝❤❛✐♥s✱ Pr♦❜✳ ❚❤✳ ❘❡❧✳ ❋✐❡❧❞s ✶✸✵ ✭✷✵✵✹✮✱ ♥♦✳ ✸✱ ✸✽✽✕✹✶✹✳ P✳ ❇❡rt❛✐❧ ❛♥❞ ❙✳ ❈❧é♠❡♥ç♦♥✱ ❘❡❣❡♥❡r❛t✐✈❡✲❜❧♦❝❦ ❜♦♦tstr❛♣ ❢♦r ▼❛r❦♦✈ ❝❤❛✐♥s✱ ❇❡r♥♦✉❧❧✐ ✶✷ ✭✷✵✵✺✮✱ ♥♦✳ ✹✳ P✳ ❇❡rt❛✐❧ ❛♥❞ ❙✳ ❈❧é♠❡♥ç♦♥✱ ❘❡❣❡♥❡r❛t✐♦♥✲❜❛s❡❞ st❛t✐st✐❝s ❢♦r ❍❛rr✐s r❡❝✉rr❡♥t ▼❛r❦♦✈ ❝❤❛✐♥s✱ Pr♦❜❛❜✐❧✐t② ❛♥❞ ❙t❛t✐st✐❝s ❢♦r ❞❡♣❡♥❞❡♥t ❞❛t❛ ✭P✳ ❇❡rt❛✐❧✱ P✳ ❉♦✉❦❤❛♥✱ ❛♥❞ P✳ ❙♦✉❧✐❡r✱ ❡❞s✳✮✱ ▲❡❝t✉r❡ ♥♦t❡s ✐♥ ❙t❛t✐st✐❝s✱ ✈♦❧✳ ✶✽✼✱ ❙♣r✐♥❣❡r✱ ✷✵✵✻✱ ♣♣✳ ✸✕✺✹✳ P✳ ❇❡rt❛✐❧ ❛♥❞ ❙✳ ❈❧é♠❡♥ç♦♥✳ ❙❤❛r♣ ❜♦✉♥❞s ❢♦r t❤❡ t❛✐❧s ♦❢ ❢✉♥❝t✐♦♥❛❧s ♦❢ ▼❛r❦♦✈ ❝❤❛✐♥s✳ ❚❤✳ Pr♦❜✳ ❆♣♣❧✳✱ ✺✹✭✸✮ ✿✺✵✺✕✺✶✺✱ ✷✵✶✵✳ P✳ ❇❡rt❛✐❧ ❛♥❞ ❙✳ ❈❧é♠❡♥ç♦♥✳ ❆ r❡♥❡✇❛❧❛♣♣r♦❛❝❤ t♦ ❯✲st❛t✐st✐❝s ❢♦r ▼❛r❦♦✈✐❛♥ ❞❛t❛✳▼❛t❤❡♠❛t✐❝❛❧ ▼❡t❤♦❞s ♦❢ ❙t❛t✐st✐❝s✱ ✷✵✭✷✮ ✿✼✾✕✶✶✺✱ ✷✵✶✶✳ ❘✳❉✳ ▼❛rt✐♥ ❛♥❞ ❱✳❏✳ ❨♦❤❛✐✳ ■♥✢✉❡♥❝❡ ❢✉♥❝t✐♦♥❛❧s ❢♦r t✐♠❡ s❡r✐❡s✳ ❆♥♥✳ ❙t❛t✳✱ ✶✹ ✿✼✽✶✕✽✶✽✱ ✶✾✽✻✳ ❙✳P✳ ▼❡②♥ ❛♥❞ ❘✳▲✳ ❚✇❡❡❞✐❡✱ ▼❛r❦♦✈ ❝❤❛✐♥s ❛♥❞ st♦❝❤❛st✐❝ st❛❜✐❧✐t②✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ✶✾✾✻✳ ❊✳ ◆✉♠♠❡❧✐♥✱ ❆ s♣❧✐tt✐♥❣ t❡❝❤♥✐q✉❡ ❢♦r ❍❛rr✐s r❡❝✉rr❡♥t ❝❤❛✐♥s✳✱ ❩✳ ❲❛❤rs❝❤✳ ❱❡r✇✳ ●❡❜✐❡t❡ ✹✸ ✭✶✾✼✽✮✱ ✸✵✾✕✸✶✽✳ ❍✳ ❘✐❡❞❡r✳ ❘♦❜✉st ❛s②♠♣t♦t✐❝ st❛t✐st✐❝s✳ ✶✾✾✹✳ ❘✳ ❙❡r✢✐♥❣✳ ❆♣♣r♦①✐♠❛t✐♦♥ ❚❤❡♦r❡♠s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❙t❛t✐st✐❝s✳ ✷✵✵✷✳ P❛tr✐❝❡ ❇❡rt❛✐❧ ✭▼❖❉❆▲✬❳✮ ✶❡r ❢é✈r✐❡r ✷✵✶✼ ✸✾ ✴ ✸✽