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❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❝♦♠♠♦♥ ✈❛❧✉❡ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ✷✵✶✺ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❆ ❧♦♥❣ t✐♠❡ ❛❣♦✱ ✐♥ ❛ ❣❛❧❛①② ❢❛r✱ ❢❛r ❛✇❛② ✳✳✳ ❆ ❧♦t ♦❢ ♣❡♦♣❧❡ tr✐❡❞ t♦ ✜♥❞ ❛♥ ♦♣t✐♠❛❧ ♠❡❝❤❛♥✐s♠ t♦ s❡❧❧ s♦♠❡t❤✐♥❣✿ ❲✐❧❧✐❛♠ ❱✐❝❦r❡② ✭✶✾✻✶✮ ✧❈♦✉♥t❡rs♣❡❝✉❧❛t✐♦♥✱ ❆✉❝t✐♦♥s✱ ❛♥❞ ❈♦♠♣❡t✐t✐✈❡ ❙❡❛❧❡❞ ❚❡♥❞❡rs✧ ❘♦❣❡r ▼②❡rs♦♥ ✭✶✾✽✶✮ ✧❖♣t✐♠❛❧ ❆✉❝t✐♦♥ ❉❡s✐❣♥✧ P❛✉❧ ▼✐❧❣r♦♠ ❛♥❞ ❘♦❜❡rt ❲❡❜❡r ✭✶✾✽✷✮ ✧❆ t❤❡♦r② ♦❢ ❛✉❝t✐♦♥s ❛♥❞ ❝♦♠♣❡t✐t✐✈❡ ❜✐❞❞✐♥❣✧ ❱✐❥❛② ❑r✐s❤♥❛ ❛♥❞ ❏♦❤♥ ▼♦r❣❛♥ ✭✶✾✾✼✮ ✧❆♥ ❛♥❛❧②s✐s ♦❢ t❤❡ ✇❛r ♦❢ ❛ttr✐t✐♦♥ ❛♥❞ t❤❡ ❛❧❧✲♣❛② ❛✉❝t✐♦♥✧ ❏❛❝q✉❡s ❈r✁ ❡♠❡r ❛♥❞ ❘✐❝❤❛r❞ ▼❝▲❡❛♥ ✭✶✾✾✽✮✱ ✧❋✉❧❧ ❡①tr❛❝t✐♦♥ ♦❢ t❤❡ s✉r♣❧✉s ✐♥ ❇❛②❡s✐❛♥ ❛♥❞ ❞♦♠✐♥❛♥t str❛t❡❣② ❛✉❝t✐♦♥s✧ ❙❡r❣✐✉ ❍❛rt✱ P❤✐❧✐♣ ❘❡♥② ❛♥❞ ◆♦❛♠ ◆✐s❛♥ ✭✷✵✶✵✕✳✳✳✮ ✧❚✇♦✭✦✮ ●♦♦❞ ❚♦ ❇❡ ❚r✉❡✧ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❙♦♠❡t❤✐♥❣ t♦ ❜❡❣✐♥ ✇✐t❤ ❈❧❛ss✐❝❛❧ ❛✉❝t✐♦♥ t❤❡♦r② s♣❡♥t ❛ ❧♦t ♦❢ ❡✛♦rt ❝♦♠♣❛r✐♥❣ ❞✐✛❡r❡♥t ❜✐❞❞✐♥❣ ♣r♦❝❡❞✉r❡s ❛♥❞ ♠❛❞❡ ❣r❡❛t ❛❞✈❛♥❝❡s ✐♥ t❤✐s ❛r❡❛✳ ❍♦✇❡✈❡r ✈❡r② ❢❡✇ ♣❛♣❡rs ❝♦♥s✐❞❡r ❝♦♠♣❛r❛t✐✈❡ st❛t✐❝s ♦❢ ❛✉❝t✐♦♥ ♦✉t❝♦♠❡ ✇✐t❤ r❡s♣❡❝t t♦ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ ❜✐❞❞❡r t②♣❡s✳ ❲❡ ❛r❣✉❡ t❤❛t t❤✐s q✉❡st✐♦♥ ✐s ❜♦t❤ ✈❡r② ♥❛t✉r❛❧ ❛♥❞ ✈❡r② ✐♠♣♦rt❛♥t✳ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❚❤❡ ♠♦❞❡❧ ❚✇♦ ❡①✲❛♥t❡ s②♠♠❡tr✐❝ ♣❧❛②❡rs✱ ❡❛❝❤ ♦♥❡ ✐s ♠❛❦✐♥❣ ❤✐s ❜✐❞✳ ❚❤❡ ♦♥❡ ♠❛❦✐♥❣ t❤❡ ❤✐❣❤❡st ❜✐❞ ✐s ❛ ✇✐♥♥❡r✱ t❤❡ ♦t❤❡r ♦♥❡ ✐s ❛ ❧♦s❡r✳ ▲❡t θ ✶ , θ ✷ ❜❡ ♣❧❛②❡rs✬ t②♣❡s ❛♥❞ ❜ ✶ , ❜ ✷ ✖ t❤❡✐r ❜✐❞s✳ ❙✉♣♣♦s❡ t❤❛t ❜ ✐ > ❜ ❥ ❀ ♣❧❛②❡r ✐ ♦❜t❛✐♥s t❤❡ ❢♦❧❧♦✇✐♥❣ ✉t✐❧✐t②✿ ❯ ✐ ✇ = ❇ ✇ ( θ ✐ , θ ❥ , ❜ ❥ ) − ❜ ✐ ❈ ✇ ( θ ✐ ) . ❚❤❡ ♦t❤❡r ❜✐❞❞❡r✬s ✭❧♦s❡r✬s✮ ✉t✐❧✐t② ❡q✉❛❧s✿ ❯ ❥ ❧ = ❇ ❧ ( θ ❥ , θ ✐ , ❜ ✐ ) − ❜ ❥ ❈ ❧ ( θ ❥ ) . ❈ ✇ ≥ ✵✱ ❈ ❧ ≥ ✵ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❊①❛♠♣❧❡s ❯ ✐ ✇ = ❇ ✇ ( θ ✐ , θ ❥ , ❜ ❥ ) − ❜ ✐ ❈ ✇ ( θ ✐ ) . ❯ ❥ ❧ = ❇ ❧ ( θ ❥ , θ ✐ , ❜ ✐ ) − ❜ ❥ ❈ ❧ ( θ ❥ ) . ✶ ❋✐rst ♣r✐❝❡ ❛✉❝t✐♦♥✿ ❈ ✇ = ✶✱ ❇ ✇ = θ ✐ ❈ ❧ = ❇ ❧ = ✵✱ ✷ ❙❡❝♦♥❞ ♣r✐❝❡ ❛✉❝t✐♦♥✿ ❈ ✇ = ✵✱ ❇ ✇ = θ ✐ − ❜ ❥ ❈ ❧ = ❇ ❧ = ✵✱ ✸ ❲❛r ♦❢ ❛ttr✐t✐♦♥✿ ❈ ✇ = ✵✱ ❇ ✇ = θ ✐ − ❜ ❥ ❈ ❧ = ✶✱ ❇ ❧ = ✵✱ ✹ ❆❧❧✲♣❛② ❛✉❝t✐♦♥✿ ❈ ✇ = ✶✱ ❇ ✇ = θ ✐ ❈ ❧ = ✶✱ ❇ ❧ = ✵✳ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❊①♣❡❝t❡❞ ✉t✐❧✐t② ❜ ( · ) ✐s ❛ ❝♦♥t✐♥✉♦✉s ❛♥❞ ♠♦♥♦t♦♥✐❝ ❡q✉✐❧✐❜r✐✉♠ str❛t❡❣②❀ ❧❡t θ ❜❡ ♣❧❛②❡r✬s tr✉❡ t②♣❡ ❛♥❞ ¯ θ ✐s t❤❡ t②♣❡ ❤❡ ♣r❡t❡♥❞s t♦ ❜❡✳ � ¯ θ ❯ (¯ ❇ ✇ ( θ, ˜ θ, ❜ (˜ θ )) ❞❋ (˜ θ | θ ) − ❜ (¯ θ ) ❈ ✇ ( θ ) ❋ (¯ θ, θ ) = θ | θ )+ ✵ � ✶ ❇ ❧ ( θ, ˜ θ, ❜ (˜ θ )) ❞❋ (˜ θ | θ ) − ❜ (¯ θ ) ❈ ❧ ( θ )( ✶ − ❋ (¯ + θ | θ )) . ¯ θ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❋❖❈ ■♥ ❡q✉✐❧✐❜r✐✉♠ t❤❡ ♣❧❛②❡rs ❛r❡ ✇✐❧❧✐♥❣ t♦ r❡♣♦rt t❤❡✐r tr✉❡ t②♣❡✱ t❤✉s ✇❡ ♦❜t❛✐♥✿ ❜ ′ ( θ )[ ❈ ✇ ( θ ) ❋ ( θ | θ ) + ❈ ❧ ( θ )( ✶ − ❋ ( θ | θ ))] = = [ ❇ ✇ ( θ, θ, ❜ ( θ )) − ❇ ❧ ( θ, θ, ❜ ( θ )) − ❜ ( θ )( ❈ ✇ ( θ ) − ❈ ❧ ( θ ))] ❢ ( θ | θ ) . ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❙♦♠❡ ♥♦t❛t✐♦♥s ❢ ( θ | θ ) ❣ ( θ, ❋ ) = ❈ ✇ ( θ ) ❋ ( θ | θ ) + ❈ ❧ ( θ )( ✶ − ❋ ( θ | θ )) = ❢ ( θ, θ ) = , � θ � ✶ ❈ ✇ ( θ ) ✵ ❢ ( ① , θ ) ❞① + ❈ ❧ ( θ ) θ ❢ ( ① , θ ) ❞① ▲ ( ❜ , θ ) = ❇ ✇ ( θ, θ, ❜ ( θ )) − ❇ ❧ ( θ, θ, ❜ ( θ )) − ❜ ( θ )( ❈ ✇ ( θ ) − ❈ ❧ ( θ )) . ❙♦ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥ ❢♦r ❡q✉✐❧✐❜r✐✉♠ str❛t❡❣② ❝❛♥ ❜❡ r❡✇r✐tt❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ ❜ ′ ( θ ) = ▲ ( ❜ , θ ) ❣ ( θ, ❋ ) , ❜ ( ✵ ) = ✵ . ◆♦t❡ t❤❛t ✐ts ♦♥❧② ❣ ( θ, ❋ ) t❤❛t ❞❡♣❡♥❞s ♦♥ ✐♥✐t✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t②♣❡s✳ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❙✉♠♠✐♥❣ ✉♣ ❚❤❡ ❋❖❈ ❜ ′ ( θ ) = ▲ ( ❜ , θ ) ❣ ( θ, ❋ ) , ❜ ( ✵ ) = ✵ . ◆♦t❡ t❤❛t ✐ts ♦♥❧② ❣ ( θ, ❋ ) t❤❛t ❞❡♣❡♥❞s ♦♥ ✐♥✐t✐❛❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t②♣❡s✳ ❚❤❡ ♣❧❛②❡r✬s ❡①♣❡❝t❡❞ ❜✐❞ ❝❛♥ ❜❡ ❡❛s✐❧② ❢♦✉♥❞ ❣✐✈❡♥ ❤✐s str❛t❡❣② ✭❋ ( θ ) ✐s ❛ ♠❛r❣✐♥❛❧ ♦❢ ❋ ( θ ✶ , θ ✷ ) ✿ � ✶ ❏ = ❜ ( θ ) ❞❋ ( θ ) ✵ ❆ss✉♠♣t✐♦♥ ❋r♦♠ ♥♦✇ ♦♥ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t t❤❡ ❤❡✉r✐st✐❝ ❛❜♦✈❡ ✏✇♦r❦s✑ ❛♥❞ t❤❛t ❜ ( · ) ✐s ✐♥❞❡❡❞ s♠♦♦t❤ ❛♥❞ ✐♥❝r❡❛s✐♥❣ ❡q✉✐❧✐❜r✐✉♠ str❛t❡❣②✳ ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞

❙♦♠❡ ♠♦r❡ t❤♦✉❣❤ts✳✳✳ � ✶  ❏ = ✵ ❜ ( θ ) ❞❋ ( θ ) ,  ❜ ′ ( θ ) = ▲ ( ❜ , θ ) ❣ , ❜ ( ✵ ) = ✵ .  ▲ ( ❜ , θ ) st❛♥❞s ❢♦r ❞✐✛❡r❡♥❝❡ ✐♥ ✉t✐❧✐t✐❡s ❢r♦♠ ✇✐♥♥✐♥❣ ❛♥❞ ❧♦s✐♥❣ ✐❢ ❜♦t❤ ♣❧❛②❡rs ❤❛✈❡ t❤❡ s❛♠❡ t②♣❡s ❛♥❞ ❜♦t❤ ❛r❡ ♣❧❛②✐♥❣ ❡q✉✐❧✐❜r✐✉♠ str❛t❡❣②✳ ■t ❛❧s♦ ❝❛♥✬t ❜❡ ♥❡❣❛t✐✈❡ s✐♥❝❡ ❜ ( θ ) ✐s ✐♥❝r❡❛s✐♥❣ ❜② ♦✉r ❛ss✉♠♣t✐♦♥ ❛♥❞ ❣ ( θ, ❋ ) ✐s ❛❧✇❛②s ♥♦♥♥❡❣❛t✐✈❡✳ ❙♦✱ t❤❡ t❤❡ ♦♥❧② t❤✐♥❣ t❤❛t ♠❛tt❡rs ❢♦r ♦✉r ❝♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐s t❤❡ s✐❣♥ ♦❢ ∆ ❣✳ ❢ ( θ, θ ) ❣ ( θ, ❋ ) = . � θ � ✶ ❈ ✇ ( θ ) ✵ ❢ ( ① , θ ) ❞① + ❈ ❧ ( θ ) θ ❢ ( ① , θ ) ❞① ❈✳ ❙♦r♦❦✐♥✱ ❊✳ ❲✐♥t❡r ❈♦♠♣❛r❛t✐✈❡ st❛t✐❝s ✐♥ ❛✉❝t✐♦♥s ❛♥❞ ❜❡②♦♥❞