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✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❋❛❝✉❧t② ♦❢ ❍❡❛❧t❤ ❙❝✐❡♥❝❡s

❘✷✲t②♣❡ ❈✉r✈❡s ❢♦r ❉②♥❛♠✐❝ Pr❡❞✐❝t✐♦♥s ❢r♦♠ ❏♦✐♥t ▲♦♥❣✐t✉❞✐♥❛❧✲❙✉r✈✐✈❛❧ ▼♦❞❡❧s

■♥❢❡r❡♥❝❡ ✫ ❛♣♣❧✐❝❛t✐♦♥ t♦ ♣r❡❞✐❝t✐♦♥ ♦❢ ❦✐❞♥❡② ❣r❛❢t ❢❛✐❧✉r❡

P❛✉❧ ❇❧❛♥❝❤❡

❥♦✐♥t ✇♦r❦ ✇✐t❤

▼✲❈✳ ❋♦✉r♥✐❡r ✫ ❊✳ ❉❛♥t❛♥ ✭◆❛♥t❡s✱ ❋r❛♥❝❡✮

❏✉❧② ✷✵✶✺ ❙❧✐❞❡ ✶✴✷✾

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♥t❡①t ✫ ▼♦t✐✈❛t✐♦♥

• ▼❡❞✐❝❛❧ r❡s❡❛r❝❤❡rs ❤♦♣❡ t♦ ✐♠♣r♦✈❡ ♣❛t✐❡♥t ♠❛♥❛❣❡♠❡♥t

✉s✐♥❣ ❡❛r❧✐❡r ❞✐❛❣♥♦s❡s

• ❙t❛t✐st✐❝✐❛♥s ❝❛♥ ❤❡❧♣ ❜② ✜tt✐♥❣ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧s
• ❚❤❡ ♠❛❦✐♥❣ ♦❢ s♦✲❝❛❧❧❡❞ ✧❞②♥❛♠✐❝✧ ♣r❡❞✐❝t✐♦♥s ❤❛s r❡❝❡♥t❧②

r❡❝❡✐✈❡❞ ❛ ❧♦t ♦❢ ❛tt❡♥t✐♦♥

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

❜❡ ✧❛❝❝✉r❛t❡✧ ❍♦✇ s❤♦✉❧❞ ✇❡ ❡✈❛❧✉❛t❡ ❞②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥ ❛❝❝✉r❛❝②❄

❙❧✐❞❡ ✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉❛t❛ ✫ ❈❧✐♥✐❝❛❧ ❣♦❛❧

◮ ❉❛t❛✿ ❉■❱❆❚ ❝♦❤♦rt ❞❛t❛ ♦❢ ❦✐❞♥❡② tr❛♥s♣❧❛♥t r❡❝✐♣✐❡♥ts

✭s✉❜s❛♠♣❧❡✱ n = ✹, ✶✶✾✮

◮ ❈❧✐♥✐❝❛❧ ❣♦❛❧✿

• ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥ ♦❢ r✐s❦ ♦❢ ❦✐❞♥❡②

❣r❛❢t ❢❛✐❧✉r❡ ✭❞❡❛t❤ ♦r r❡t✉r♥ t♦ ❞✐❛❧②s✐s✮

• ❯s✐♥❣ r❡♣❡❛t❡❞ ♠❡❛s✉r❡♠❡♥ts ♦❢

s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ ❞❛t❛ s❛♠♣❧❡ ✭♥❂✹✱✶✶✾✮

• ❋r❡♥❝❤ ❝♦❤♦rt
• ❆❞✉❧t r❡❝✐♣✐❡♥ts
• ❚r❛♥s♣❧❛♥t❡❞ ❛❢t❡r ✷✵✵✵
• ❈r❡❛t✐♥✐♥❡ ♠❡❛s✉r❡❞ ❡✈❡r② ②❡❛r

✻ ❝❡♥t❡rs ✭✇✇✇✳❞✐✈❛t✳❢r✮

❙❧✐❞❡ ✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ ❞❛t❛ s❛♠♣❧❡ ✭♥❂✹✱✶✶✾✮

• ❋r❡♥❝❤ ❝♦❤♦rt
• ❆❞✉❧t r❡❝✐♣✐❡♥ts
• ❚r❛♥s♣❧❛♥t❡❞ ❛❢t❡r ✷✵✵✵
• ❈r❡❛t✐♥✐♥❡ ♠❡❛s✉r❡❞ ❡✈❡r② ②❡❛r
• ✻ ❝❡♥t❡rs

✭✇✇✇✳❞✐✈❛t✳❢r✮

❙❧✐❞❡ ✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙t❛t✐st✐❝❛❧ ❝❤❛❧❧❡♥❣❡s ❞✐s❝✉ss❡❞

❍♦✇ t♦ ❡✈❛❧✉❛t❡ ❛♥❞✴♦r ❝♦♠♣❛r❡ ❞②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s❄ ◮ ❯s✐♥❣ ❝♦♥❝❡♣ts ♦❢✿

• ❉✐s❝r✐♠✐♥❛t✐♦♥
• ❈❛❧✐❜r❛t✐♦♥

◮ ❆❝❝♦✉♥t✐♥❣ ❢♦r✿

• ❉②♥❛♠✐❝ s❡tt✐♥❣
• ❈❡♥s♦r✐♥❣ ✐ss✉❡

❙❧✐❞❡ ✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s ❈♦♥❝❡♣ts✿

❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

◮ ❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ◮ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

◮ ❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ◮ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❇❛s✐❝ ✐❞❡❛ ✫ ❝♦♥❝❡♣ts ❢♦r ❡✈❛❧✉❛t✐♥❣ ♣r❡❞✐❝t✐♦♥s

❇❛s✐❝ ✐❞❡❛✿ ❝♦♠♣❛r✐♥❣ ♣r❡❞✐❝t✐♦♥s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s (simple!) ❈♦♥❝❡♣ts✿

◮ ❉✐s❝r✐♠✐♥❛t✐♦♥✿ ❆ ♠♦❞❡❧ ❤❛s ❤✐❣❤ ❞✐s❝r✐♠✐♥❛t✐✈❡ ♣♦✇❡r ✐❢ t❤❡ r❛♥❣❡ ♦❢ ♣r❡❞✐❝t❡❞ r✐s❦s ✐s ✇✐❞❡ ❛♥❞ s✉❜❥❡❝ts ✇✐t❤ ❧♦✇ ✭❤✐❣❤✮ ♣r❡❞✐❝t❡❞ r✐s❦ ❛r❡ ♠♦r❡ ✭❧❡ss✮ ❧✐❦❡❧② t♦ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t✳ ◮ ❈❛❧✐❜r❛t✐♦♥✿ ❆ ♠♦❞❡❧ ✐s ❝❛❧✐❜r❛t❡❞ ✐❢ ✇❡ ❝❛♥ ❡①♣❡❝t t❤❛t ① s✉❜❥❡❝ts ♦✉t ♦❢ ✶✵✵ ❡①♣❡r✐❡♥❝❡ t❤❡ ❡✈❡♥t ❛♠♦♥❣ ❛❧❧ s✉❜❥❡❝ts t❤❛t r❡❝❡✐✈❡ ❛ ♣r❡❞✐❝t❡❞ r✐s❦ ♦❢ ①✪ ✭✏✇❡❛❦✑ ❞❡✜♥✐t✐♦♥✮✳

❙❧✐❞❡ ✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮ ✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

• s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮

✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years s = 4 years Landmark time ''s''

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

• s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮

✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years

• s = 4 years

Landmark time ''s''

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

• s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮
• t✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years

• s = 4 years

s+t = 9 years Landmark time ''s'' Horizon ''t'' Horizon ''t''

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥

• s✿ ▲❛♥❞♠❛r❦ t✐♠❡ ❛t ✇❤✐❝❤ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡ ✭✈❛r✐❡s✮
• t✿ ♣r❡❞✐❝t✐♦♥ ❤♦r✐③♦♥ ✭✜①❡❞✮

follow−up time (years) Creatinine (mol/L) 100 150 200 250 300 s = 4 years s+t = 9 years

• 0 %

100 % 1−Pr(Graft Failure in (s,s+t]) Landmark time ''s'' Horizon ''t'' 1 − πi(s, t) = 64%

❙❧✐❞❡ ✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t ❝❡♥s♦r❡❞ ✇✐t❤✐♥ t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t ❝❡♥s♦r❡❞ ✇✐t❤✐♥ t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t ❝❡♥s♦r❡❞ ✇✐t❤✐♥ t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❘✐❣❤t ❝❡♥s♦r✐♥❣ ✐ss✉❡

▲❛♥❞♠❛r❦ t✐♠❡ s

✭✇❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ♠❛❞❡✮

❚✐♠❡ s + t

✭❡♥❞ ♦❢ ♣r❡❞✐❝t✐♦♥ ✇✐♥❞♦✇✮

t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❋♦r s✉❜❥❡❝t i ❝❡♥s♦r❡❞ ✇✐t❤✐♥ (s, s + t] t❤❡ st❛t✉s Di(s, t) = ✶ ✶{❡✈❡♥t ♦❝❝✉rs ✐♥ (s, s + t]} ✐s ✉♥❦♥♦✇♥✳

❙❧✐❞❡ ✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ ✶ ❳

✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) ✶ ❳

✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

• ✶

❳

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

• Di(s, t) = ✶
• ❳

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮ ✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ ❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

• Di(s, t) = ✶
• Ti > s, Yi(s),

❳

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮

• Yi(s)✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ s

❳ ✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

◆♦t❛t✐♦♥s ❢♦r ♣♦♣✉❧❛t✐♦♥ ♣❛r❛♠❡t❡rs

◮ ■♥❞✐❝❛t♦r ♦❢ ❡✈❡♥t ✐♥ (s, s + t]✿ Di(s, t) = ✶ ✶{s < Ti ≤ s + t}

✇❤❡r❡ Ti ✐s t❤❡ t✐♠❡✲t♦✲❡✈❡♥t✳

◮ ❉②♥❛♠✐❝ ♣r❡❞✐❝t✐♦♥s✿ πi(s, t) = P

ξ

• Di(s, t) = ✶
• Ti > s, Yi(s), ❳i

ξ✿ ♣r❡✈✐♦✉s❧② ❡st✐♠❛t❡❞ ♣❛r❛♠❡t❡rs ✭❢r♦♠ ✐♥❞❡♣❡♥❞❡♥t tr❛✐♥✐♥❣ ❞❛t❛✮

• Yi(s)✿ ♠❛r❦❡r ♠❡❛s✉r❡♠❡♥ts ♦❜s❡r✈❡❞ ❜❡❢♦r❡ t✐♠❡ s
• ❳i✿ ❜❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s

❙❧✐❞❡ ✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

Pr❡❞✐❝t✐✈❡ ❛❝❝✉r❛❝②

❍♦✇ ❝❧♦s❡ ❛r❡ t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s πi(s, t) t♦ t❤❡ ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦ P

• ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
• ✐♥❢♦r♠❛t✐♦♥ ❛t s
• ❄

Pr❡❞✐❝t✐♦♥ ❊rr♦r✿ P❊

✷

t❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r P❊ ❇✐❛s✷ ✰ ❱❛r✐❛♥❝❡ ❡✈❛❧✉❛t❡s ❜♦t❤ ❈❛❧✐❜r❛t✐♦♥ ❛♥❞ ❉✐s❝r✐♠✐♥❛t✐♦♥ ❞❡♣❡♥❞s ♦♥

❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪ ❛t r✐s❦ ❛t s

♦❢t❡♥ ❝❛❧❧❡❞ ✧❊①♣❡❝t❡❞ ❇r✐❡r ❙❝♦r❡✧

❙❧✐❞❡ ✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

Pr❡❞✐❝t✐✈❡ ❛❝❝✉r❛❝②

❍♦✇ ❝❧♦s❡ ❛r❡ t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s πi(s, t) t♦ t❤❡ ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦ P

• ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
• ✐♥❢♦r♠❛t✐♦♥ ❛t s
• ❄

◮ Pr❡❞✐❝t✐♦♥ ❊rr♦r✿ P❊π(s, t) = E

• D(s, t) − π(s, t)

✷

• T > s
• t❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r

P❊ ❇✐❛s✷ ✰ ❱❛r✐❛♥❝❡ ❡✈❛❧✉❛t❡s ❜♦t❤ ❈❛❧✐❜r❛t✐♦♥ ❛♥❞ ❉✐s❝r✐♠✐♥❛t✐♦♥ ❞❡♣❡♥❞s ♦♥

❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪ ❛t r✐s❦ ❛t s

♦❢t❡♥ ❝❛❧❧❡❞ ✧❊①♣❡❝t❡❞ ❇r✐❡r ❙❝♦r❡✧

❙❧✐❞❡ ✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

Pr❡❞✐❝t✐✈❡ ❛❝❝✉r❛❝②

❍♦✇ ❝❧♦s❡ ❛r❡ t❤❡ ♣r❡❞✐❝t❡❞ r✐s❦s πi(s, t) t♦ t❤❡ ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦ P

• ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
• ✐♥❢♦r♠❛t✐♦♥ ❛t s
• ❄

◮ Pr❡❞✐❝t✐♦♥ ❊rr♦r✿ P❊π(s, t) = E

• D(s, t) − π(s, t)

✷

• T > s
• t❤❡ ❧♦✇❡r t❤❡ ❜❡tt❡r
• P❊ ≈ ❇✐❛s✷ ✰ ❱❛r✐❛♥❝❡
• ❡✈❛❧✉❛t❡s ❜♦t❤ ❈❛❧✐❜r❛t✐♦♥ ❛♥❞ ❉✐s❝r✐♠✐♥❛t✐♦♥
• ❞❡♣❡♥❞s ♦♥ P
• ❡✈❡♥t ♦❝❝✉rs ✐♥ ✭s✱s✰t❪
• ❛t r✐s❦ ❛t s
• ♦❢t❡♥ ❝❛❧❧❡❞ ✧❊①♣❡❝t❡❞ ❇r✐❡r ❙❝♦r❡✧

❙❧✐❞❡ ✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) =E

• D(s, t) − π(s, t)

✷

• T > s
• E
• D(s, t)+ E
• D(s, t)
• Hπ(s)
• ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦

−π(s, t)

• T > s
• ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t

t✐♠❡ ✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) =E

• D(s, t) − E
• D(s, t)
• Hπ(s)

✷ E

• D(s, t) + E
• D(s, t)
• Hπ(s)
• ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦

− π(s, t) ✷

• T > s
• ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t

t✐♠❡ ✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) =E

• D(s, t) − E
• D(s, t)
• Hπ(s)

✷

• T > s
• + E
• E
• D(s, t)
• Hπ(s)
• ✏tr✉❡ ✉♥❞❡r❧②✐♥❣✑ r✐s❦

− π(s, t) ✷

• T > s
• Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t

t✐♠❡ s✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

• D(s, t) − E
• D(s, t)
• Hπ(s)

✷

• T > s
• ■♥s❡♣❛r❛❜✐❧✐t②

+ E

• E
• D(s, t)
• Hπ(s)
• − π(s, t)

✷

• T > s
• ❇✐❛s✴❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

• ❱❛r
• D(s, t)
• Hπ(s)
• T > s
• ❉✐s❝r✐♠✐♥❛t✐♦♥

+ E

• E
• D(s, t)
• Hπ(s)
• − π(s, t)

✷

• T > s
• ❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ t❤❡ s♠❛❧❧❡r ❱❛r ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

• ❱❛r
• D(s, t)
• Hπ(s)
• T > s
• ❉✐s❝r✐♠✐♥❛t✐♦♥

+ E

• E
• D(s, t)
• Hπ(s)
• − π(s, t)

✷

• T > s
• ❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ ◮ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ Hπ(s) t❤❡ s♠❛❧❧❡r ❱❛r

• D(s, t)
• Hπ(s)
• ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

• ❱❛r
• D(s, t)
• Hπ(s)
• T > s
• ❉✐s❝r✐♠✐♥❛t✐♦♥

+ E

• E
• D(s, t)
• Hπ(s)
• − π(s, t)

✷

• T > s
• ❈❛❧✐❜r❛t✐♦♥

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ ◮ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ Hπ(s) t❤❡ s♠❛❧❧❡r ❱❛r

• D(s, t)
• Hπ(s)
• ◮ E
• D(s, t)
• Hπ(s)
• − π(s, t) ≡ ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❍♦✇ ❞♦❡s t❤❡ P❊ r❡❧❛t❡ t♦ ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♦♥❄

P❊π(s, t) = E

• ❱❛r
• D(s, t)
• Hπ(s)
• T > s
• ❉♦❡s ◆❖❚ ❞❡♣❡♥❞ ♦♥ π(s,t)

+ E

• E
• D(s, t)
• Hπ(s)
• − π(s, t)

✷

• T > s
• ❉❡♣❡♥❞s ♦♥ π(s,t)

Hπ(s) = {X π(s), T > s} ❞❡♥♦t❡s t❤❡ s✉❜❥❡❝t✲s♣❡❝✐✜❝ ❤✐st♦r② ❛t t✐♠❡ s✳ ◮ t❤❡ ♠♦r❡ ❞✐s❝r✐♠✐♥❛t✐♥❣ Hπ(s) t❤❡ s♠❛❧❧❡r ❱❛r

• D(s, t)
• Hπ(s)
• ◮ E
• D(s, t)
• Hπ(s)
• − π(s, t) ≡ ✵ ❞❡✜♥❡s ✧str♦♥❣✧ ❝❛❧✐❜r❛t✐♦♥✳

❙❧✐❞❡ ✶✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π✲t②♣❡ ❝r✐t❡r✐♦♥ ◮ ❇❡♥❝❤♠❛r❦ PE✵

❚❤❡ ❜❡st ✧♥✉❧❧✧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ✭♠❛r❣✐♥❛❧✮ ♣r❡❞✐❝t❡❞ r✐s❦ S(s + t|s) = E

• D(s, t)
• H✵(s)
• ,

H✵(s) = {T > s} t♦ ❛❧❧ s✉❜❥❡❝ts ❧❡❛❞s t♦

PE✵(s, t) = ❱❛r{D(s, t)|T > s} = S(s + t|s)

• ✶ − S(s + t|s)
• .

❙✐♠♣❧❡ ✐❞❡❛

✷

✶ P❊ P❊✵

❙❧✐❞❡ ✶✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π✲t②♣❡ ❝r✐t❡r✐♦♥ ◮ ❇❡♥❝❤♠❛r❦ PE✵

❚❤❡ ❜❡st ✧♥✉❧❧✧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ✭♠❛r❣✐♥❛❧✮ ♣r❡❞✐❝t❡❞ r✐s❦ S(s + t|s) = E

• D(s, t)
• H✵(s)
• ,

H✵(s) = {T > s} t♦ ❛❧❧ s✉❜❥❡❝ts ❧❡❛❞s t♦

PE✵(s, t) = ❱❛r{D(s, t)|T > s} = S(s + t|s)

• ✶ − S(s + t|s)
• .

◮ ❙✐♠♣❧❡ ✐❞❡❛ R✷

π(s, t) = ✶ − P❊π(s, t)

P❊✵(s, t)

❙❧✐❞❡ ✶✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❲❤② ❜♦t❤❡r❄

◮ R✷

π(s, t) ❛✐♠s t♦ ❝✐r❝✉♠✈❡♥t t❤❡ ❞✐✣❝✉❧t ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢✿

• t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ PE(s, t) ✐s ♠❡❛s✉r❡❞
• ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r tr❡♥❞ ♦❢ PE(s, t) ✈s s

❇❡❝❛✉s❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ P❊✭s✱t✮ ✐s ♠❡❛s✉r❡❞ ❝❤❛♥❣❡s ✇✐t❤ s✱ ❛♥ ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ tr❡♥❞ ❝❛♥ ❜❡ ❞✉❡ t♦ ❝❤❛♥❣❡s ✐♥✿ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s ❛♥❞✴♦r t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❙❧✐❞❡ ✶✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❲❤② ❜♦t❤❡r❄

◮ R✷

π(s, t) ❛✐♠s t♦ ❝✐r❝✉♠✈❡♥t t❤❡ ❞✐✣❝✉❧t ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢✿

• t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ PE(s, t) ✐s ♠❡❛s✉r❡❞
• ✐♥t❡r♣r❡t❛t✐♦♥ ❢♦r tr❡♥❞ ♦❢ PE(s, t) ✈s s

◮ ❇❡❝❛✉s❡ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s❝❛❧❡ ♦♥ ✇❤✐❝❤ P❊✭s✱t✮ ✐s ♠❡❛s✉r❡❞ ❝❤❛♥❣❡s ✇✐t❤ s✱ ❛♥ ✐♥❝r❡❛s✐♥❣✴❞❡❝r❡❛s✐♥❣ tr❡♥❞ ❝❛♥ ❜❡ ❞✉❡ t♦ ❝❤❛♥❣❡s ✐♥✿

• t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s

❛♥❞✴♦r • t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❙❧✐❞❡ ✶✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s

✧❊ss❡♥t✐❛❧❧②✱ ❛❧❧ ♠♦❞❡❧s ❛r❡ ✇r♦♥❣✱ ❜✉t s♦♠❡ ❛r❡ ✉s❡❢✉❧✳✧✱ ●✳ ❇♦① ❚❤❡ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ ♣r❡❞✐❝t✐♦♥s ❝❛♥ ❜❡ ✏♠♦r❡ ✇r♦♥❣✑ ❢♦r s♦♠❡ t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳ ❈❛❧✐❜r❛t✐♦♥ t❡r♠ ♦❢ ❝❤❛♥❣❡s ✇✐t❤ ❲❡ ❝❛♥ ✇♦r❦ ♦♥ ✐t✦

❙❧✐❞❡ ✶✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ q✉❛❧✐t② ♦❢ t❤❡ ♣r❡❞✐❝t✐♦♥s

✧❊ss❡♥t✐❛❧❧②✱ ❛❧❧ ♠♦❞❡❧s ❛r❡ ✇r♦♥❣✱ ❜✉t s♦♠❡ ❛r❡ ✉s❡❢✉❧✳✧✱ ●✳ ❇♦① ◮ ❚❤❡ ♣r❡❞✐❝t✐♦♥ ♠♦❞❡❧ ❢r♦♠ ✇❤✐❝❤ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ t❤❡ ♣r❡❞✐❝t✐♦♥s ❝❛♥ ❜❡ ✏♠♦r❡ ✇r♦♥❣✑ ❢♦r s♦♠❡ s t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳

• ❈❛❧✐❜r❛t✐♦♥ t❡r♠ ♦❢ PE(s, t) ❝❤❛♥❣❡s ✇✐t❤ s
• ❲❡ ❝❛♥ ✇♦r❦ ♦♥ ✐t✦

❙❧✐❞❡ ✶✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❆♥ ❡①❛♠♣❧❡✿

• P❛t✐❡♥ts ✇✐t❤ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ✭❈❱✮ ❛❧❧ ❞✐❡ ❡❛r❧②✳
• ❖♥❧② t❤♦s❡ ✇✐t❤♦✉t ❈❱ r❡♠❛✐♥ ❛t r✐s❦ ❢♦r ❧❛t❡ s✳
• ❚❤❡♥✿
• t❤❡ ❡❛r❧✐❡r s t❤❡ ♠♦r❡ ❤♦♠♦❣❡♥❡♦✉s t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥
• ❈❱ ✐s ✉s❡❢✉❧ ❢♦r ♣r❡❞✐❝t✐♦♥ ❢♦r ❡❛r❧② s ❜✉t ✉s❡❧❡ss ❢♦r ❧❛t❡ s✳

❚❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ♠♦r❡ ✐♥❢♦r♠❛t✐✈❡ ❢♦r s♦♠❡ s t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳ ❉✐s❝r✐♠✐♥❛t✐♥❣ t❡r♠ ♦❢ ❝❤❛♥❣❡s ✇✐t❤ ❚❤✐s ✐s ❥✉st ❤♦✇ ✐t ✐s✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇❡ ❝❛♥ ❞♦✦

✭✇❡ ❝❛♥ ♦♥❧② ✇♦r❦ ✇✐t❤ t❤❡ ❞❛t❛ ✇❡ ❤❛✈❡✮

❙❧✐❞❡ ✶✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❤❛♥❣❡s ✐♥ t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥

❆♥ ❡①❛♠♣❧❡✿

• P❛t✐❡♥ts ✇✐t❤ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ✭❈❱✮ ❛❧❧ ❞✐❡ ❡❛r❧②✳
• ❖♥❧② t❤♦s❡ ✇✐t❤♦✉t ❈❱ r❡♠❛✐♥ ❛t r✐s❦ ❢♦r ❧❛t❡ s✳
• ❚❤❡♥✿
• t❤❡ ❡❛r❧✐❡r s t❤❡ ♠♦r❡ ❤♦♠♦❣❡♥❡♦✉s t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥
• ❈❱ ✐s ✉s❡❢✉❧ ❢♦r ♣r❡❞✐❝t✐♦♥ ❢♦r ❡❛r❧② s ❜✉t ✉s❡❧❡ss ❢♦r ❧❛t❡ s✳

◮ ❚❤❡ ❛✈❛✐❧❛❜❧❡ ✐♥❢♦r♠❛t✐♦♥ ❝❛♥ ❜❡ ♠♦r❡ ✐♥❢♦r♠❛t✐✈❡ ❢♦r s♦♠❡ s t❤❛♥ ❢♦r s♦♠❡ ♦t❤❡rs✳

• ❉✐s❝r✐♠✐♥❛t✐♥❣ t❡r♠ ♦❢ PE(s, t) ❝❤❛♥❣❡s ✇✐t❤ s
• ❚❤✐s ✐s ❥✉st ❤♦✇ ✐t ✐s✱ t❤❡r❡ ✐s ♥♦t❤✐♥❣ ✇❡ ❝❛♥ ❞♦✦

✭✇❡ ❝❛♥ ♦♥❧② ✇♦r❦ ✇✐t❤ t❤❡ ❞❛t❛ ✇❡ ❤❛✈❡✮

❙❧✐❞❡ ✶✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿

✷

❱❛r ❱❛r ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ ❈♦rr✷ ❝♦rr❡❧❛t✐♦♥ ✶ ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡ ✵

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

◮ ❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿ R✷

π(s, t) = ❱❛r{π(s, t)|T > s}

❱❛r{D(s, t)|T > s} ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ ❈♦rr✷ ❝♦rr❡❧❛t✐♦♥ ✶ ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡ ✵

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

◮ ❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿ R✷

π(s, t) = ❱❛r{π(s, t)|T > s}

❱❛r{D(s, t)|T > s} ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ = ❈♦rr✷ D(s, t), π(s, t)

• T > s
• ❝♦rr❡❧❛t✐♦♥

✶ ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡ ✵

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✐♥t❡r♣r❡t❛t✐♦♥ ◮ ❆❧✇❛②s tr✉❡✿

▼❡❛s✉r❡ ♦❢ ❤♦✇ t❤❡ ♣r❡❞✐❝t✐♦♥ t♦♦❧ π(s, t) ♣❡r❢♦r♠s ❝♦♠♣❛r❡❞ t♦ t❤❡ ❜❡♥❝❤♠❛r❦ ♥✉❧❧ ♣r❡❞✐❝t✐♦♥ t♦♦❧✱ ✇❤✐❝❤ ❣✐✈❡s t❤❡ s❛♠❡ ♣r❡❞✐❝t❡❞ r✐s❦ t♦ ❛❧❧ s✉❜❥❡❝ts ✭♠❛r❣✐♥❛❧ r✐s❦✮✳

◮ ❲❤❡♥ ♣r❡❞✐❝t✐♦♥s ❛r❡ ❝❛❧✐❜r❛t❡❞ ✭str♦♥❣❧②✮✿ R✷

π(s, t) = ❱❛r{π(s, t)|T > s}

❱❛r{D(s, t)|T > s} ❡①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥ = ❈♦rr✷ D(s, t), π(s, t)

• T > s
• ❝♦rr❡❧❛t✐♦♥

= E

• π(s, t)
• D(s, t) = ✶, T > s
• ♠❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡

− E

• π(s, t)
• D(s, t) = ✵, T > s
• .

✭❛❢t❡r ❧✐tt❧❡ ❛❧❣❡❜r❛✮

❙❧✐❞❡ ✶✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

• , i = ✶, . . . , n
• ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ ✿

✶ ✶ ✶ ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✵

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

✶

✶ ✷

❛♥❞

✷

✶

✵

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

• , i = ✶, . . . , n
• ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

• Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✵

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

✶

✶ ✷

❛♥❞

✷

✶

✵

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

• , i = ✶, . . . , n
• ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

• Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✵

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

◮ ■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

• PE π(s, t) = ✶

n

n

• i=✶
• Di(s, t) − πi(s, t)

✷ ❛♥❞

✷

✶

✵

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

• , i = ✶, . . . , n
• ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

• Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✵

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

◮ ■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

• PE π(s, t) = ✶

n

n

• i=✶
• Wi(s, t)
• Di(s, t) − πi(s, t)

✷ ❛♥❞

✷

✶

✵

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❖❜s❡r✈❛t✐♦♥s ✫ ■P❈❲ P❊ ❡st✐♠❛t♦r

◮ ❖❜s❡r✈❛t✐♦♥s ✭✐✳✐✳❞✳✮

Ti, ∆i, πi(·, ·)

• , i = ✶, . . . , n
• ✇❤❡r❡

Ti = Ti ∧ Ci, ∆i = ✶ ✶{Ti ≤ Ci}

◮ ■♥❞✐❝❛t♦r ♦❢ ✏♦❜s❡r✈❡❞ ❡✈❡♥t ♦❝❝✉rr❡♥❝❡✑ ✐♥ (s, s + t]✿

• Di(s, t) = ✶

✶{s < Ti ≤ s + t, ∆i = ✶} =    ✶

✿ ❡✈❡♥t ♦❝❝✉rr❡❞

✵

✿ ❡✈❡♥t ❞✐❞ ♥♦t ♦❝❝✉r ♦r ❝❡♥s♦r❡❞ ♦❜s✳

◮ ■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤t✐♥❣ ✭■P❈❲✮ ❡st✐♠❛t♦r✿

• PE π(s, t) = ✶

n

n

• i=✶
• Wi(s, t)
• Di(s, t) − πi(s, t)

✷ ❛♥❞

• R✷

π(s, t) = ✶ −

• PE π(s, t)
• PE ✵(s, t)

❙❧✐❞❡ ✶✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

• Wi(s, t) =

✶ ✶ + ✶ ✶ + ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

• Wi(s, t) = ✶

✶{s < Ti ≤ s + t}∆i

• G(

Ti|s) + ✶ ✶ + ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

• Wi(s, t) = ✶

✶{s < Ti ≤ s + t}∆i

• G(

Ti|s) + ✶ ✶{ Ti > s + t}

• G(s + t|s)

+ ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

■♥✈❡rs❡ Pr♦❜❛❜✐❧✐t② ♦❢ ❈❡♥s♦r✐♥❣ ❲❡✐❣❤ts

• Wi(s, t) = ✶

✶{s < Ti ≤ s + t}∆i

• G(

Ti|s) + ✶ ✶{ Ti > s + t}

• G(s + t|s)

+ ✵

✇✐t❤ G(u|s) t❤❡ ❑❛♣❧❛♥✲▼❡✐❡r ❡st✐♠❛t♦r ♦❢ P(C > u|C > s)✳ ▲❛♥❞♠❛r❦ t✐♠❡ s ❚✐♠❡ s + t t✐♠❡ ✿ ✉♥❝❡♥s♦r❡❞ ✿ ❝❡♥s♦r❡❞

❙❧✐❞❡ ✶✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❆s②♠♣t♦t✐❝ ✐✳✐✳❞✳ r❡♣r❡s❡♥t❛t✐♦♥

▲❡♠♠❛✿ ❆ss✉♠❡ t❤❛t t❤❡ ❝❡♥s♦r✐♥❣ t✐♠❡ C ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ (T, η, π(·, ·)) ❛♥❞ ❧❡t θ ❞❡♥♦t❡ ❡✐t❤❡r PEπ✱ R✷

π ♦r ❛ ❞✐✛❡r❡♥❝❡ ✐♥ P❊

♦r R✷

π✱ t❤❡♥

√n

• θ(s, t) − θ(s, t)
• =

✶ √n

n

• i=✶

■❋θ( Ti, ∆i, πi(s, t), s, t) + op (✶) ✇❤❡r❡ ■❋θ( Ti, ∆i, πi(s, t), s, t) ❜❡✐♥❣ ✿ ◮ ③❡r♦✲♠❡❛♥ ✐✳✐✳❞✳ t❡r♠s ◮ ❡❛s② t♦ ❡st✐♠❛t❡ ✭✉s✐♥❣ ◆❡❧s♦♥✲❆❛❧❡♥ ✫ ❑❛♣❧❛♥✲▼❡✐❡r✮

❙❧✐❞❡ ✶✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

P♦✐♥t✇✐s❡ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧ ✭✜①❡❞ s✮

• ❆s②♠♣t♦t✐❝ ♥♦r♠❛❧✐t②✿

√n

• θ(s, t) − θ(s, t)
• D

− → N

• ✵, σ✷

s,t

• ✾✺✪ ❝♦♥✜❞❡♥❝❡ ✐♥t❡r✈❛❧✿
• θ(s, t) ± z✶−α/✷
• σs,t

√n

• ✇❤❡r❡ z✶−α/✷ ✐s t❤❡ ✶ − α/✷ q✉❛♥t✐❧❡ ♦❢ N(✵, ✶)✳
• ❱❛r✐❛♥❝❡ ❡st✐♠❛t♦r✿
• σ✷

s,t = ✶

n

n

• i=✶
• ■❋θ(

Ti, ∆i, πi(s, t), s, t) ✷

❙❧✐❞❡ ✶✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢

✶

❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r

✶ ✱ s❛② ✹✵✵✵✱ ❞♦✿

✶ ●❡♥❡r❛t❡

✶

❢r♦♠ ✐✳✐✳❞✳ ✵ ✶ ✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠

✭ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡

✶

❢r♦♠ ✐✳✐✳❞✳ ✵ ✶ ✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡

✶

❢r♦♠ ✐✳✐✳❞✳ ✵ ✶ ✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r ■❋

✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✶

√n

n

• i=✶

ωb

i

• ■❋θ(

Ti, ∆i, πi(s, t), s, t)

• σs,t
• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✶

√n

n

• i=✶

ωb

i

• ■❋θ(

Ti, ∆i, πi(s, t), s, t)

• σs,t
• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✶

√n

n

• i=✶

ωb

i

• ■❋θ(

Ti, ∆i, πi(s, t), s, t)

• σs,t
• ✷ ❈♦♠♣✉t❡

✶

❛s t❤❡ ✶✵✵ ✶ t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

✶

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✐♠✉❧t❛♥❡♦✉s ❝♦♥✜❞❡♥❝❡ ❜❛♥❞ ♦✈❡r s ∈ S

• θ(s, t) ±

q(S,t)

✶−α

• σs,t

√n

• ,

s ∈ S ❈♦♠♣✉t❛t✐♦♥ ♦❢ q(S,t)

✶−α ❜② t❤❡ s✐♠✉❧❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✭≈ ❲✐❧❞ ❇♦♦tstr❛♣✮✿

✶ ❋♦r b = ✶, . . . , B✱ s❛② B = ✹✵✵✵✱ ❞♦✿ ✶ ●❡♥❡r❛t❡ {ωb

✶, . . . , ωb n} ❢r♦♠ n ✐✳✐✳❞✳ N(✵, ✶)✳

✷ ❯s✐♥❣ t❤❡ ♣❧✉❣✲✐♥ ❡st✐♠❛t♦r

■❋θ(·)✱ ❝♦♠♣✉t❡✿ Υb = s✉♣

s∈S

• ✶

√n

n

• i=✶

ωb

i

• ■❋θ(

Ti, ∆i, πi(s, t), s, t)

• σs,t
• ✷ ❈♦♠♣✉t❡

q(S,t)

✶−α ❛s t❤❡ ✶✵✵(✶ − α)t❤ ♣❡r❝❡♥t✐❧❡ ♦❢

• Υ✶, . . . , ΥB

✭❈♦♥❞✐t✐♦♥❛❧ ♠✉❧t✐♣❧✐❡r ❝❡♥tr❛❧ ❧✐♠✐t t❤❡♦r❡♠✮

❙❧✐❞❡ ✷✵✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ s❛♠♣❧❡

• P♦♣✉❧❛t✐♦♥ ❜❛s❡❞ st✉❞② ♦❢ ❦✐❞♥❡② r❡❝✐♣✐❡♥ts ✭♥❂✹✱✶✶✾✮
• ❙♣❧✐t t❤❡ ❞❛t❛ ✐♥t♦ tr❛✐♥✐♥❣ ✭✷✴✸✮ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✭✶✴✸✮ s❛♠♣❧❡s

✿ t✐♠❡ ❢r♦♠ ✶✲②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥ t♦ ❣r❛❢t ❢❛✐❧✉r❡ ✇❤✐❝❤ ✐s✿ ❉❡❛t❤ ❘❡t✉r♥ t♦ ❞✐❛❧②s✐s ❖❘ ❈❡♥s♦r✐♥❣ ❞✉❡ t♦✿ ❞❡❧❛②❡❞ ❡♥tr✐❡s✿ ✷✵✵✵✲✷✵✶✸ ❡♥❞ ♦❢ ❢♦❧❧♦✇✲✉♣✿ ✷✵✶✹ ❇❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s✿ ❛❣❡✱ s❡①✱ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ▲♦♥❣✐t✉❞✐♥❛❧ ❜✐♦♠❛r❦❡r ✭②❡❛r❧②✮✿ s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ s❛♠♣❧❡

• P♦♣✉❧❛t✐♦♥ ❜❛s❡❞ st✉❞② ♦❢ ❦✐❞♥❡② r❡❝✐♣✐❡♥ts ✭♥❂✹✱✶✶✾✮
• ❙♣❧✐t t❤❡ ❞❛t❛ ✐♥t♦ tr❛✐♥✐♥❣ ✭✷✴✸✮ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✭✶✴✸✮ s❛♠♣❧❡s
• T✿ t✐♠❡ ❢r♦♠ ✶✲②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥ t♦ ❣r❛❢t ❢❛✐❧✉r❡ ✇❤✐❝❤ ✐s✿

❉❡❛t❤ ❘❡t✉r♥ t♦ ❞✐❛❧②s✐s ❖❘ ❈❡♥s♦r✐♥❣ ❞✉❡ t♦✿ ❞❡❧❛②❡❞ ❡♥tr✐❡s✿ ✷✵✵✵✲✷✵✶✸ ❡♥❞ ♦❢ ❢♦❧❧♦✇✲✉♣✿ ✷✵✶✹ ❇❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s✿ ❛❣❡✱ s❡①✱ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r② ▲♦♥❣✐t✉❞✐♥❛❧ ❜✐♦♠❛r❦❡r ✭②❡❛r❧②✮✿ s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉■❱❆❚ s❛♠♣❧❡

• P♦♣✉❧❛t✐♦♥ ❜❛s❡❞ st✉❞② ♦❢ ❦✐❞♥❡② r❡❝✐♣✐❡♥ts ✭♥❂✹✱✶✶✾✮
• ❙♣❧✐t t❤❡ ❞❛t❛ ✐♥t♦ tr❛✐♥✐♥❣ ✭✷✴✸✮ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✭✶✴✸✮ s❛♠♣❧❡s
• T✿ t✐♠❡ ❢r♦♠ ✶✲②❡❛r ❛❢t❡r tr❛♥s♣❧❛♥t❛t✐♦♥ t♦ ❣r❛❢t ❢❛✐❧✉r❡ ✇❤✐❝❤ ✐s✿

❉❡❛t❤ ❘❡t✉r♥ t♦ ❞✐❛❧②s✐s ❖❘

• ❈❡♥s♦r✐♥❣ ❞✉❡ t♦✿
• ❞❡❧❛②❡❞ ❡♥tr✐❡s✿ ✷✵✵✵✲✷✵✶✸
• ❡♥❞ ♦❢ ❢♦❧❧♦✇✲✉♣✿ ✷✵✶✹
• ❇❛s❡❧✐♥❡ ❝♦✈❛r✐❛t❡s✿ ❛❣❡✱ s❡①✱ ❝❛r❞✐♦✈❛s❝✉❧❛r ❤✐st♦r②
• ▲♦♥❣✐t✉❞✐♥❛❧ ❜✐♦♠❛r❦❡r ✭②❡❛r❧②✮✿ s❡r✉♠ ❝r❡❛t✐♥✐♥❡

❙❧✐❞❡ ✷✶✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉❡s❝r✐♣t✐✈❡ st❛t✐st✐❝s ✫ ❝❡♥s♦r✐♥❣ ✐ss✉❡

• s ∈ S = {✵, ✵.✺, . . . , ✺}
• t = ✺ ②❡❛rs

s=0 s=1 s=2 s=3 s=4 s=5 Censored in (s,s+5] Known as event−free at s+5 Observed failure in (s,s+5] landmark time (year)

• No. of subjects

200 600 1000 1400

❙❧✐❞❡ ✷✷✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❏♦✐♥t ♠♦❞❡❧

◮ ▲♦♥❣✐t✉❞✐♥❛❧ log

• Yi(tij)
• = (β✵ + b✵i) + β✵,❛❣❡❆●❊i + β✵,s❡①❙❊❳i

+

• β✶ + b✶i + β✶,❛❣❡❆●❊i
• × tij + ǫij

= ♠i(t) + εij ❙✉r✈✐✈❛❧ ✭❤❛③❛r❞✮

✵

❡①♣

❛❣❡❆●❊ ❈❱❈❱ ✶♠ ✷

♠

✭✜tt❡❞ ✉s✐♥❣ ♣❛❝❦❛❣❡ ❏▼✮

❙❧✐❞❡ ✷✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❏♦✐♥t ♠♦❞❡❧

◮ ▲♦♥❣✐t✉❞✐♥❛❧ log

• Yi(tij)
• = (β✵ + b✵i) + β✵,❛❣❡❆●❊i + β✵,s❡①❙❊❳i

+

• β✶ + b✶i + β✶,❛❣❡❆●❊i
• × tij + ǫij

= ♠i(t) + εij ◮ ❙✉r✈✐✈❛❧ ✭❤❛③❛r❞✮ hi(t) = h✵(t) ❡①♣

• γ❛❣❡❆●❊i + γ❈❱❈❱i

+ α✶♠i(t) + α✷ d♠i(t) dt

• ✭✜tt❡❞ ✉s✐♥❣

♣❛❝❦❛❣❡ ❏▼✮

❙❧✐❞❡ ✷✸✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

R✷

π(s, t) ✈s s ✭t❂✺ ②❡❛rs✮

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

• 95% pointwise CI

95% simultaneous CB

❙❧✐❞❡ ✷✹✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

• T ~ Age + CV + m(t) + m'(t) (JM)

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

• T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

• T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV + Y(t=0) T ~ Age + CV

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s (years) Rπ

2(s, t)

1 2 3 4 5 0 % 10 % 20 % 30 %

• T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV + Y(t=0)

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s Difference in Rπ

2(s,t)

• −5 %

5 % 10 % 20 % 0 % 1 2 3 4 5

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈♦♠♣❛r✐♥❣ R✷

π(s, t) ✈s s ❢♦r ❞✐✛❡r❡♥t π(s, t)

Landmark time s Difference in Rπ

2(s,t)

• −5 %

5 % 10 % 20 % 0 % 1 2 3 4 5

❙❧✐❞❡ ✷✺✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❈❛❧✐❜r❛t✐♦♥ ♣❧♦t ✭❡①❛♠♣❧❡ ❢♦r s = ✸ ②❡❛rs✮

(0,10) (0;3.5] (10,20) (3.5;5.3] (20,30) (5.3;8] (30,40) (8;10.4] (40,50) (10.4;13.1] (50,60) (13.1;16.5] (60,70) (16.5;22.3] (70,80) (22.3;33] (80,90) (33;55.1] (90,100) (55.1;100] Predicted Observed

Risk groups (quantile groups, in %) 5−year risk (%)

20 40 60 80 100 2% 8(4)% 4% 15(6)% 7% 15(5)% 9% 13(5)% 12% 16(6)% 15% 21(7)% 19% 26(6)% 27% 29(7)% 42% 36(9)% 77% 74(8)%

❙❧✐❞❡ ✷✻✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❆r❡❛ ✉♥❞❡r t❤❡ ❘❖❈(s, t) ❝✉r✈❡ ✈s s

Landmark time s (years) AUCπ(s, t) 1 2 3 4 5 50% 60% 70% 80% 100%

• T ~ Age + CV + m(t) + m'(t) (JM)

T ~ Age + CV + Y(t=0) T ~ Age + CV

❙❧✐❞❡ ✷✼✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❙✉♠♠✐♥❣ ✉♣

◮ ❘✷✲t②♣❡ ❝✉r✈❡

• s✉♠♠❛r✐③❡s ❝❛❧✐❜r❛t✐♦♥ ❛♥❞ ❞✐s❝r✐♠✐♥❛t✐♥❣ s✐♠✉❧t❛♥❡♦✉s❧②
• ❤❛s ❛♥ ✉♥❞❡rst❛♥❞❛❜❧❡ tr❡♥❞

◮ ❙✐♠♣❧❡ ♠♦❞❡❧ ❢r❡❡ ✐♥❢❡r❡♥❝❡

• ♣r❡❞✐❝t✐♦♥s ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ ❛♥② ♠♦❞❡❧
• ✇❡ ❞♦ ♥♦t ❛ss✉♠❡ ❛♥② ♠♦❞❡❧ t♦ ❤♦❧❞
• ❛❧❧♦✇s ❢❛✐r ❝♦♠♣❛r✐s♦♥s ♦❢ ❞✐✛❡r❡♥t ♣r❡❞✐❝t✐♦♥s

◮ ❚❤❡ ♠❡t❤♦❞ ❛❝❝♦✉♥ts ❢♦r✿

• ❈❡♥s♦r✐♥❣
• ❉②♥❛♠✐❝ s❡tt✐♥❣ ✭t❤❡ ❛t r✐s❦ ♣♦♣✉❧❛t✐♦♥ ❝❤❛♥❣❡s✮

❙❧✐❞❡ ✷✽✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉✐s❝✉ss✐♦♥

◮ ❚❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❛❧❧♦✇s ❞✐✛❡r❡♥t ✐♥t❡r❡st✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥s✿

• ❊①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥
• ❈♦rr❡❧❛t✐♦♥
• ▼❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡

◮ ❯♥❢♦rt✉♥❛t❡❧②

• t❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥♥♦t ❜❡ ❝❤❡❝❦❡❞

✭❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✮

◮ ❍♦✇❡✈❡r

• ✇❡❛❦ ❛♥❞ str♦♥❣ ❞❡✜♥✐t✐♦♥s ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✿
• str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♠♣❧✐❡s ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥
• ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ✏♦❢t❡♥✑ ❜❡ s❡❡♥ ❛s ❛ r❡❛s♦♥❛❜❧❡

❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♥ ♣r❛❝t✐❝❡

• ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ❜❡ ❛ss❡ss❡❞ ✭♣❧♦ts✮

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❙❧✐❞❡ ✷✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳

✉ ♥ ✐ ✈ ❡ r s ✐ t ② ♦ ❢ ❝ ♦ ♣ ❡ ♥ ❤ ❛ ❣ ❡ ♥ ❞ ❡ ♣ ❛ r t ♠ ❡ ♥ t ♦ ❢ ❜ ✐ ♦ s t ❛ t ✐ s t ✐ ❝ s

❉✐s❝✉ss✐♦♥

◮ ❚❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❛ss✉♠♣t✐♦♥ ❛❧❧♦✇s ❞✐✛❡r❡♥t ✐♥t❡r❡st✐♥❣ ✐♥t❡r♣r❡t❛t✐♦♥s✿

• ❊①♣❧❛✐♥❡❞ ✈❛r✐❛t✐♦♥
• ❈♦rr❡❧❛t✐♦♥
• ▼❡❛♥ r✐s❦ ❞✐✛❡r❡♥❝❡

◮ ❯♥❢♦rt✉♥❛t❡❧②

• t❤❡ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥♥♦t ❜❡ ❝❤❡❝❦❡❞

✭❝✉rs❡ ♦❢ ❞✐♠❡♥s✐♦♥❛❧✐t②✮

◮ ❍♦✇❡✈❡r

• ✇❡❛❦ ❛♥❞ str♦♥❣ ❞❡✜♥✐t✐♦♥s ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✿
• str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♠♣❧✐❡s ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥
• ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ✏♦❢t❡♥✑ ❜❡ s❡❡♥ ❛s ❛ r❡❛s♦♥❛❜❧❡

❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ str♦♥❣ ❝❛❧✐❜r❛t✐♦♥ ✐♥ ♣r❛❝t✐❝❡

• ✇❡❛❦ ❝❛❧✐❜r❛t✐♦♥ ❝❛♥ ❜❡ ❛ss❡ss❡❞ ✭♣❧♦ts✮

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❙❧✐❞❡ ✷✾✴✷✾ ✖ P ❇❧❛♥❝❤❡ ❡t ❛❧✳