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❉✐❣✐t❛❧ ❲❛t❡r♠❛r❦ ❉❡t❡❝t✐♦♥ ✐♥ ❱✐s✉❛❧ ▼✉❧t✐♠❡❞✐❛ ❈♦♥t❡♥t P❤❉ ❚❤❡s✐s ❉❡❢❡♥s❡ P❡t❡r ▼❡❡r✇❛❧❞ ❉❡♣t✳ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡s✱ ❯♥✐✈❡rs✐t② ♦❢ ❙❛❧③❜✉r❣ ❙❡♣t❡♠❜❡r ✷✵✶✵

❲❛t❡r♠❛r❦✐♥❣ ◮ ❲❛t❡r♠❛r❦✐♥❣ ✐s ❛ t❡❝❤♥♦❧♦❣② t♦ ❡♠❜❡❞ ✐♥❢♦r♠❛t✐♦♥ ✐♥t♦ ♠✉❧t✐♠❡❞✐❛ ❝♦♥t❡♥t ✐♥ ❛♥ ✐♠♣❡r❝❡♣t✐❜❧❡✱ ②❡t ❞❡t❡❝t❛❜❧❡ ✇❛②✳ ❬❈♦① ❡t ❛❧✳✱ ✷✵✵✼❪ ◮ ❆♣♣❧✐❝❛t✐♦♥s✿ ❈♦♣②r✐❣❤t ♣r♦t❡❝t✐♦♥✱ ✜♥❣❡r♣r✐♥t✐♥❣ ✭tr❛✐t♦r tr❛❝✐♥❣✮✱ ✳ ✳ ✳

▼♦t✐✈❛t✐♦♥ ❛♥❞ ❖✉t❧✐♥❡ ❚❤❡s✐s s✉♣♣♦rt❡❞ ❜② ❆✉str✐❛♥ ❙❝✐❡♥❝❡ ❋✉♥❞ ✭❋❲❋✮ ♣r♦❥❡❝t P✶✾✶✺✾ ♦♥ ✏❆❞❛♣t✐✈❡ ❙tr❡❛♠✐♥❣ ♦❢ ❙❡❝✉r❡ ❙❝❛❧❛❜❧❡ ❲❛✈❡❧❡t✲❜❛s❡❞ ❱✐❞❡♦✑✳ ✶✳ ❊✣❝✐❡♥t s♣r❡❛❞✲s♣❡❝tr✉♠ ✇❛t❡r♠❛r❦ ❞❡t❡❝t✐♦♥ ✷✳ ❲❛t❡r♠❛r❦✐♥❣ ♦❢ s❝❛❧❛❜❧❡ ♠✉❧t✐♠❡❞✐❛ ❢♦r♠❛ts ❖t❤❡r t♦♣✐❝s ◮ ❲❛t❡r♠❛r❦ ❞❡t❡❝t✐♦♥ ✐♥ r❛✇ ❛♥❞ ❞❡♠♦s❛✐❝❦❡❞ ✐♠❛❣❡s ◮ ❆tt❛❝❦s ♦♥ q✉❛♥t✐③❛t✐♦♥✲❜❛s❡❞ ✇❛t❡r♠❛r❦✐♥❣ s❝❤❡♠❡s ◮ ❲❛t❡r♠❛r❦ ❞❡t❡❝t✐♦♥ ✐♥ t❤❡ ❉✉❛❧ ❚r❡❡ ❈♦♠♣❧❡① ❲❛✈❡❧❡t ❚r❛♥s❢♦r♠ ✭❉❚✲❈❲❚✮ ❞♦♠❛✐♥ ◮ ❲❛t❡r♠❛r❦✐♥❣ ♦❢ ✷❉ ✈❡❝t♦r ❣r❛♣❤✐❝s

❉❡t❡❝t✐♦♥ Pr♦❜❧❡♠ ◮ ❉❡t❡❝t✐♦♥ ♣r♦❜❧❡♠ ❢♦r ❛❞❞✐t✐✈❡ s♣r❡❛❞✲s♣❡❝tr✉♠ ✇❛t❡r♠❛r❦✐♥❣ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❛ ❤②♣♦t❤❡s✐s t❡st t♦ ❞❡❝✐❞❡ ❜❡t✇❡❡♥ ❛❜s❡♥❝❡ ✭ H ✵ ✮ ♦r ♣r❡s❡♥❝❡ ✭ H ✶ ✮ ♦❢ t❤❡ ✇❛t❡r♠❛r❦ ✇ ✐♥ t❤❡ r❡❝❡✐✈❡❞ s✐❣♥❛❧ ② ♦❢ ❧❡♥❣t❤ ◆ H ✵ : ② [ t ] = ① [ t ] t = ✶ , . . . , ◆ H ✶ : ② [ t ] = ① [ t ] + α ✇ [ t ] t = ✶ , . . . , ◆ ◮ ▲✐❦❡❧✐❤♦♦❞✲❘❛t✐♦ ❚❡st ✭▲❘❚✮ ♠✐♥✐♠✐③❡s t❤❡ ♣r♦❜❛❜✐❧✐t② ♦❢ ♠✐ss ❣✐✈❡♥ ❛ ♣r♦❜❛❜✐❧✐t② ♦❢ ❢❛❧s❡✲❛❧❛r♠ ❬❑❛②✱ ✶✾✾✽❪ � ♣ ( ② ; H ✶ ) � ▲ ( ② ) := ❧♦❣ > ❧♦❣ ( τ ) =: ❚ ♣ ( ② ; H ✵ ) ✇❤❡r❡ ♣ ( · ) ❞❡♥♦t❡s t❤❡ Pr♦❜❛❜✐❧✐t② ❉❡♥s✐t② ❋✉♥❝t✐♦♥ ✭P❉❋✮ ♦❢ t❤❡ s✐❣♥❛❧ ❛♥❞ ❚ ✐s t❤❡ ❞❡t❡❝t✐♦♥ t❤r❡s❤♦❧❞ ◮ ✭❯♥r❡❛❧✐st✐❝✮ ❛ss✉♠♣t✐♦♥✿ ❝♦♠♣❧❡t❡ ❦♥♦✇❧❡❞❣❡ ♦❢ t❤❡ ❤♦st s✐❣♥❛❧ P❉❋ ❛♥❞ t❤❡ ❡♠❜❡❞❞✐♥❣ str❡♥❣t❤ α > ✵

❲❛t❡r♠❛r❦ ❉❡t❡❝t♦r ■♥❣r❡❞✐❡♥ts ■♥❣r❡❞✐❡♥ts ✶✳ ❍♦st s✐❣♥❛❧ ♠♦❞❡❧ ✭✇❤✐❝❤❄✮ ✇✐t❤ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ✭❤♦✇❄✮ ✷✳ ❉❡t❡❝t✐♦♥ st❛t✐st✐❝ ✭❜❛s❡❞ ♦♥ ▲❘❚ ♦r ❘❛♦ ❚❡st✮ ❞❡♣❡♥❞✐♥❣ ♦♥ ❤♦st s✐❣♥❛❧ ♠♦❞❡❧ ✭❝♦♠♣✉t❛t✐♦♥❛❧❧② ❡✣❝✐❡♥t❄✮ ✸✳ ❉❡t❡❝t✐♦♥ t❤r❡s❤♦❧❞ ❢♦r ❛ ❣✐✈❡♥ ❢❛❧s❡✲❛❧❛r♠ r❛t❡✱ ❡✳❣✳ ✶✵ − ✻ ◮ ❋♦r t❤❡ ▲❘❚ ✇❡ ♥❡❡❞ ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❞❡t❡❝t✐♦♥ st❛t✐st✐❝ ✉♥❞❡r H ✵ ◮ ❘❛♦ t❡sts ❧❡❛❞ t♦ ❝♦♥st❛♥t ❢❛❧s❡✲❛❧❛r♠ r❛t❡ ✭❈❋❆❘✮ ❞❡t❡❝t♦rs✱ t❤❡ t❤r❡s❤♦❧❞ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ s✐❣♥❛❧ ♦r ❡♠❜❡❞❞✐♥❣ str❡♥❣t❤ α ◮ ❘❡❧✐❛❜❧❡❄

❘❡s❡❛r❝❤ ◗✉❡st✐♦♥s ◮ ❍♦✇ ❞♦ ❤♦st s✐❣♥❛❧ ♠♦❞❡❧ ❛♥❞ ♣❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ❛♣♣r♦❛❝❤❡s ❝❤❛♥❣❡ ❞❡t❡❝t✐♦♥ ♣❡r❢♦r♠❛♥❝❡❄ ◮ ❲❤❛t ✐s t❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❡✛♦rt ❢♦r ❡st✐♠❛t✐♦♥✱ ❡✈❛❧✉❛t✐♦♥ ♦❢ t❤❡ ❞❡t❡❝t✐♦♥ st❛t✐st✐❝ ❛♥❞ t❤r❡s❤♦❧❞ ❞❡t❡r♠✐♥❛t✐♦♥❄ ◮ ❈❛♥ ✇❡ ✐❞❡♥t✐❢② ❛ ♠♦r❡ ✬♣r❛❝t✐❝❛❧✬ ❞❡t❡❝t♦r t❤❛♥ ▲❘❚ ✇✐t❤ ❛ ●❡♥❡r❛❧✐③❡❞ ●❛✉ss✐❛♥ ✭●●✮ ♠♦❞❡❧ ❬❍❡r♥á♥❞❡③ ❡t ❛❧✳✱ ✷✵✵✵❪ ❄

❍♦st ❙✐❣♥❛❧ ▼♦❞❡❧✐♥❣ ❉❈❚ ❛♥❞ ❉❲❚ ❝♦❡✣❝✐❡♥ts ♦❢ ♥❛t✉r❛❧ ✐♠❛❣❡s ❛r❡ ♥♦♥✲●❛✉ss✐❛♥ ❬❇✐r♥❡② ❛♥❞ ❋✐s❝❤❡r✱ ✶✾✾✺❪ 0.09 0.08 c = 0.5 γ = 4 c = 1.1 γ = 7 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 −100 −50 0 50 100 −100 −50 0 50 100 ❉❲❚ s✉❜❜❛♥❞ ❤✐st♦❣r❛♠s ●● ❈❛✉❝❤② ●● ❞✐str✐❜✉t✐♦♥ ❈❛✉❝❤② ❞✐str✐❜✉t✐♦♥ � ❝ � ♣ ( ① | γ, δ ) = ✶ γ ❝ � � ① � � ♣ ( ① | ❛ , ❝ ) = ✷ ❛ Γ( ✶ / ❝ ) ❡①♣ − γ ✷ +( ① − δ ) ✷ ❛ π s❝❛❧❡ ♣❛r❛♠❡t❡r ❛ > ✵ ❧♦❝❛t✐♦♥ ♣❛r❛♠❡t❡r δ ✭ ≈ ✵✮ s❤❛♣❡ ♣❛r❛♠❡t❡r ❝ > ✵ s❤❛♣❡ ♣❛r❛♠❡t❡r γ > ✵

P❛r❛♠❡t❡r ✭●● ❛ , ❝ ✱ ❈❛✉❝❤② δ, γ ✮ ❊st✐♠❛t✐♦♥ ❖♣t✐♦♥s ◮ ▼❛①✐♠✉♠ ▲✐❦❡❧✐❤♦♦❞ ❊st✐♠❛t✐♦♥ ✭▼▲❊✮ ❬❱❛r❛♥❛s✐ ❛♥❞ ❆❛③❤❛♥❣✱ ✶✾✽✾❪ ◮ ❆♣♣r♦①✐♠❛t✐✈❡ ♠❡t❤♦❞s ❬❑r✉♣✐♥s❦✐ ❛♥❞ P✉r❝③②♥s❦✐✱ ✷✵✵✻✱ ❚s✐❤r✐♥t③✐s ❛♥❞ ◆✐❦✐❛s✱ ✶✾✾✻❪ ◮ ❋✐①❡❞ s❡tt✐♥❣s ✭❡✳❣✳ ❝ = ✵ . ✽✱ γ = ✽✮ Cauchy GGD 120 180 MLE MLE 160 Approximation Approximation 100 140 80 120 Histogram Histogram 100 60 80 40 60 40 20 20 0 0 0 10 20 30 40 0 0.5 1 1.5 2 2.5 Shape parameter range Shape parameter range ●● ❛♥❞ ❈❛✉❝❤② s❤❛♣❡ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s ♦✈❡r ❉❲❚ ❞❡t❛✐❧ s✉❜❜❛♥❞s ♦❢ ✶✵✵✵ ♥❛t✉r❛❧ ✐♠❛❣❡s

❉❡t❡❝t✐♦♥ ❙t❛t✐st✐❝s Pr♦♣♦s❡❞ ❘❛♦✲❈❛✉❝❤② ❞❡t❡❝t✐♦♥ st❛t✐st✐❝ � ◆ � ✷ ✽ γ ✷ ② [ t ] ✇ [ t ] � ρ ❘❛♦ - ❈ = γ ✷ + ② [ t ] ✷ ◆ t = ✶ Pr✐♦r ❲♦r❦ ◆ ◆ ρ ▲❈ = ✶ ρ ▲❘❚ - ●● = ✶ ( | ② [ t ] | ❝ − | ② [ t ] − α ✇ [ t ] | ❝ ) � � ② [ t ] ✇ [ t ] ❛ ❝ ◆ t = ✶ t = ✶ t = ✶ s❣♥ ( ② [ t ]) ✇ [ t ] | ② [ t ] | ❝ � ✷ �� ◆ ◆ γ ✷ + ② [ t ] ✷ � � � ρ ▲❘❚ - ❈ = ❧♦❣ ρ ❘❛♦ - ●● = γ ✷ + ( ② [ t ] − α ✇ [ t ]) ✷ � ◆ t = ✶ | ② [ t ] | ✷ ❝ t = ✶

◆✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ❖♣❡r❛t✐♦♥s + , − × , ÷ ♣♦✇ , ❧♦❣ | · | , s❣♥ ▲❈ ◆ ◆ ▲❘❚✲●● ✸◆ ◆ ✷◆ ✷◆ ▲❘❚✲❈ ✹◆ ✹◆ ◆ ❘❛♦✲●● ✷◆ ✸◆ ◆ ✷◆ ❘❛♦✲❈ ✷◆ ✸◆ ❆r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s t♦ ❝♦♠♣✉t❡ t❤❡ ❞❡t❡❝t✐♦♥ st❛t✐st✐❝ ✭s✐❣♥❛❧ ❧❡♥❣t❤ ◆ ✮ + , × ❡①❡❝✉t❡ ✐♥ s✐♥❣❧❡ ❝②❝❧❡❀ ♣♦✇ , ❧♦❣ t❛❦❡ ❤✉♥❞r❡❞s ♦❢ ❝②❝❧❡s

■♠♣❛❝t ♦❢ ❍♦st ❙✐❣♥❛❧ P❛r❛♠❡t❡r ❊st✐♠❛t❡s Lena Lena LRT−GG LRT−C Rao−GG Rao−C Probability of Miss Probability of Miss −50 −50 10 10 −100 10 −100 10 −150 10 0.5 1 1.5 2 2.5 3 3.5 4 10 20 30 40 Shape parameter range Shape parameter range Pr♦❜❛❜✐❧✐t② ♦❢ ♠✐ss ✭ P ♠ ✮ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ●● ❛♥❞ ❈❛✉❝❤② s❤❛♣❡ ♣❛r❛♠❡t❡r ✭ ❝ ❛♥❞ γ ✱ r❡s♣✳✮ ❛t ✶✻ ❞❇ ❉❲❘ ❛♥❞ P ❢ = ✶✵ − ✻ ✳ ❈✐r❝❧❡ ✭ ◦ ✮ ❛♥❞ ❞✐❛♠♦♥❞ ✭ ⋄ ✮ ❞❡♥♦t❡ ▼▲ ❛♥❞ ❛♣♣r♦①✐♠❛t❡ ♣❛r❛♠❡t❡r ❡st✐♠❛t❡s✳