Tests for sphericity in multivariate garch models

Munich Personal RePEc Archive

Tests for sphericity in multivariate garch models

Francq, Christian and Jiménez Gamero, Maria Dolores and Meintanis, Simos

September 2015

Online at https://mpra.ub.uni-muenchen.de/67411/

MPRA Paper No. 67411, posted 23 Oct 2015 12:29 UTC

❚❡sts ❢♦r s♣❤❡r✐❝✐t② ✐♥ ♠✉❧t✐✈❛r✐❛t❡ ❣❛r❝❤ ♠♦❞❡❧s

❈✳ ❋r❛♥❝q

✱ ▼✳❉✳ ❏✐♠é♥❡③✕●❛♠❡r♦

✱ ❙✳●✳ ▼❡✐♥t❛♥✐s

a❈❘❊❙❚ ❛♥❞ ❯♥✐✈❡rs✐t② ▲✐❧❧❡ ✸✱ ❇P ✻✵✶✹✾✱ ✺✾✻✺✸ ❱✐❧❧❡♥❡✉✈❡ ❞✬❆s❝q ❝❡❞❡①✱ ❋r❛♥❝❡

b❉❡♣❛rt♠❡♥t ♦❢ ❊❝♦♥♦♠✐❝s✱ ◆❛t✐♦♥❛❧ ❛♥❞ ❑❛♣♦❞✐str✐❛♥ ❯♥✐✈❡rs✐t② ♦❢ ❆t❤❡♥s

❆t❤❡♥s✱ ●r❡❡❝❡

c❯♥✐t ❢♦r ❇✉s✐♥❡ss ▼❛t❤❡♠❛t✐❝s ❛♥❞ ■♥❢♦r♠❛t✐❝s✱ ◆♦rt❤✕❲❡st ❯♥✐✈❡rs✐t② P♦t❝❤❡❢str♦♦♠✱ ❙♦✉t❤ ❆❢r✐❝❛

d❉❡♣❛rt♠❡♥t ♦❢ ❙t❛t✐st✐❝s ❛♥❞ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ❯♥✐✈❡rs✐t② ♦❢ ❙❡✈✐❧❧❛

❙❡✈✐❧❧❛✱ ❙♣❛✐♥

❆❜str❛❝t✳ ❚❡sts ❢♦r s♣❤❡r✐❝❛❧ s②♠♠❡tr② ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ❛r❡ ♣r♦♣♦s❡❞ ✐♥

♠✉❧t✐✈❛r✐❛t❡ ●❆❘❈❍ ♠♦❞❡❧s✳ ❚❤❡ ♥❡✇ t❡sts ❛r❡ ♦❢ ❑♦❧♠♦❣♦r♦✈✕❙♠✐r♥♦✈ ❛♥❞ ❈r❛♠ér✕✈♦♥

▼✐s❡s✕t②♣❡ ❛♥❞ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❝♦♠♠♦♥ ❣❡♦♠❡tr② ✉♥❞❡r❧②✐♥❣ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢

❛♥② s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥✳ ❚❤❡ ❛s②♠♣t♦t✐❝ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡st st❛t✐st✐❝s

❛s ✇❡❧❧ ❛s t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ t❡sts ✐s ✐♥✈❡st✐❣❛t❡❞ ✉♥❞❡r ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥s✳ ■t ✐s s❤♦✇♥

t❤❛t ❜♦t❤ t❤❡ ✜♥✐t❡ s❛♠♣❧❡ ❛♥❞ t❤❡ ❛s②♠♣t♦t✐❝ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥ ❞❡♣❡♥❞ ♦♥ t❤❡ ✉♥❦♥♦✇♥

❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥s✳ ❚❤❡r❡❢♦r❡ ❛ ❝♦♥❞✐t✐♦♥❛❧ ▼♦♥t❡ ❈❛r❧♦

♣r♦❝❡❞✉r❡ ✐s ✉s❡❞ t♦ ❛❝t✉❛❧❧② ❝❛rr② ♦✉t t❤❡ t❡sts✳ ❚❤❡ ✈❛❧✐❞✐t② ♦❢ t❤✐s r❡s❛♠♣❧✐♥❣ s❝❤❡♠❡ ✐s

❢♦r♠❛❧❧② ❥✉st✐✜❡❞✳ ❘❡s✉❧ts ♦♥ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ t❡st ✐♥ ✜♥✐t❡✕s❛♠♣❧❡s ❛r❡ ✐♥❝❧✉❞❡❞✱ ❛s ✇❡❧❧

❛s ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦♥ ✜♥❛♥❝✐❛❧ ❞❛t❛✳

❑❡②✇♦r❞s✳ ❊①t❡♥❞❡❞ ❈❈❈✲●❆❘❈❍❀ ❙♣❤❡r✐❝❛❧ s②♠♠❡tr②❀ ❊♠♣✐r✐❝❛❧ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥❀

❈♦♥❞✐t✐♦♥❛❧ ▼♦♥t❡ ❈❛r❧♦ t❡st✳

❏❊▲ ❝❧❛ss✐✜❝❛t✐♦♥ ✿ ❈✶✷✱ ❈✶✺✱ ❈✸✷✱ ❈✺✽

✶❖♥ s❛❜❜❛t✐❝❛❧ ❧❡❛✈❡ ❢r♦♠ t❤❡ ❯♥✐✈❡rs✐t② ♦❢ ❆t❤❡♥s

✶

✶ ■♥tr♦❞✉❝t✐♦♥

❋♦r d ≥ 1✱ ❝♦♥s✐❞❡r t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ♦❜s❡r✈❛t✐♦♥ ✈❡❝t♦r y_t = (y_1t, . . . , y_dt)^′✱ ❢r♦♠ t❤❡

♠♦❞❡❧

y_t=C^1/2

t ε_t,

✭✶✳✶✮

✇❤❡r❡ t❤❡ ✭✉♥♦❜s❡r✈❡❞✮ r❛♥❞♦♠ ❡rr♦rs {ε_t}^t ✭❛❧s♦ r❡❢❡rr❡❞ t♦ ❛s ✐♥♥♦✈❛t✐♦♥s✮✱ ❛r❡ ✐♥✲

❞❡♣❡♥❞❡♥t ❛♥❞ ❢♦❧❧♦✇ ❛♥ ✉♥s♣❡❝✐✜❡❞ ❞✐str✐❜✉t✐♦♥ ✇❤✐❝❤ r❡♠❛✐♥s ✐♥✈❛r✐❛♥t ✇✐t❤ r❡s♣❡❝t t♦ t✐♠❡ t✱ ❛♥❞ ❤❛s ♠❡❛♥ ③❡r♦ ❛♥❞ ✐❞❡♥t✐t② ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✳ ❲❡ ❛ss✉♠❡ t❤❛t ❣✐✈❡♥

t❤❡ ✐♥❢♦r♠❛t✐♦♥ s❡t ❛✈❛✐❧❛❜❧❡ ❛t t✐♠❡ t✱ t❤❡ ❝♦♥❞✐t✐♦♥❛❧ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ y_t ❡q✉❛❧s C_t✱ C_t ❜❡✐♥❣ ❛ (d×d) s②♠♠❡tr✐❝ ❛♥❞ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ ♠❛tr✐①✳ ❚❤✐s ✐s t❤❡ s❡tt✐♥❣ ♦❢

t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ●❆❘❈❍ ✭▼●❆❘❈❍✮ ♠♦❞❡❧✱ ❛♥❞ ✉♥❞❡r t❤✐s ♠♦❞❡❧ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞

✐♥ t❡st✐♥❣ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ♦❢ s♣❤❡r✐❝❛❧ s②♠♠❡tr② ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥♥♦✈❛✲

t✐♦♥s✳ ❙♣❡❝✐✜❝❛❧❧②✱ ❛♥❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ ♦❜s❡r✈❛t✐♦♥s {y_t, t = 1, . . . , T} ❞r✐✈❡♥ ❜② t❤❡

❡q✉❛t✐♦♥ ✭✶✳✶✮✱ ✇❡ ✇✐s❤ t♦ t❡st t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s

H⁰ : t❤❡ ❧❛✇ ♦❢ {ε_t}^t ❜❡❧♦♥❣s t♦ t❤❡ ❢❛♠✐❧② ♦❢ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ❧❛✇s∈R^d,

❛❣❛✐♥st ❣❡♥❡r❛❧ ❛❧t❡r♥❛t✐✈❡s✳ ◆♦t❡ t❤❛t t❤❡ ❤②♣♦t❤❡s✐s t❤❛t {ε_t}t ❜❡❧♦♥❣s t♦ t❤❡ ❝❧❛ss

♦❢ s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ❞✐str✐❜✉t✐♦♥s ✭❙❙❉✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ss✉♠✐♥❣ t❤❛t t❤❡ ❝♦rr❡✲

s♣♦♥❞✐♥❣ ❞✐str✐❜✉t✐♦♥ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❣r♦✉♣ ♦❢ tr❛♥s❢♦r♠❛t✐♦♥sε_t7→Hε_t✱ ✇❤❡r❡

H ✐s ❛♥② ♦rt❤♦❣♦♥❛❧(d×d)✕♠❛tr✐①✳

❚❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H0 ✐♠♣❧✐❡s ❛ ♠♦❞❡❧ t❤❛t ❧✐❡s s♦♠❡✇❤❡r❡ ❜❡t✇❡❡♥ ❛ ❢✉❧❧② ♣❛r❛✲

♠❡tr✐❝ ▼●❆❘❈❍✱ ❛♥❞ ❛♥ ▼●❆❘❈❍ ♠♦❞❡❧ ✇✐t❤ ❛ ❝♦♠♣❧❡t❡❧② ✉♥s♣❡❝✐✜❡❞ ✐♥♥♦✈❛t✐♦♥

❞✐str✐❜✉t✐♦♥✳ ❖❢ ❝♦✉rs❡ ✐♥ t❤❡ ✐✳✐✳❞✳ s❡tt✐♥❣✱ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤❡ ❝❧❛ss ♦❢ ❙❙❉ ✐s

✇❡❧❧ ❦♥♦✇♥✿ ❙❡✈❡r❛❧ ♥♦t✐♦♥s ❛♥❞ ♣r♦❝❡❞✉r❡s ❡①t❡♥❞ ♥✐❝❡❧② ❢r♦♠ t❤❡ ❝❧❛ss✐❝❛❧ ●❛✉ss✐❛♥

❝♦♥t❡①t t♦ s♣❤❡r✐❝❛❧ s②♠♠❡tr②❀ s❡❡✱ ❢♦r ✐♥st❛♥❝❡✱ ❏♦♥❡s ✭✷✵✵✽✮✱ ❈❛❝♦✉❧❧♦s ✭✷✵✶✹✮✱ ❩✉♦

❛♥❞ ❙❡r✢✐♥❣ ✭✷✵✵✵✮✱ ❍❛❧❧✐♥ ❛♥❞ P❛✐♥❞❛✈❡✐♥❡ ✭✷✵✵✷✮✱ ❛♥❞ ❍❛❧❧✐♥ ❛♥❞ ❲❡r❦❡r ✭✷✵✵✸✮✳ ❖♥

t❤❡ ♦t❤❡r ❤❛♥❞✱ ❛♥❞ ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❞②♥❛♠✐❝ ♠♦❞❡❧s✱ ✐t ♠❛② ❜❡❡♥ s❤♦✇♥✱ s❡❡ ❡✳❣✳

❊♠❜r❡❝❤ts ❡t ❛❧✳ ✭✷✵✵✷✮ ❛♥❞ ❇❡r❦ ✭✶✾✾✼✮✱ t❤❛t ❛♥ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ❜❡❧♦♥❣✐♥❣

t♦ t❤❡ ❙❙❉ ❝❧❛ss r❡♥❞❡rs ♠♦❞❡❧ ✭✶✳✶✮ ❝♦♥✈❡♥✐❡♥t❧② ❛♠❡♥❛❜❧❡ t♦ st❛♥❞❛r❞ ❛♣♣r♦❛❝❤❡s

♦❢ r✐s❦ ♠❛♥❛❣❡♠❡♥t s✉❝❤ ❛s ❱❛❧✉❡✕❛t✕❘✐s❦ ❛♥❞ t❤❡ ♠❡❛♥✕✈❛r✐❛♥❝❡ ❛♣♣r♦❛❝❤ t♦ r✐s❦

♠❛♥❛❣❡♠❡♥t ❛♥❞ ♣♦rt❢♦❧✐♦ ♦♣t✐♠✐③❛t✐♦♥✳ ❍❡♥❝❡ s♣❤❡r✐❝❛❧ s②♠♠❡tr② ❤❛s ♦❢t❡♥ ❜❡❡♥ ❛

✷

♣♦✐♥t ♦❢ ❞❡♣❛rt✉r❡ ❢♦r ✜♥❛♥❝✐❛❧ ❞❛t❛✳ ■♥ ❢❛❝t✱ ♠❛♥② ❢✉❧❧② ♣❛r❛♠❡tr✐❝ ✈❡rs✐♦♥s ♠❛❦❡ ✉s❡

♦❢ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥s ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❙❙❉✳ ❊①❛♠♣❧❡s ❛r❡ t❤❡ ●❛✉ss✐❛♥

✭▼✮●❆❘❈❍ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ ❇❛✐ ❛♥❞ ❈❤❡♥ ✭✷✵✵✽✮✱ ▲❡❡ ❡t ❛❧✳ ✭✷✵✶✵✮✱ ❛♥❞ ▲❡❡ ❡t ❛❧✳

✭✷✵✶✹✮✱ ♦r ✐ts ❙t✉❞❡♥t✕t ❝♦✉♥t❡r♣❛rt✳ ❋♦r ❢✉rt❤❡r ❢❛♠✐❧✐❡s ❛♥❞ ❢♦r st❛t✐st✐❝❛❧ ♣r♦❝❡❞✉r❡s

✇✐t❤✐♥ ❞②♥❛♠✐❝ ♠♦❞❡❧s ✐♥✈♦❧✈✐♥❣ ❙❙❉s s❡❡ ❆♠❡♥❣✉❛❧ ❛♥❞ ❙❡♥t❛♥❛ ✭✷✵✶✶✮✱ ❛♥❞ ▲✐✉ ❡t

❛❧✳ ✭✷✵✶✶✮✳ ❋♦r ♠♦r❡ ❣❡♥❡r❛❧ s♣❡❝✐✜❝❛t✐♦♥ t❡sts ✐♥ ❝♦♥❞✐t✐♦♥❛❧ ♠♦❞❡❧s t❤❡ r❡❛❞❡r ✐s r❡✲

❢❡rr❡❞ t♦ ❉❡❧❣❛❞♦ ❛♥❞ ❙t✉t❡ ✭✷✵✵✽✮ ❛♥❞ ❑♦✉❧ ❛♥❞ ❙t✉t❡ ✭✶✾✾✾✮✳ ❆s ❛❧r❡❛❞② ♠❡♥t✐♦♥❡❞✱

❛♥ ▼●❆❘❈❍ ♠♦❞❡❧ ✇✐t❤ ❛ ❝♦♠♣❧❡t❡❧② ✉♥s♣❡❝✐✜❡❞ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ♠❛② ❛❧s♦

❜❡ ❡♥t❡rt❛✐♥❡❞❀ s❡❡ ❍❛❢♥❡r ❛♥❞ ❘♦♠❜♦✉ts ✭✷✵✵✼✮ ❢♦r ✐♥st❛♥❝❡✳ ❍♦✇❡✈❡r ❡✈❡♥ ✐♥ t❤✐s

❝❛s❡✱ ❍❛❢♥❡r ❛♥❞ ❘♦♠❜♦✉ts ✭✷✵✵✼✮ ❛ss✉♠❡ ❛♥ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ ✐♥ t❤❡ ❙❙❉ ❝❧❛ss

❢♦r t❤❡✐r ♥♦♥♣❛r❛♠❡tr✐❝ ❡st✐♠❛t♦r ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ t♦ ❛✈♦✐❞ t❤❡ ❵❝✉rs❡ ♦❢

❞✐♠❡♥s✐♦♥❛❧✐t②✬ ❛♥❞ ❝❛♣t✉r❡ t❤❡ ✉♥✐✈❛r✐❛t❡ ❝♦♥✈❡r❣❡♥❝❡ r❛t❡✳ ◆❡✈❡rt❤❡❧❡ss ❛♥❞ ❞❡s♣✐t❡

t❤❡ ♣♦♣✉❧❛r✐t② ♦❢ t❤❡ ❙❙❉ ❝❧❛ss✱ t❤❡r❡ ✐s r❡❝❡♥t❧② ❛ str♦♥❣ t❡♥❞❡♥❝② t♦ ❛❧❧♦✇ ❢♦r s❦❡✇✲

♥❡ss ✐♥ ●❆❘❈❍ ♠♦❞❡❧s ❢♦r ✜♥❛♥❝✐❛❧ r❡t✉r♥s✱ ❛♥❞ ♦♥❡ ✇❛② t♦ ❞♦ s♦ ✐s ✈✐❛ t❤❡ ❝♦♥❞✐t✐♦♥❛❧

❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s❀ s❡❡ ▼✐tt♥✐❦ ❛♥❞ P❛♦❧❡❧❧❛ ✭✷✵✵✵✮✱ ❇❛✉✇❡♥s ❛♥❞ ▲❛✉r❡♥t

✭✷✵✵✺✮✱ ❉❡ ▲✉❝❛ ❡t ❛❧✳ ✭✷✵✵✻✮✱ ❚r✐♥❞❛❞❡ ❛♥❞ ❩❤✉ ✭✷✵✵✼✮✱ ❍❛❛s ❡t ❛❧✳ ✭✷✵✵✾✮✱ ❛♥❞ ❈❤❡♥

❡t ❛❧✳ ✭✷✵✶✷✮✳ ❚❤✐s r❡❝❡♥t t❡♥❞❡♥❝② ✐♥ ❝♦♥❥✉♥❝t✐♦♥ ✇✐t❤ t❤❡ ❡❛r❧✐❡r ❜✐❛s t♦✇❛r❞s ❛ ❙❙❉

❢♦r t❤❡ ✐♥♥♦✈❛t✐♦♥s ♣r♦✈✐❞❡s t❤❡ ❣r♦✉♥❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ ✇❤✐❝❤ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H⁰

❝♦✉❧❞ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❤✐❣❤❧② r❡❧❡✈❛♥t✱ ♣❛rt✐❝✉❧❛r❧② ✐♥ st❛t✐st✐❝❛❧ ♠♦❞❡❧❧✐♥❣ ✇✐t❤ ❛ ✈✐❡✇

t♦✇❛r❞s ✜♥❛♥❝✐❛❧ ❛♣♣❧✐❝❛t✐♦♥s✳

❋♦r ✐✳✐✳❞✳ ❞❛t❛✱ t❤❡r❡ ❡①✐st s❡✈❡r❛❧ ✇♦r❦s ♦♥ t❡st✐♥❣ s♣❤❡r✐❝❛❧ s②♠♠❡tr②❀ s❡❡ ❢♦r

✐♥st❛♥❝❡ ❑♦❧t❝❤✐♥s❦✐✐ ❛♥❞ ▲✐ ✭✶✾✾✽✮✱ ❇❛r✐♥❣❤❛✉s ✭✶✾✾✶✮✱ ❑❛r✐②❛ ❛♥❞ ❊❛t♦♥ ✭✶✾✼✼✮ ❛♥❞

t❤❡ r❡✈✐❡✇ ❛rt✐❝❧❡ ❜② ▼❡✐♥t❛♥✐s ❛♥❞ ◆❣❛t❝❤♦✉✕❲❛♥❞❥✐ ✭✷✵✶✷✮✳ ❚❡sts ❢♦r ❝♦♥❞✐t✐♦♥❛❧

s②♠♠❡tr② ♠❛②❜❡ ❢♦✉♥❞ ✐♥ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✶✮ ❛♥❞ ❉❡❧❣❛❞♦ ❛♥❞ ❊s❝❛♥❝✐❛♥♦ ✭✷✵✵✼✮✳

❚❤❡ ♠❡t❤♦❞ ♣r❡s❡♥t❡❞ ❤❡r❡ ❤♦✇❡✈❡r ✐s r❡❧❛t❡❞ ♠♦r❡ ✇✐t❤ t❤❡ ❛♣♣r♦❛❝❤❡s s✉❣❣❡st❡❞

❜② ●❤♦s❤ ❛♥❞ ❘✉②♠❣❛❛rt ✭✶✾✾✷✮✱ ❉✐❦s ❛♥❞ ❚♦♥❣ ✭✶✾✾✾✮✱ ❩❤✉ ❛♥❞ ◆❡✉❤❛✉s ✭✷✵✵✵✮✱

❩❤✉ ✭✷✵✵✺✮ ❛♥❞ ❍❡♥③❡ ❡t ❛❧✳ ✭✷✵✶✹✮✳ ❚❤❡ ❝♦♠♠♦♥ t❤❡♠❡ ✐♥ ❛❧❧ t❤❡s❡ ✇♦r❦s ✐s t❤❛t t❤❡ ❛✉t❤♦rs ✉s❡ s♣❡❝✐✜❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ♦❢ ❙❙❉s ✐♥ t❤❡✐r t❡st st❛t✐st✐❝s✳

❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ ❡①t❡♥❞ t❤❡ t❡st ♣r♦❝❡❞✉r❡ ♦❢ ❍❡♥③❡ ❡t ❛❧✳ ✭✷✵✶✹✮

❢r♦♠ t❤❡ ✐✳✐✳❞✳ ❝♦♥t❡①t t♦ ♠♦❞❡❧s ✐♥✈♦❧✈✐♥❣ ❞❡♣❡♥❞❡♥❝❡✱ ✇✐t❤ s♣❡❝✐❛❧ ❡♠♣❤❛s✐s ♦♥

✸

▼●❆❘❈❍ ♠♦❞❡❧s✳ ■♥ ❞♦✐♥❣ s♦ ✇❡ ❞❡r✐✈❡ t❤❡ ❧✐♠✐t ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ♣r♦❝❡❞✉r❡ ✉♥❞❡r

●❆❘❈❍✲t②♣❡ ❞❡♣❡♥❞❡♥❝❡✳ ■♥ ❛❞❞✐t✐♦♥ ✇❡ s✉❣❣❡st ❛♥❞ s❤♦✇ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ ❛ ♠♦❞✲

✐✜❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ r❡s❛♠♣❧✐♥❣ ❝♦✉♥t❡r♣❛rt ♦❢ t❤❡ t❡st st❛t✐st✐❝ ❡♠♣❧♦②❡❞ ✐♥ ❍❡♥③❡ ❡t ❛❧✳

✭✷✵✶✹✮✳ ❆❧t❤♦✉❣❤ ✐♥ t❤❡ ♣r♦♦❢s ✇❡ ♠❛❦❡ ✉s❡ ♦❢ ❝♦♥st❛♥t ❝♦rr❡❧❛t✐♦♥s✱ ♦✉r s✐♠✉❧❛t✐♦♥s

❛❧s♦ ✐♥❝❧✉❞❡ t✐♠❡✕❞❡♣❡♥❞❡♥t ❝♦rr❡❧❛t✐♦♥s✳

■♥ ♦r❞❡r t♦ ✐♥tr♦❞✉❝❡ t❤❡ ♣r♦♣♦s❡❞ ♣r♦❝❡❞✉r❡✱ ❧❡tX ∈R^d ❜❡ ❛♥ ❛r❜✐tr❛r② r❛♥❞♦♠

✈❛r✐❛❜❧❡ ✇✐t❤ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❤❛r❛❝t❡r✐st✐❝ ❢✉♥❝t✐♦♥ ✭❈❋✮ ϕ(u) = E[exp(iu^′X)], u ∈ R^d✳ ❲❡ ✇✐❧❧ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ❙❙❉✿ ❚❤❡ ❈❋ ϕ(u)✐s t❤❡ ❈❋

♦❢ ❛ ❙❙❉ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts s♦♠❡ ❢✉♥❝t✐♦♥φ :R→R s✉❝❤ t❤❛t ϕ(u) = φ(kuk²),

✭✶✳✷✮

✇❤❡r❡ k·k ✐s t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ✐♥ R^d✳ ❈❤❛r❛❝t❡r✐③❛t✐♦♥ ✭✶✳✷✮ ♠❛② ❜❡ ❢♦✉♥❞ ✐♥ ❋❛♥❣

❡t ❛❧✳ ✭✶✾✾✵✮✱ t♦❣❡t❤❡r ✇✐t❤ ❛ ✇❡❛❧t❤ ♦❢ ♠❛t❡r✐❛❧ ♦♥ ❙❙❉✳

❆❧♦♥❣ t❤❡ ❧✐♥❡s ♣r♦♣♦s❡❞ ❜② ❍❡♥③❡ ❡t ❛❧✳ ✭✷✵✶✹✮✱ ✇❡ s✉❣❣❡st t♦ ✉s❡ t❤❡ ♣r♦❝❡ss

∆_T(u,v) = ϕ_T(u)−ϕ_T(v), u,v∈R^d,

✇❤❡r❡

ϕT(u) = 1 T

XT t=1

e^iu′εt,

✐s t❤❡ ❡♠♣✐r✐❝❛❧ ❈❋ ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥s ε_t = C^−1/2

t y_t, t = 1, . . . , T✳ ❚❤❡♥✱ ✐♥ ✈✐❡✇ ♦❢

❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✭✶✳✷✮ ❛♥❞ t❤❡ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ❡♠♣✐r✐❝❛❧ ❈❋✱ ✇❡ ❡①♣❡❝t t❤❛t ❢♦r ❧❛r❣❡

T✱ t❤❡ ✈❛❧✉❡ ♦❢∆T(u,v)s❤♦✉❧❞ ❜❡ ❝❧♦s❡ t♦ ③❡r♦ ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐sH⁰ ♣r♦✈✐❞❡❞

t❤❛t t❤✐s ✈❛❧✉❡ ✐s ❝♦♠♣✉t❡❞ ♦✈❡r ♣❛✐rs ♦❢ ♣♦✐♥tsu,vs✉❝❤ t❤❛t kuk=kvk✳

❙✐♥❝❡ε₁,· · · ,ε_T✱ ❛r❡ ✉♥♦❜s❡r✈❡❞✱ ❛♥② ❞❡❝✐s✐♦♥ r❡❣❛r❞✐♥❣ t❤❡ ✐♥♥♦✈❛t✐♦♥✕❞✐str✐❜✉t✐♦♥

s❤♦✉❧❞ ♥❛t✉r❛❧❧② ❜❡ ❜❛s❡❞ ♦♥ t❤❡ r❡s✐❞✉❛❧s e

ε_t =Ce^−1/2

t y_t, t= 1, . . . , T,

✇❤❡r❡ Ce_t ❞❡♥♦t❡s ❛♥ ❛♣♣r♦♣r✐❛t❡ ❡st✐♠❛t♦r ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① C_t t❤❛t ✇✐❧❧ ❜❡

❞❡t❛✐❧❡❞ ❧❛t❡r✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ❝♦♥s✐❞❡r t❡st st❛t✐st✐❝s ✐♥✈♦❧✈✐♥❣ t❤❡ ♣r♦❝❡ss

✭✶✳✸✮ DT(u,v) = ϕeT(u)−ϕeT(v),

✹

✇❤❡r❡

e

ϕT(u) = 1 T

XT t=1

e^iu′eεt,

✐s t❤❡ ❡♠♣✐r✐❝❛❧ ❈❋ ♦❢ t❤❡ r❡s✐❞✉❛❧seε_t, t= 1, . . . , T✳

❚❤❡ r❡♠❛✐♥❞❡r ♦❢ t❤✐s ♣❛♣❡r ✐s ♦✉t❧✐♥❡❞ ❛s ❢♦❧❧♦✇s✳ ■♥ ❙❡❝t✐♦♥ ✷✱ t❤❡ t❡st st❛t✐st✐❝s

❛r❡ ❞❡✜♥❡❞✱ ✇❤✐❧❡ ❙❡❝t✐♦♥ ✸ ✇❡ ❞✐s❝✉ss ♣r♦❝❡❞✉r❡s ♦❢ ❡st✐♠❛t✐♥❣ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① C_t✉♥❞❡r s♣❡❝✐✜❝ ✈❡rs✐♦♥s ♦❢ ▼●❆❘❈❍ ♠♦❞❡❧s✳ ■♥ ❙❡❝t✐♦♥ ✹ ❧❛r❣❡✕s❛♠♣❧❡ ♣r♦♣❡rt✐❡s ♦❢

t❤❡ ♣r♦♣♦s❡❞ t❡sts ❛r❡ st✉❞✐❡❞✱ ✇❤✐❧❡ ✐♥ ❙❡❝t✐♦♥ ✺ ✇❡ ✐♥tr♦❞✉❝❡ ❛♥❞ ♣r♦✈❡ t❤❡ ✈❛❧✐❞✐t② ♦❢

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

♦♥ ❛s②♠♣t♦t✐❝s ✐♥ ♦r❞❡r t♦ ❛❝t✉❛❧❧② ❝❛rr② ♦✉t t❤❡ t❡sts✳ ❙✐♠✉❧❛t✐♦♥s ❛♥❞ ❛ r❡❛❧ ❞❛t❛

❛♣♣❧✐❝❛t✐♦♥ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ ❙❡❝t✐♦♥ ✻✱ ✇❤✐❧❡ ✐♥ t❤❡ ❧❛st ♣❛rt ♦❢ t❤❡ ♣❛♣❡r ✐♥ ❙❡❝t✐♦♥ ✼

✇❡ ❞r❛✇ s♦♠❡ ❝♦♥❝❧✉s✐♦♥s ❛♥❞ ❝♦♥s✐❞❡r ♣♦ss✐❜❧❡ ❡①t❡♥s✐♦♥s✳ ❆❧❧ ♣r♦♦❢s✱ ❛s ✇❡❧❧ ❛s s♦♠❡

✐♥t❡r♠❡❞✐❛t❡ r❡s✉❧ts✱ ❛r❡ s❦❡t❝❤❡❞ ✐♥ ❙❡❝t✐♦♥ ✽✳

✷ ❚❡st st❛t✐st✐❝s

❲❡ ❝♦♥s✐❞❡r ❑♦❧♠♦❣♦r♦✈✲❙♠✐r♥♦✈ ✭❑❙✮ ❛♥❞ ❈r❛♠ér✕✈♦♥✕▼✐s❡s t②♣❡ ✭❈▼✮ t❡st st❛t✐s✲

t✐❝s ✐♥✈♦❧✈✐♥❣ t❤❡ ♣r♦❝❡ssDT(u,v)✳ ❙♣❡❝✐✜❝❛❧❧② ❛♥❞ s✐♥❝❡DT(·,·)✐s ❛ ❝♦♠♣❧❡① ❢✉♥❝t✐♦♥✱

❢♦r t❤❡ ♣✉r♣♦s❡ ♦❢ t❡st✐♥❣ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H⁰ ✇❡ s❤❛❧❧ ♠♦♥✐t♦r t❤❡ ✈❛❧✉❡s ♦❢ t❤❡

❢✉♥❝t✐♦♥ |DT(u,v)|✱ ♦✈❡r ♣❛✐rs ♦❢ ♣♦✐♥ts (u,v) ∈ R^d×R^d ✇❤✐❝❤ ❛r❡ ❡q✉✐❞✐st❛♥t ❢r♦♠

t❤❡ ♦r✐❣✐♥✳ ■♥t✉✐t✐✈❡❧② ❛♥❞ ✐♥ ✈✐❡✇ ♦❢ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ✭✶✳✷✮✱ ✇❡ ❡①♣❡❝t t❤❡s❡ ✈❛❧✉❡s t♦

❜❡ ❵s♠❛❧❧✬ ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ❛♥❞ ❛s T → ∞✱ ❛♥❞ ❝♦♥s❡q✉❡♥t❧② ❧❛r❣❡ ✈❛❧✉❡s ♦❢

t❤✐s ❢✉♥❝t✐♦♥ s❤♦✉❧❞ ❧❡❛❞ t♦ r❡❥❡❝t✐♦♥ ♦❢ H0✳ ❍♦✇❡✈❡r✱ ❛♥② t❡st st❛t✐st✐❝ ❝♦♥❞✉❝t❡❞ ♦♥

t❤❡ ❜❛s✐s ♦❢ t❤✐s ❝❤❛r❛❝t❡r✐③❛t✐♦♥ s❤♦✉❧❞✱ ❛t ❧❡❛st ✐♥ ♣r✐♥❝✐♣❧❡✱ t❛❦❡ ✐♥t♦ ❛❝❝♦✉♥t t❤❡

❢✉❧❧ ✈❛r✐❛t✐♦♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ♦✈❡r ❛❧❧ ♣♦ss✐❜❧❡ s✉❝❤ ♣❛✐rs(u,v)✳ ❆s ❛ ❝♦♠♣r♦♠✐s❡✱ ✇❡

❝❤♦♦s❡ ❢♦r ❛ ✜①❡❞ ✐♥t❡❣❡rK ≥1✱ ❛ ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥

{u₁, . . . ,u_K} ∈S

◦,

♦❢ ♣♦✐♥ts ❧②✐♥❣ ✐♥ t❤❡ ✉♥✐t s♣❤❡r❡ S

◦ := {u ∈ R^d : kuk = 1}✱ ❛♥❞ ✇❤✐❝❤ ❛r❡ s❝❛tt❡r❡❞

❛s ✉♥✐❢♦r♠❧② ❛s ♣♦ss✐❜❧❡ ♦✈❡r S_◦✳ ❲❡ s❤❛❧❧ ❜❛s❡ ♦✉r t❡st st❛t✐st✐❝s ♦♥ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢

|DT(u,v)|r❡❛❧✐③❡❞ ♦✈❡r ♣❛✐rs (u,v)∈R^d×R^d❧②✐♥❣ ✐♥ ❞✐r❡❝t✐♦♥s ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥

✺

s♣❡❝✐✜❡❞ ❜② t❤✐s ❝♦❧❧❡❝t✐♦♥✳ ❋♦r t❤❡ ❈▼ t❡st st❛t✐st✐❝ ✇❡ ❞♦ ♥♦t ❧✐♠✐t t❤❡ ❡①t❡♥❞ t♦

✇❤✐❝❤ t❤❡ ♣♦✐♥ts (u,v)✇✐❧❧ str❡t❝❤ ❛✇❛② ❢r♦♠ t❤❡ ♦r✐❣✐♥✳ ❋♦r t❤❡ ❑❙ st❛t✐st✐❝ ❤♦✇❡✈❡r

✇❡ r❡str✐❝t t❤✐s r❛♥❣❡ ❜② ❞❡✜♥✐♥❣✱ ❢♦r ❛ ✜①❡❞ ✐♥t❡❣❡rL≥1✱ ❛♥♦t❤❡r ✜♥✐t❡ ❝♦❧❧❡❝t✐♦♥ ♦❢

♣♦✐♥ts

0< ρ1 < ρ2 < . . . ρL <∞,

❛♥❞ ❜② ❝♦♥s✐❞❡r✐♥❣ t❤❡ ✈❛r✐❛t✐♦♥ ♦❢ |DT(u,v)| ♦✈❡r ♣♦✐♥ts (u,v) s✉❝❤ t❤❛t✱ kuk = kvk=ρℓ✱ ∀ℓ∈ {1, . . . , L}✳

❇❛s❡❞ ♦♥ t❤❡ ❛❜♦✈❡ ♥♦t❛t✐♦♥ ❛♥❞ r❡❛s♦♥✐♥❣ ✇❡ s✉❣❣❡st t♦ r❡❥❡❝t t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H0 ❢♦r ❧❛r❣❡ ✈❛❧✉❡s ♦❢ t❤❡ ❑❙ t❡st st❛t✐st✐❝

KST =√

T max

l=1,...,L max

j,m=1,...,K|DT(ρlu_j, ρlu_m)|.

▲✐❦❡✇✐s❡ t❤❡ ♣r♦♣♦s❡❞ ❈▼ t❡st st❛t✐st✐❝ ✐s ❞❡✜♥❡❞ ❛s✱

CMT =T Z _∞

0

XK j,m=1

|DT(ρu_j, ρu_m)|²

!

ω(ρ)dρ,

✇❤❡r❡ ω(·) ❞❡♥♦t❡s ❛ ♥♦♥♥❡❣❛t✐✈❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥ s❛t✐s❢②✐♥❣

✭✷✳✶✮ Z _∞

0

ω(ρ)dρ <∞,

Z _∞

0

ρ²ω(ρ)dρ <∞.

■♥ ❢❛❝t✱ ✐❢ ✇❡ ❧❡teε_st =eε_s−eε_t ❛♥❞

Iω(z) :=

Z _∞

0

cos(ρz)ω(ρ)dρ,

t❤❡♥ ❞✐r❡❝t ❝♦♠♣✉t❛t✐♦♥ ②✐❡❧❞s CM_T = 1

T XK j,m=1

XT s,t=1

I_ω(u^′

jeε_st) +I_ω(u^′

meε_st)−2I_ω(u^′

meε_s−u^′

jeε_t) ,

✇❤✐❝❤ s❤♦✇s t❤❛t ❛ s✉✐t❛❜❧❡ ❝❤♦✐❝❡ ♦❢ t❤❡ ✇❡✐❣❤t ❢✉♥❝t✐♦♥✱ s✉❝❤ ❛s ω(ρ) = e^−aρb✱ ✇✐t❤

a >0✱ ❛♥❞b = 1♦rb = 2✱ r❡♥❞❡rs t❤❡ ❈▼ t❡st st❛t✐st✐❝ ✐♥ ❛ ❝❧♦s❡❞ ❢♦r♠ ❝♦♥✈❡♥✐❡♥t ❢♦r

❝♦♠♣✉t❛t✐♦♥s✳ ◆♦t❡ t❤❛t ❜♦t❤ t❡st st❛t✐st✐❝s ❛r❡ ❝♦♠♣✉t❡❞ ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ r❡s✐❞✉❛❧s

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

❡st✐♠❛t❡ Ce_t ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① C_t✱ ❛s t❤✐s ❡st✐♠❛t❡ ✐s ✉s❡❞ ✐♥ ✭✶✳✸✮ ✐♥ ♦r❞❡r t♦

♦❜t❛✐♥ t❤❡ r❡s✐❞✉❛❧s✳ ❊st✐♠❛t✐♦♥ ♦❢C_t✇✐❧❧ ❜❡ ❝❛rr✐❡❞ ♦✉t ♥❡①t ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ s♣❡❝✐✜❝

▼●❆❘❈❍ str✉❝t✉r❡s✳

✻

✸ ❊st✐♠❛t✐♦♥ ✉♥❞❡r ●❆❘❈❍ ♠♦❞❡❧s

❚❤❡r❡ ❡①✐st s❡✈❡r❛❧ ✈❡rs✐♦♥s ♦❢ ▼●❆❘❈❍ ♠♦❞❡❧s✳ ❚❤❡ r❡❛❞❡r ✐s r❡❢❡rr❡❞ t♦ ❚s❛②

✭✷✵✶✹✮✱ ❋r❛♥❝q ❛♥❞ ❩❛❦♦ï❛♥ ✭✷✵✶✵✮ ❛♥❞ ❙✐❧✈❡♥♥♦✐♥❡♥ ❛♥❞ ❚❡räs✈✐rt❛ ✭✷✵✵✾✮ ❢♦r s♦♠❡

r❡❝❡♥t ❛❝❝♦✉♥ts✳ ❚♦ ✐♥tr♦❞✉❝❡ ▼●❆❘❈❍ ❝♦♥s✐❞❡r t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✐♥ ✭✶✳✶✮ ❛♥❞

✇r✐t❡ C_t := C_t(ϑ) t♦ ✐♥❞✐❝❛t❡ t❤❛t t❤✐s ♠❛tr✐① ❞❡♣❡♥❞s ♦♥ ❛ ♣❛r❛♠❡t❡r ✈❡❝t♦r ϑ✳

❉✐✛❡r❡♥t ✈❡rs✐♦♥s ♦❢ ▼●❆❘❈❍ ❞❡✈✐❛t❡ ✐♥ t❤❡ s♣❡❝✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❞❡♣❡♥❞❡♥❝❡ str✉❝t✉r❡

♦❢ C_t ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♣❛st✱ ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ ✐ss✉❡s ❜❡✐♥❣ t❤❡ ❞✐♠❡♥s✐♦♥❛❧✐t② ♦❢ t❤❡

♣❛r❛♠❡t❡r ϑ ✇✐t❤ ✐♥❝r❡❛s✐♥❣ ❞✐♠❡♥s✐♦♥ d✳ ❆ s♣❡❝✐✜❝ ✐♥st❛♥❝❡ ♦❢ ▼●❆❘❈❍ ✇❤✐❝❤

✐s ❜♦t❤ ✐♥t✉✐t✐✈❡❧② ❛♥❞ ❝♦♠♣✉t❛t✐♦♥❛❧❧② ❛ttr❛❝t✐✈❡ ✐s t❤❡ s♦✕❝❛❧❧❡❞ ❡①t❡♥❞❡❞ ❝♦♥st❛♥t

❝♦♥❞✐t✐♦♥❛❧ ❝♦rr❡❧❛t✐♦♥ ✭❊✮❈❈❈✕●❆❘❈❍✭p, q✮ ♠♦❞❡❧✳ ❚❤✐s s♣❡❝✐✜❝❛t✐♦♥ ✐s ❞❡✜♥❡❞ ❜② C_t =D_tRD_t,

✭✸✳✶✮

✇❤❡r❡D_t❛♥❞R❛r❡(d×d)♠❛tr✐❝❡s ✇✐t❤D_t❞✐❛❣♦♥❛❧ ❛♥❞R❜❡✐♥❣ ❛ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐①✳

■❢ A ✐s ❛ sq✉❛r❡ ♠❛tr✐①✱ t❤❡♥ ❞✐❛❣(A) ❞❡♥♦t❡s t❤❡ ✈❡❝t♦r ♦❢ t❤❡ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥ts ♦❢

A✳ ■❢ a✐s ❛ ✈❡❝t♦r✱ t❤❡♥ ❞✐❛❣(a)❞❡♥♦t❡s t❤❡ ❞✐❛❣♦♥❛❧ ♠❛tr✐① ✇❤♦s❡ ❞✐❛❣♦♥❛❧ ✐sa✳ ❚❤❡

♠❛tr✐① D_t ✐s r❡❧❛t❡❞ t♦ ❛ ✈♦❧❛t✐❧✐t② ✈❡❝t♦r σ_t=❞✐❛❣(D²

t) ❜②

✭✸✳✷✮ σ_t=b+ Xp

j=1

B_jy⁽²⁾_t−j + Xq

j=1

Γ_jσ_t−j,

y⁽²⁾_t = y_t⊙y_t✱ ⊙ ❞❡♥♦t✐♥❣ t❤❡ ❍❛❞❛♠❛r❞ ♣r♦❞✉❝t✱ t❤❛t ✐s✱ t❤❡ ❡❧❡♠❡♥t ❜② ❡❧❡♠❡♥t

♣r♦❞✉❝t✳ ■♥ ✭✸✳✷✮✱ t❤❡ ✈❡❝t♦r b ✐s ♦❢ ❞✐♠❡♥s✐♦♥ d ❛♥❞ ❤❛s ♣♦s✐t✐✈❡ ❡❧❡♠❡♥ts✱ ✇❤✐❧❡

{B_j}^pj=1 ❛♥❞ {Γ_j}^qj=1 ❛r❡ (d ×d) ♠❛tr✐❝❡s ✇✐t❤ ♥♦♥✕♥❡❣❛t✐✈❡ ❡❧❡♠❡♥ts✳ ❚❤❡ ❈❈❈✲

●❆❘❈❍ ♠♦❞❡❧ ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❜② ❇♦❧❧❡rs❧❡✈ ✭✶✾✾✵✮ ✇✐t❤ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s B_j

❛♥❞ Γ_j✳ ❲❡ ❝♦♥s✐❞❡r ❤❡r❡ t❤❡ ❡①t❡♥❞❡❞ ✈❡rs✐♦♥ ♦❢ ❏❡❛♥t❤❡❛✉ ✭✶✾✾✽✮✱ ✐♥ ✇❤✐❝❤ t❤❡

♠❛tr✐❝❡s ♦❢ ✭✸✳✷✮ ❛r❡ ❛❧❧♦✇❡❞ t♦ ❜❡ ♥♦♥❞✐❛❣♦♥❛❧✳

❚❤❡ ♠♦❞❡❧ ✭✸✳✷✮ ❝♦✉❧❞ ❜❡ ❡①t❡♥❞❡❞ ❜② ✐♥tr♦❞✉❝✐♥❣ ❛s②♠♠❡tr✐❡s✱ ❛s ✐♥ ❋r❛♥❝q ❛♥❞

❩❛❦♦ï❛♥ ✭✷✵✶✷✮✳ ❚❤✐s ✇♦✉❧❞ ♥♦t ❝❤❛♥❣❡ t❤❡ r❡s❛♠♣❧✐♥❣ s❝❤❡♠❡ t❤❛t ✇❡ ♣r♦♣♦s❡ ✐♥

❙❡❝t✐♦♥ ✺ ❜❡❧♦✇✱ ❜✉t ✇♦✉❧❞ ❡♥t❛✐❧ ❤❡❛✈✐❡r ♥♦t❛t✐♦♥ ❛♥❞ ❛❞❞✐t✐♦♥❛❧ t❡❝❤♥✐❝❛❧ ❞✐✣❝✉❧t✐❡s✳

❲❡ t❤❡r❡❢♦r❡ ❝♦♥❝❡♥tr❛t❡ ♦♥ t❤❡ ❢♦r♠✉❧❛t✐♦♥ ✭✸✳✷✮ ❢♦r t❤❡ t❤❡♦r❡t✐❝❛❧ r❡s✉❧ts✱ ❜✉t ✇❡

✇✐❧❧ ❝♦♥s✐❞❡r ❛❧t❡r♥❛t✐✈❡ ●❆❘❈❍ ❢♦r♠❛t✐♦♥s ✐♥ t❤❡ ❛♣♣❧✐❝❛t✐♦♥s✳

❆s ♦❜s❡r✈❡❞ ❜❡❢♦r❡✱ ❛♥ ❡st✐♠❛t♦r ♦❢ t❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s C_t✱ t = 1, . . . , T✱ ✐s r❡q✉✐r❡❞ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ r❡s✐❞✉❛❧s✳ ◆♦t❡ t❤❛t C_t ❞❡♣❡♥❞s ♦♥ {y_k, t −p ≤

✼

k ≤t−1} ❛♥❞ {σ_k, t−q≤k ≤t−1}✱ ✇❤❡r❡❛s ✇❡ ♦♥❧② ♦❜s❡r✈❡ y₁, . . . ,y_T✳ ❇❡❝❛✉s❡

♦❢ t❤✐s r❡❛s♦♥ ✐♥✐t✐❛❧ ✈❛❧✉❡s (y_1−p, . . . ,y₀)❛♥❞ (σe_1−q, . . . ,σe₀) ❛r❡ ♥❡❝❡ss❛r② ✐♥ ♦r❞❡r t♦

st❛rt t❤❡ r❡❝✉rs✐♦♥ ✐♠♣❧✐❡❞ ❜② ✭✸✳✶✮ ❛♥❞ ✭✸✳✷✮✱ ❛♥❞ ✇❡ s❤❛❧❧ ❞❡♥♦t❡ ❜②Ce_tt❤❡ ❝♦✈❛r✐❛♥❝❡

♠❛tr✐① ❝♦♠♣✉t❡❞ r❡❝✉rs✐✈❡❧② ♦♥ t❤❡ ❜❛s✐s ♦❢ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ✐♥✐t✐❛❧ ✈❛❧✉❡s✳

❚❤❡ st❛♥❞❛r❞ ❡st✐♠❛t✐♦♥ ♠❡t❤♦❞ ❢♦r t❤❡ ●❛✉ss✐❛♥ ▼●❆❘❈❍ ♠♦❞❡❧ ✐s ♠❛①✐♠✉♠

❧✐❦❡❧✐❤♦♦❞✳ ❍♦✇❡✈❡r✱ ✐t ❤❛s ❜❡❡♥ s❤♦✇♥ t❤❛t ❡✈❡♥ ✇✐t❤ ♥♦♥✕●❛✉ss✐❛♥ ✐♥♥♦✈❛t✐♦♥s✱ ✉♥❞❡r q✉✐t❡ ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥s✱ ♠❛①✐♠✐③✐♥❣ t❤❡ ●❛✉ss✐❛♥ ❧✐❦❡❧✐❤♦♦❞ ❧❡❛❞s t♦ ❛ ❝♦♥s✐st❡♥t ❛♥❞

❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ❡st✐♠❛t♦r ✭s❡❡✱ ❡✳❣✳✱ ❋r❛♥❝q ❛♥❞ ❩❛❦♦ï❛♥✱ ✷✵✶✵✮✳ ❚❤✐s ❡st✐♠❛t♦r

✐s ❝❛❧❧❡❞ t❤❡ q✉❛s✐✕▼▲❊ ✭◗▼▲❊✮✱ ❛♥❞ ✐s ❢♦r♠❛❧❧② ❞❡✜♥❡❞ ❛s b

ϑ_T =❛r❣ ♠❛①

ϑ∈Θ LT(ϑ),

✇❤❡r❡ Θ❞❡♥♦t❡s t❤❡ ♣❛r❛♠❡t❡r s♣❛❝❡✱

LT(ϑ) = −1 2

XT t=1

eℓ_t,

❛♥❞

eℓ_t :=eℓ_t(ϑ) = y^′_tCe⁻¹

t y_t+ log eC_t .

◆♦t❡ t❤❛t✱ ❛s ✐t ✐s ✇❡❧❧ ❦♥♦✇♥✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡s(y_1−p, . . . ,y₀)❛♥❞ (σe_1−q, . . . ,σe₀)❤❛✈❡

♥♦ ✐♥✢✉❡♥❝❡ ♦♥ t❤❡ ❛s②♠♣t♦t✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ◗▼▲❊✳

✹ ❆s②♠♣t♦t✐❝ ♣r♦♣❡rt✐❡s

❍❡r❡ ❛♥❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s✱ t❤❡ ♥♦t❛t✐♦♥−→^D ♠❡❛♥s ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ❞✐str✐❜✉t✐♦♥ ♦❢ r❛♥❞♦♠

❡❧❡♠❡♥ts ❛♥❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✱ o_P(1) st❛♥❞s ❢♦r ❛ ✈❡❝t♦r ❝♦♥s✐st✐♥❣ ♦❢ oP(1) ❡❧❡♠❡♥ts

❛♥❞ ❛❧❧ ❧✐♠✐ts ❛r❡ t❛❦❡♥ ✇❤❡♥ T → ∞✳ ❲❡ ♥♦✇ ❞✐st✐♥❣✉✐s❤ t❤❡ tr✉❡ ✈❛❧✉❡ ϑ₀ ♦❢ t❤❡

♣❛r❛♠❡t❡r ❛♥❞ ❛ ❣❡♥❡r✐❝ ❡❧❡♠❡♥t ϑ ♦❢ t❤❡ ♣❛r❛♠❡t❡r s♣❛❝❡ Θ✳ ❉❡♥♦t✐♥❣ ❜② rℓj t❤❡

❡❧❡♠❡♥t ♦❢ t❤❡ r♦✇ ℓ ❛♥❞ ❝♦❧✉♠♥j ♦❢ t❤❡ ♠❛tr✐① R✱ ✇❡ ❝❛♥ ✇r✐t❡

ϑ= (ϑ1, . . . , ϑs0)^′ = (b^′,✈❡❝^′(B₁), . . . ,✈❡❝^′(Γ_q),r^′)^′,

✇❤❡r❡r^′ = (r21, . . . , rd1, r32, . . . , rd,d−1)∈R^s2✱ s0 =s1+s2 ✇✐t❤ s1 =d+ (p+q)d² ❛♥❞

s2 =d(d−1)/2✳ ■❢ ♥❡❝❡ss❛r②✱ ✇❡ ✇r✐t❡Ce_t(ϑ) ♦rCe_t(y_t−1, . . . ,y₁;ϑ)✐♥st❡❛❞ ♦❢Ce_t✱ ❜✉t

✽

✇❡ ❦❡❡♣ t❤❡ s✐♠♣❧❡st ♥♦t❛t✐♦♥ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ ❝♦♥❢✉s✐♦♥✳ ❆ s✐♠✐❧❛r ❝♦♥✈❡♥t✐♦♥ ✐s ✉s❡❞

❢♦r ♦t❤❡r t❡r♠s✱ s✉❝❤ ❛s D_t(ϑ) ♦r Df_t(ϑ)✳ ❋♦r ❛♥② ♠❛tr✐① A = (aℓj)✱ ✇❡ ✇✐❧❧ ✉s❡ t❤❡

♥♦r♠ ❞❡✜♥❡❞ ❜② kAk=P

ℓ,j|aℓj|❀ ✐❢A ✐s ❛ ✈❡❝t♦r✱ kAk ❞❡♥♦t❡s t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠✳

❲❡ ♦❜t❛✐♥ t❤❡ ❛s②♠♣t♦t✐❝ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡st st❛t✐st✐❝s ✉♥❞❡r ❛♥ ❛r❜✐tr❛r②

❡st✐♠❛t♦rϑb_T ♦❢ t❤❡ ♣❛r❛♠❡t❡rϑ₀✳ ■♥ ❞♦✐♥❣ s♦ ✇❡ ❛ss✉♠❡ ❛♥ ❛s②♠♣t♦t✐❝ r❡♣r❡s❡♥t❛t✐♦♥

❢♦r ϑb_T ✇❤✐❝❤ ✐s r❡❧❛t✐✈❡❧② ❣❡♥❡r❛❧ ❛♥❞ ❛♣♣❧✐❡s t♦ ♠♦st ❡st✐♠❛t♦rs ♦❢ ✐♥t❡r❡st✱ s✉❝❤ ❛s t❤❡ ◗▼▲❊ ✭s❡❡ ▲❡♠♠❛ ✹✳✷✮✳ ❆❧s♦ ✐♥ ♦r❞❡r t♦ ♦❜t❛✐♥ ❝♦♥s✐st❡♥❝② ♦❢ t❤❡ ♣r♦♣♦s❡❞ t❡sts

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

t♦ t❤❡ ❢❛♠✐❧② ♦❢ ❙❙❉✳ ■♥ ♣❛rt✐❝✉❧❛r✿

✭❆✳✶✮ ❆ss✉♠❡ t❤❛t t❤❡ ❡st✐♠❛t♦r ✐s str♦♥❣❧② ❝♦♥s✐st❡♥t ❛♥❞ ❛❞♠✐ts t❤❡ ❢♦❧❧♦✇✐♥❣ r❡♣r❡✲

s❡♥t❛t✐♦♥✱

√T(ϑb_T −ϑ₀) = 1

√T XT

t=1

ψ_0,t−1g_0t+o_P(1),

✇❤❡r❡g_0t:=g(ϑ0,ε_t) ✐s ❛ ✈❡❝t♦r ♦❢d² ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛tE(g_0t) = 0 ❛♥❞ E(g^′_0tg_0t)² < ∞✱ ❛♥❞ ψ_0t := ψ(ϑ0;ε_t,ε_t−1, . . .) ✐s ❛ s0 × d² ♠❛tr✐① ♦❢

♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥s s✉❝❤ t❤❛t Ekψ_0tψ^′_0tk² <∞✳

✭❆✳✷✮ ❆ss✉♠❡ t❤❛t ✉♥❞❡r ❛ ✜①❡❞ ❛❧t❡r♥❛t✐✈❡ ❞✐str✐❜✉t✐♦♥✱ t❤❡ ❈❋ ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥

❞✐str✐❜✉t✐♦♥ s❛t✐s✜❡s ϕ(ρ₀u₀)6=ϕ(ρ₀v₀),

❢♦r s♦♠❡ u₀,v₀ ∈ {u₁, . . . ,u_K}❛♥❞ s♦♠❡ ρ0 ∈ {ρ1, . . . , ρL}✳

❚❤❡ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ✭✉♥✐q✉❡✮ str✐❝t❧② st❛t✐♦♥❛r②

✭❛♥❞ ♥♦♥ ❛♥t✐❝✐♣❛t✐✈❡✮ s♦❧✉t✐♦♥ t♦ t❤❡ ❈❈❈✲●❆❘❈❍ ♠♦❞❡❧ ❞❡✜♥❡❞ ❜② ✭✶✳✶✮ ❛♥❞ ✭✸✳✶✮✲

✭✸✳✷✮ ✐s γ0 < 0✱ ✇❤❡r❡ γ0 ✐s t❤❡ t♦♣✲▲②❛♣♦✉♥♦✈ ❡①♣♦♥❡♥t ♦❢ t❤❡ ♠♦❞❡❧ ✭❛s ❞❡✜♥❡❞

❜② ✭✷✳✷✸✮ ❛♥❞ ✭✶✶✳✸✻✮ ✐♥ ❋r❛♥❝q ❛♥❞ ❩❛❦♦ï❛♥✱ ✷✵✶✵✮✳ ❚❤❡ ♥✉♠❜❡r γ0 ❞❡♣❡♥❞s✱ ✐♥ ❛

♥♦♥ ❡①♣❧✐❝✐t ✇❛②✱ ♦♥ ϑ₀ ❛♥❞ ♦♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ε_t ❛♥❞ ✐ts ✈❛❧✉❡ ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞

❜② ▼♦♥t❡ ❈❛r❧♦ s✐♠✉❧❛t✐♦♥s✳ ❲❡ ✇✐❧❧ ❛❧s♦ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛r❛♠❡t❡r ✐s ✐❞❡♥t✐✜❛❜❧❡

✭✉♥✐q✉❡♥❡ss ♦❢ t❤❡ ♣❛r❛♠❡tr✐③❛t✐♦♥✮✳ ❙❡✈❡r❛❧ t②♣❡s ♦❢ ❝♦♥❞✐t✐♦♥s ❝❛♥ ❜❡ ✉s❡❞ t♦ ❡♥s✉r❡

✐t✳ ❍❡r❡ ✇❡ ✇✐❧❧ ❛ss✉♠❡ t❤❛t ❆ss✉♠♣t✐♦♥ ❆✹ ❜❡❧♦✇ ❤♦❧❞s✳ ❆❧t❤♦✉❣❤ ❛ ❜✐t r❡str✐❝t✐✈❡✱

✾

✐t ✐s q✉✐t❡ s✐♠♣❧❡❀ ❢♦r ✇❡❛❦❡r ❛❧t❡r♥❛t✐✈❡ ❝♦♥❞✐t✐♦♥s ❡♥s✉r✐♥❣ t❤❡ ✐❞❡♥t✐✜❛❜✐❧✐t② s❡❡ ❢♦r

✐♥st❛♥❝❡✱ ❘❡✐♥s❡❧✱ ✶✾✾✼✱ ♣✳ ✸✼✕✹✵✳ ❉❡♥♦t❡ ❜② I_d t❤❡ d×d ✐❞❡♥t✐t② ♠❛tr✐①✱ ❛♥❞ ❜② e_j t❤❡j✲t❤ ❝♦❧✉♠♥ ♦❢I_d✳ ▲❡tB^ϑ(z) = Pp

j=1B_jz^j ✇❤❡♥p >0❛♥❞G^ϑ(z) = I_d−Pq

j=1Γ_jz^j

✇❤❡♥ q >0✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ ❛ss✉♠♣t✐♦♥s ✇✐❧❧ ❜❡ ❛ss✉♠❡❞ t♦ ❞❡r✐✈❡ ❛❧❧ r❡s✉❧ts ✐♥ t❤✐s s❡❝t✐♦♥✳

❆✶✿ ϑ₀ ∈Θ❛♥❞Θ✐s ❛ ❝♦♠♣❛❝t s✉❜s❡t ♦❢(0,+∞)^d×[0,+∞)^d2(p+q)×(−1,1)^d(d−1)/2✳

❆✷✿ γ0 <0 ❛♥❞ ∀ϑ ∈Θ, |G^ϑ(z)|= 0⇒ |z|>1.

❆✸✿ ❋♦rj = 1, . . . , dt❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢e^′

jε₁ ✐s ♥♦t ❝♦♥❝❡♥tr❛t❡❞ ♦♥ t✇♦ ♣♦✐♥ts ❛♥❞

P(e^′

jε₁ >0)∈(0,1)✳

❆✹✿ ■❢ q >0 t❤❡♥ B^ϑ0(1) 6= 0✱ t❤❡ ♣♦❧②♥♦♠✐❛❧s B^ϑ0(z) ❛♥❞ G^ϑ0(z)❛r❡ ❧❡❢t ❝♦♣r✐♠❡

❛♥❞ t❤❡ ♠❛tr✐①[B_0pΓ_0q] ❤❛s ❢✉❧❧ r❛♥❦d✳

❆✺✿ R ✐s ❛ ♣♦s✐t✐✈❡✲❞❡✜♥✐t❡ ❝♦rr❡❧❛t✐♦♥ ♠❛tr✐① ❢♦r ❛❧❧ ϑ∈Θ.

❆✻✿ ϑ₀ ∈Θ✱ ✇❤❡r❡^◦ Θ^◦ ✐s t❤❡ ✐♥t❡r✐♦r ♦❢ Θ✳

❆✼✿ Ekε_tε^′_tk² <∞.

❚❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❣✐✈❡s t❤❡ ❛s②♠♣t♦t✐❝ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡st st❛t✐st✐❝s KST ❛♥❞ CMT✳

❚❤❡♦r❡♠ ✹✳✶ ▲❡t y₁, . . . ,y_T✱ ❢♦❧❧♦✇ ❛ ▼●❆❘❈❍ ♠♦❞❡❧ ❛s s♣❡❝✐✜❡❞ ❜② ✭✶✳✶✮✱ ✭✸✳✶✮

❛♥❞ ✭✸✳✷✮✱ ❛♥❞ ❛ss✉♠❡ t❤❛t ✭✷✳✶✮ ❛♥❞ ❆✶✲❆✼ ❤♦❧❞✳ ❆ss✉♠❡ t❤❛t ϑb_T s❛t✐s✜❡s ✭❆✳✶✮✳

❚❤❡♥ ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H0✱

✭✹✳✶✮ KST D

−→ max

l=1,...,L max

j,m=1,...,K|W(ρlu_j, ρlu_m)|

❛♥❞

✭✹✳✷✮ CM_T −→^D Z _∞

0

XK j,m=1

|W(ρu_j, ρu_m)|²

!

ω(ρ)dρ,

✇❤❡r❡ W = {W(u,v); u,v ∈ R^d} ✐s ❛ ❝♦♠♣❧❡① ✈❛❧✉❡❞ ③❡r♦✕♠❡❛♥ ●❛✉ss✐❛♥ r❛♥❞♦♠

✜❡❧❞ ✇✐t❤ ❝♦✈❛r✐❛♥❝❡ ❦❡r♥❡❧ ❡q✉❛❧ t♦ t❤❛t ♦❢

✭✹✳✸✮ g1(u)−g1(v),

✶✵

✇❤❡r❡ gt(u) = e^iu′εt +g^′_0tψ^′_0,t−1ϕ(˙ u) ❛♥❞ ϕ(˙ ·) ✐s ❛ r❡❛❧ ✈❡❝t♦r ❞❡✜♥❡❞ ✐♥ t❤❡ ♣r♦♦❢✳

❇❡❝❛✉s❡ ♦❢ ✐ts ❝♦♥✈❡♥✐❡♥t ♣r♦♣❡rt✐❡s✱ ❛ ❝♦♠♠♦♥❧② ✉s❡❞ ❡st✐♠❛t♦r ♦❢ϑ✐s t❤❡ ◗▼▲❊✳

❚❤❡ ♥❡①t ▲❡♠♠❛ s❤♦✇s t❤❛t ✐t s❛t✐s✜❡s ✭❆✳✶✮✳

▲❡♠♠❛ ✹✳✷ ❯♥❞❡r ❆ss✉♠♣t✐♦♥s ❆✶✲❆✼✱ t❤❡ ◗▼▲❊ ϑb_T s❛t✐s✜❡s ✭❆✳✶✮✳

❚❤❡ ❧❛st r❡s✉❧t ✐♥ t❤✐s s❡❝t✐♦♥ ❣✐✈❡s t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ t❡st st❛t✐st✐❝s KST ❛♥❞ CMT ✉♥❞❡r ❛❧t❡r♥❛t✐✈❡s✳

❚❤❡♦r❡♠ ✹✳✸ ❙✉♣♣♦s❡ t❤❛t t❤❡ ❛ss✉♠♣t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✹✳✶ ❛r❡ s❛t✐s✜❡❞✱ ❜✉t ✐♥st❡❛❞

♦❢ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H0✱ ❝♦♥s✐❞❡r ❛♥② ✜①❡❞ ❛❧t❡r♥❛t✐✈❡ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ s❛t✐s✲

❢②✐♥❣ ✭❆✳✷✮✳ ❚❤❡♥ ✇❡ ❤❛✈❡

✭✹✳✹✮ lim inf KST

√T ≥ |ϕ(ρ0u₀)−ϕ(ρ0v₀)|,

❛♥❞

✭✹✳✺✮ lim inf CMT

T ≥

Z _∞

0 |ϕ(ρu₀)−ϕ(ρv₀)|²ω(ρ)dρ,

❛❧♠♦st s✉r❡❧②✳

❘❡♠❛r❦ ✹✳✹ ❆s ❛ r❡s✉❧t ♦❢ ❚❤❡♦r❡♠s ✹✳✶ ❛♥❞ ✹✳✸✱ t❤❡ t❡st ✇❤✐❝❤ r❡❥❡❝ts t❤❡ ♥✉❧❧

❤②♣♦t❤❡s✐s H0 ❢♦r ❧❛r❣❡ ✈❛❧✉❡s ♦❢ t❤❡ t❡st st❛t✐st✐❝ KST ✭r❡s♣✳ CMT✮ ✐s ❝♦♥s✐st❡♥t

❛❣❛✐♥st ❡❛❝❤ ♥♦♥✕s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝ ❛❧t❡r♥❛t✐✈❡ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ s❛t✐s❢②✐♥❣

✭❆✳✷✮✳

✺ ❆ ❝♦♥❞✐t✐♦♥❛❧ r❡s❛♠♣❧✐♥❣ s❝❤❡♠❡

❇♦t❤ t❤❡ ✜♥✐t❡✕s❛♠♣❧❡ ❛♥❞ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ t❡st st❛t✐st✐❝s ✉♥❞❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s H0 ♦❢ s♣❤❡r✐❝❛❧ s②♠♠❡tr② ❞❡♣❡♥❞ ♦♥ t❤❡ ✉♥❦♥♦✇♥ ❞✐str✐❜✉t✐♦♥

♦❢ t❤❡ ❊✉❝❧✐❞❡❛♥ ♥♦r♠ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ r❛♥❞♦♠ ✈❡❝t♦r ε_t✳ ❋r♦♠ ❚❤❡♦r❡♠ ✹✳✶ ✐t ✐s

❛❧s♦ ❝❧❡❛r t❤❛t t❤❡s❡ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥s ❛❧s♦ ❞❡♣❡♥❞ ♦♥ t❤❡ ❡st✐♠❛t♦r ♦❢ ϑ ❡♠♣❧♦②❡❞✳

❆ ✇❡❧❧ ❦♥♦✇♥ r❡s✉❧t ✇❤✐❝❤ ✇✐❧❧ ❜❡ ✉s❡❞ ❜❡❧♦✇ ✐s t❤❛t ε_t = kε_tk(ε_t/kε_tk) ❛♥❞ t❤❛t

✉♥❞❡r H0✱ kε_tk ❛♥❞ ε_t/kε_tk ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ❛♥❞ t❤❡ ❧❛tt❡r r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❢♦❧❧♦✇s

❛ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥ ♦✈❡r t❤❡ ✉♥✐t s♣❤❡r❡ S_◦✳ ■♥ ✈✐❡✇ ♦❢ t❤❡s❡ ♦❜s❡r✈❛t✐♦♥s✱ ✇❡

✶✶

❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥❛❧ r❡s❛♠♣❧✐♥❣ s❝❤❡♠❡✱ ❣✐✈❡♥ t❤❡ ❞❛t❛y₁, . . . ,y_T✱ ✇❤❡r❡

❢♦r s✐♠♣❧✐❝✐t② ✇❡ ✇r✐t❡T ❢♦r t❤❡ t❡st st❛t✐st✐❝✿

✭✐✮ ❈❛❧❝✉❧❛t❡ ϑb_T = ϑb_T(y₁, . . . ,y_T)✱ t❤❡ r❡s✐❞✉❛❧s eε₁, . . . ,eε_T ❛♥❞ t❤❡ t❡st st❛t✐st✐❝

T :=T(eε₁, . . . ,eε_T)✳

✭✐✐✮ ●❡♥❡r❛t❡ ✈❡❝t♦rss^∗_t, t= 1, . . . , T✱ t❤❛t ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞

♦♥ S

◦✱ ✐♥❞❡♣❡♥❞❡♥t❧② ❣❡♥❡r❛t❡ ✈❡❝t♦rs ε^∗_t, t = 1, . . . , T✱ t❤❛t ❛r❡ ✐♥❞❡♣❡♥❞❡♥t

❛♥❞ ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ {eε₀₁, . . . ,eε_0T}✱ ✇❤❡r❡ eε_0j = S_T^−1/2(eε_j −eε_.)✱ eε_. ✐s t❤❡ s❛♠♣❧❡ ♠❡❛♥ ♦❢ t❤❡ r❡s✐❞✉❛❧s ❛♥❞ ST ✐s t❤❡ s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢

t❤❡ r❡s✐❞✉❛❧s✱ ❝♦♠♣✉t❡ ε^∗_t = kε^∗_tks^∗_t ❛♥❞ ❧❡t y^∗_t = Ce^1/2_t (y^∗_t−1, . . . ,y^∗₁;ϑb_T)ε^∗_t, t = 1, . . . , T✳

✭✐✐✐✮ ❈❛❧❝✉❧❛t❡ ϑb^∗_T =ϑb_T(y^∗₁, . . . ,y^∗_T)✱ t❤❡ r❡s❛♠♣❧✐♥❣ r❡s✐❞✉❛❧s e

ε^∗_t =Ce⁻_t^1/2(y^∗_t−1, . . . ,y^∗₁;ϑb^∗_T)y^∗_t, t= 1, . . . , T

❛♥❞ t❤❡ t❡st st❛t✐st✐❝ T^∗ =T(eε^∗₁, . . . ,eε^∗_T)✳

✭✐✈✮ ❘❡♣❡❛t st❡♣s ✭✐✐✮ ❛♥❞ ✭✐✐✐✮ ❛ ♥✉♠❜❡r ♦❢ t✐♠❡s B ❛♥❞ ❝❛❧❝✉❧❛t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

t❡st st❛t✐st✐❝ ✈❛❧✉❡s T1^∗, . . . ,TB^∗✳

✭✈✮ ❘❡❥❡❝t t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ✐❢ T > T(B^∗−[αB])✱ ✇❤❡r❡ T(1)^∗ ≤ . . . ≤ T(B)^∗ ❞❡♥♦t❡ t❤❡

❝♦rr❡s♣♦♥❞✐♥❣ ♦r❞❡r st❛t✐st✐❝s ❛♥❞ α ❞❡♥♦t❡s t❤❡ ♣r❡s❝r✐❜❡❞ ❧❡✈❡❧ ♦❢ s✐❣♥✐✜❝❛♥❝❡✳

▲❡t ✉s ✇r✐t❡ε^∗_t,T ❛♥❞ y^∗_t,T ✐♥st❡❛❞ ♦❢ε^∗_t ❛♥❞ y^∗_t ✇❤❡♥ ✐t ✐s ✉s❡❢✉❧ t♦ ❡♠♣❤❛s✐③❡ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥s ♦❢ t❤❡s❡ r❛♥❞♦♠ ✈❡❝t♦rs ❞❡♣❡♥❞ ♦♥ T✳ ❖❜s❡r✈❡ t❤❛t E ε^∗_1,T |y

= E(ε1) = 0 ❛♥❞ E ε^∗_1,Tε^∗_1,T^′ |y

=E(ε1ε^′₁) =I_d✳

◆♦t❡ t❤❛t✱ ❛t ❧❡❛st ✐♥ t❤❡ ❝❛s❡ ✇❤❡r❡ϑb_T ✐s t❤❡ ◗▼▲❊✱ ♦♥❡ ❝❛♥ ❛✈♦✐❞ t❤❡ ❝♦st❧② ♦♣✲

t✐♠✐③❛t✐♦♥s r❡q✉✐r❡❞ ✐♥ st❡♣ ✭✐✐✐✮✳ ■♥❞❡❡❞ t❤❡ ◗▼▲❊ ✐s ♦❜t❛✐♥❡❞ ❜② ✐t❡r❛t✐♥❣ ❛ ◆❡✇t♦♥✲

❘❛♣❤s♦♥ ❡q✉❛t✐♦♥✳ ❆ st❛♥❞❛r❞ s♦❧✉t✐♦♥ ❢♦r ❛✈♦✐❞✐♥❣ t❤❡ ♦♣t✐♠✐③❛t✐♦♥ ✭s❡❡ ❡✳❣✳ ❑r❡✐ss

❡t ❛❧✳ ✭✷✵✶✶✮✱ ❙❤✐♠✐③✉ ✭✷✵✶✸✮✱ ❋r❛♥❝q ❡t ❛❧✳ ✭✷✵✶✹✮✱ ❛♥❞ t❤❡ r❡❢❡r❡♥❝❡s t❤❡r❡✐♥✮ ❝♦♥s✐sts

✐♥ ❜♦♦tstr❛♣♣✐♥❣ ❛ s✐♥❣❧❡ ◆❡✇t♦♥✲❘❛♣❤s♦♥ ✐t❡r❛t✐♦♥✳ ■♥ t❤❡ ♣r❡s❡♥t ❢r❛♠❡✇♦r❦✱ ✉s✐♥❣

t❤❡ ♥♦t❛t✐♦♥s ✐♥ t❤❡ ♣r♦♦❢ ♦❢ ▲❡♠♠❛ ✹✳✷✱ t❤✐s ❧❡❛❞s t♦ s❡t

✭✺✳✶✮ ϑb^∗_T =ϑb_T + 1 T

XT t=1

ψe_t−1eg^∗_t, ψe_t−1 =−J⁻¹

T ∆e_t−1,

✶✷

✇❤❡r❡

J_T = 1 T

XT t=1

∂²ℓet(ϑb_T)

∂ϑ∂ϑ^′ , ge^∗_t =✈❡❝

I_d−Rb^1/2ε^∗_tε^∗_t^′Rb^−1/2 , e^′

j∆e_t

−1 = 2✈❡❝

b D⁻¹

t Db^(j)

t

❢♦r j = 1, . . . , s₁✱ ❛♥❞ e^′

j∆e_t

−1 = ✈❡❝

b

R⁻¹Rb^(j)

❢♦r j = s1+ 1, . . . , s0✱ ✇✐t❤Rb =Re(ϑb_T)✱Db_t=De_t(ϑb_T)❛♥❞ s✐♠✐❧❛r ♥♦t❛t✐♦♥s ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡s✳

❚❤❡ ♥❡①t r❡s✉❧t ❣✐✈❡s t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♣r♦♣♦s❡❞ t❡st st❛t✐st✐❝s ✇❤❡♥

❡✈❛❧✉❛t❡❞ ♦♥ t❤❡ ❜♦♦tstr❛♣ r❡s✐❞✉❛❧s✱ s❛② KS^∗_T ❛♥❞ CM^∗_T✳ ▲❡t ε₀₁ ❜❡ ❞✐str✐❜✉t❡❞ ❛s kε₁ks✱ ✇✐t❤ s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ S_◦ ❛♥❞ ✐♥❞❡♣❡♥❞❡♥t ♦❢ kε₁k✳ ❖❜s❡r✈❡ t❤❛t ✐❢

t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ❤♦❧❞s t❤❡♥ ε₁ ❛♥❞ ε₀₁ ❜♦t❤ ❤❛✈❡ t❤❡ s❛♠❡ ❞✐str✐❜✉t✐♦♥✳ ▲❡tϕ0(u)

❞❡♥♦t❡ t❤❡ ❈❋ ♦❢ ε₀₁✳

❚❤❡♦r❡♠ ✺✳✶ ▲❡ty₁, . . . ,y_T ❢♦❧❧♦✇ ❛ ▼●❆❘❈❍ ♠♦❞❡❧ ❛s s♣❡❝✐✜❡❞ ❜② ✭✶✳✶✮✱ ✭✸✳✶✮ ❛♥❞

✭✸✳✷✮✱ ❛♥❞ ❛ss✉♠❡ t❤❛t ✭✷✳✶✮ ❛♥❞ ❆✶✲❆✼ ❤♦❧❞✳ ❆ss✉♠❡ t❤❛t t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ kε₁k

❛❞♠✐ts ❛ ❜♦✉♥❞❡❞ ❞❡♥s✐t② f ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ▲❡❜❡s❣✉❡ ♠❡❛s✉r❡✳ ❆ss✉♠❡ t❤❛t ϑb_T ✐s t❤❡ ◗▼▲❊✱ ❛♥❞ t❤❛t t❤❡ ❝♦♥❞✐t✐♦♥ ϑb^∗_T = ϑb_T(y^∗₁, . . . ,y^∗_T) ✐♥ ✭✐✐✐✮ ✐s r❡♣❧❛❝❡❞ ❜② ✭✺✳✶✮✳

❚❤❡♥ ❢♦r ❛❧♠♦st ❛❧❧ s❡q✉❡♥❝❡s y={y_t}s❛t✐s❢②✐♥❣ ✭✶✳✶✮✱ ✭✸✳✶✮✱ ✭✸✳✷✮✱ ❛♥❞ ❝♦♥❞✐t✐♦♥❛❧❧②

♦♥ y✱

✭✺✳✷✮ KS^∗_T −→^D max

l=1,...,L max

j,m=1,...,K|W0(ρlu_j, ρlu_m)|,

❛♥❞

✭✺✳✸✮ CM^∗_T −→^D Z _∞

0

XK j,m=1

|W0(ρu_j, ρu_m)|²

!

ω(ρ)dρ,

✇❤❡r❡ W0 ✐s ❛s ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✹✳✶ ✇❤❡♥ t❤❡ ✐♥♥♦✈❛t✐♦♥s ❛r❡ ❞✐str✐❜✉t❡❞ ❛sε₀₁✳

❚❤❡ r❡s✉❧t ✐♥ ❚❤❡♦r❡♠ ✺✳✶ ✐s ✈❛❧✐❞ ✇❤❡t❤❡r t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ✐s tr✉❡ ♦r ♥♦t✳

❲❤❡♥ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ✐s tr✉❡✱ t❤❡ ❧✐♠✐ts ✐♥ ❚❤❡♦r❡♠s ✹✳✶ ❛♥❞ ✺✳✶ ❝♦✐♥❝✐❞❡✱ ❛♥❞ t❤✉s t❤❡ ♣r♦♣♦s❡❞ ❜♦♦tstr❛♣ ♣r♦✈✐❞❡s ❛ ❝♦♥s✐st❡♥t ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ♥✉❧❧ ❞✐str✐❜✉t✐♦♥ ♦❢

t❤❡ t❡st st❛t✐st✐❝s✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ t❤❡ ♥✉❧❧ ❤②♣♦t❤❡s✐s ✐s ♥♦t tr✉❡✱ ✇❡ st✐❧❧ ❤❛✈❡

t❤❛t KS^∗_T = OP(1) ❛♥❞ CM^∗_T = OP(1)✳ ■♥ ✈✐❡✇ ♦❢ t❤❡ r❡s✉❧t ✐♥ ❚❤❡♦r❡♠ ✹✳✸✱ ✐t ✐s

❝♦♥❝❧✉❞❡❞ t❤❛t t❤❡ ❜♦♦tstr❛♣ t❡st ✐s ❝♦♥s✐st❡♥t ❛❣❛✐♥st ❡❛❝❤ ♥♦♥✕s♣❤❡r✐❝❛❧❧② s②♠♠❡tr✐❝

❛❧t❡r♥❛t✐✈❡ ✐♥♥♦✈❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ s❛t✐s❢②✐♥❣ ✭❆✳✷✮✳

✶✸

✻ ▼♦♥t❡ ❈❛r❧♦ r❡s✉❧ts ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥

❚❤✐s s❡❝t✐♦♥ ✐s ❞❡✈♦t❡❞ t♦ t❤❡ st✉❞② ♦❢ t❤❡ ✜♥✐t❡✕s❛♠♣❧❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ ♣r♦♣♦s❡❞

t❡sts ✐♥ t❡r♠s ♦❢ ❧❡✈❡❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❛♥❞ ♣♦✇❡r✳ ❲✐t❤ t❤✐s ❛✐♠ ❛ ▼♦♥t❡ ❈❛r❧♦ s✐♠✉✲

❧❛t✐♦♥ st✉❞② ✇❛s ❝♦♥❞✉❝t❡❞✳ ❲❡ ✜rst ❝♦♥s✐❞❡r❡❞ ❛ ❜✐✈❛r✐❛t❡ ❈❈❈✕●❆❘❈❍✭✶✱✶✮ ♠♦❞❡❧

✇✐t❤

b=



 0.1 0.1



, B₁ =



 0.3 0.1 0.1 0.2



, Γ₁ =



 0.2 0.1 0.01 0.3



, R=



 1 r r 1



,

❢♦r r = 0,0.3✳ ❋♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ✐♥♥♦✈❛t✐♦♥s ✇❡ t♦♦❦ ε₁, . . . ,ε_T✱ ✐✳✐✳❞✳ ❢r♦♠

t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ε ✇✐t❤ ε ❛s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✿

✭✐✮ ε∼N2(0,I₂)✱

✭✐✐✮✲✭✐✈✮ ε = |t_ν|R✱ ✇❤❡r❡ R = (R₁, R₂)^′ ✐s ✉♥✐❢♦r♠❧② ❞✐str✐❜✉t❡❞ ♦♥ t❤❡ ✉♥✐t ❝✐r❝❧❡ ❛♥❞

t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡tν ❤❛s t✲❞✐str✐❜✉t✐♦♥ ✇✐t❤ν ❞✳❢✳✱ ν= 5,6,7✱

✭✈✮ ε = (Z1, Z2)^′✱ ✇✐t❤ Z1, Z2 ✐✳✐✳❞✳ ❢r♦♠ ❛♥ ❛s②♠♠❡tr✐❝ ❡①♣♦♥❡♥t✐❛❧ ♣♦✇❡r ❞✐str✐✲

❜✉t✐♦♥ ✭❩❤✉ ❛♥❞ ❩✐♥❞❡✲❲❛❧s❤✱ ✷✵✵✾✮ ✇✐t❤ ♣❛r❛♠❡t❡rs α = 0.4✱ p₁ = 1.182 ❛♥❞

p2 = 1.820 ✭µ= 0✱ σ= 1✮✳

✭✈✐✮ ε= (Z1, Z2)^′✱ ✇✐t❤ Z1, Z2 ✐♥❞❡♣❡♥❞❡♥t✱Z1 ∼N(0,1)✱ Z2 ∼t5✳

✭✈✐✐✮ t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ ε ✐s ❛♥ ❡q✉❛❧ ♠✐①t✉r❡ ♦❢ t✇♦ ❜✐✈❛r✐❛t❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥s

✇✐t❤ ✉♥✐t ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ❛♥❞ ♠❡❛♥s (−1.5,0)^′ ❛♥❞ (1.5,0)^′✱

✭✈✐✐✐✮ ε ❤❛s ❛ s❦❡✇✲♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ❞✐r❡❝t ♣❛r❛♠❡t❡rs ✭❆r❡❧❧❛♥♦✲❱❛❧❧❡ ❛♥❞ ❆③✲

③❛❧✐♥✐✱ ✷✵✵✽✮µ= (0,0)✱Σ = I₂ ❛♥❞ α= (0,0.25)✳

❊❛❝❤ ❞✐str✐❜✉t✐♦♥ ✇❛s s✉✐t❛❜❧② ♠♦❞✐✜❡❞ s♦ t❤❛t E(εt) = 0 ❛♥❞ V ar(εt) = I₂✳ ❚❤❡

❝❛s❡s ✭✐✮✕✭✐✈✮ ♦❜❡② H0✱ ✇❤✐❧❡ ❝❛s❡s ✭✈✮✕✭✈✐✐✐✮ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ ❛❧t❡r♥❛t✐✈❡ ❤②♣♦t❤❡s✐s✳

❚❤❡r❡ ✐s ❛ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs t❤❛t s❤♦✉❧❞ ❜❡ s♣❡❝✐✜❡❞ ❢♦r t❤❡ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ t❤❡

t❡st st❛t✐st✐❝s✳ ❚❤❡s❡ ♣❛r❛♠❡t❡rs ❛✛❡❝t t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ t❤❡ t❡sts✳ ❍❡r❡ ❤♦✇❡✈❡r ✇❡

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

❜② ❍❡♥③❡ ❡t ❛❧✳ ✭✷✵✶✹✮ ❢♦r t❤❡ ✈❛❧✉❡s ♦❢ t❤❡s❡ ✉s❡r ♣❛r❛♠❡t❡rs✳ ■♥ ♣❛rt✐❝✉❧❛r ❢♦r t❤❡

✶✹