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❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❱♦❧❦♦✈❛ ❑s❡♥✐❛ ❙❛✐♥t✲P❡t❡rs❜✉r❣ ❯♥✐✈❡rs✐t②✱ ❘✉ss✐❛ ▼❛r❝❤ ✶✺✱ ✷✵✶✵

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ■♥tr♦❞✉❝t✐♦♥✳ ■♥tr♦❞✉❝t✐♦♥ ■♥ t❤✐s t❛❧❦ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r t✇♦ ❡①❛♠♣❧❡s ♦❢ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts✿ t❡sts ♦❢ ♥♦r♠❛❧✐t② ❛♥❞ t❡sts ♦❢ ❡①♣♦♥❡♥t✐❛❧✐t②✳ ❋♦r ❡❛❝❤ ❡①❛♠♣❧❡ ✇❡ ✇✐❧❧ ❉❡s❝r✐❜❡ t❤❡ ❧✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥s ♦❢ st❛t✐st✐❝s ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s✐s✳ ❋✐♥❞ t❤❡✐r ❧♦❣❛r✐t❤♠✐❝ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ❛s②♠♣t♦t✐❝s ✉♥❞❡r ♥✉❧❧ ❤②♣♦t❤❡s✐s✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❧♦❝❛❧ ❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ st❛t✐st✐❝s ✉♥❞❡r s♦♠❡ ❝♦♠♠♦♥ ♣❛r❛♠❡tr✐❝ ❛❧t❡r♥❛t✐✈❡s✳

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s ■♥ ✶✾✻✹ ▲✳❙❤❡♣♣ ❞✐s❝♦✈❡r❡❞ t❤❛t ✐❢ X ❛♥❞ Y ❛r❡ t✇♦ ✐♥❞❡♣❡♥❞❡♥t ❝❡♥tr❡❞ ♥♦r♠❛❧ r✈✬s ✇✐t❤ s♦♠❡ ✈❛r✐❛♥❝❡ σ 2 > 0 , t❤❡♥ t❤❡ r✈ ♣ X 2 + Y 2 k ( X, Y ) := 2 XY/ ❤❛s ❛❣❛✐♥ t❤❡ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ N (0 , σ 2 ) ✭❙❤❡♣♣ ♣r♦♣❡rt②✳✮ ❏✳●❛❧❛♠❜♦s ❛♥❞ ■✳❙✐♠♦♥❡❧❧✐ ♣r♦✈❡❞ ✐♥ ✷✵✵✸✱ t❤❛t t❤❡ ❙❤❡♣♣ ♣r♦♣❡rt② ✐♠♣❧✐❡s t❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ♥♦r♠❛❧ ❧❛✇ ✐♥ ❛ ❜r♦❛❞ ❝❧❛ss ♦❢ ❞✐str✐❜✉t✐♦♥s✳ ❈♦♥s✐❞❡r t❤❡ ❝❧❛ss F , ♦❢ ❞❢✬s F s❛t✐s❢②✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s✿ ✐✮ 0 < F (0) < 1 ❛♥❞ ✐✐✮ F ( x ) − F ( − x ) ✐s r❡❣✉❧❛r❧② ✈❛r②✐♥❣ ✐♥ ③❡r♦ ✇✐t❤ t❤❡ ❡①♣♦♥❡♥t ✶✳ ❚❤❡♦r❡♠ ▲❡t X ❛♥❞ Y ❜❡ ✐♥❞❡♣❡♥❞❡♥t r✈✬s ✇✐t❤ ❝♦♠♠♦♥ ❞❢ F ❢r♦♠ t❤❡ ❝❧❛ss F . ❚❤❡♥ d = k ( X, Y ) ✐s ✈❛❧✐❞ ✐✛ X ∈ N (0 , σ 2 ) ❢♦r s♦♠❡ t❤❡ ❡q✉❛❧✐t② ✐♥ ❞✐str✐❜✉t✐♦♥ X ✈❛r✐❛♥❝❡ σ 2 > 0 .

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳ ■♥tr♦❞✉❝t✐♦♥ ▲❡t X 1 , . . . , X n ❜❡ ✐♥❞❡♣❡♥❞❡♥t ♦❜s❡r✈❛t✐♦♥s ✇✐t❤ ③❡r♦ ♠❡❛♥ ❛♥❞ ❞❢ F, ❛♥❞ ❧❡t F n ❜❡ t❤❡ ✉s✉❛❧ ❡♠♣✐r✐❝❛❧ ❞❢ ❜❛s❡❞ ♦♥ t❤✐s s❛♠♣❧❡✳ ❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❡st✐♥❣ t❤❡ ❝♦♠♣♦s✐t❡ ❤②♣♦t❤❡s✐s H 0 : F ∈ N (0 , σ 2 ) ❢♦r s♦♠❡ ✉♥❦♥♦✇♥ ✈❛r✐❛♥❝❡ σ 2 > 0 ❛❣❛✐♥st t❤❡ ❛❧t❡r♥❛t✐✈❡s H 1 , ✉♥❞❡r ✇❤✐❝❤ t❤❡ ❤②♣♦t❤❡s✐s H 0 ✐s ❢❛❧s❡✳ ❲❡ ❜✉✐❧❞ t❤❡ U ✲ ❡♠♣✐r✐❝❛❧ ❞❢ H n ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛ n ) − 1 ❳ H n ( t ) = ( C 2 ✶ { k ( X i , X j ) < t } , t ∈ R 1 . 1 ≤ i<j ≤ n ❈♦♥s✐❞❡r t✇♦ st❛t✐st✐❝s✿ ❩ I n = R 1 ( H n ( t ) − F n ( t )) dF n ( t ) , D n = sup t ∈ R 1 | H n ( t ) − F n ( t ) | .

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳ ▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ I n ■t ✐s ✇❡❧❧✲❦♥♦✇♥ t❤❛t ♥♦♥✲❞❡❣❡♥❡r❛t❡ U ✲ ❛♥❞ V ✲st❛t✐st✐❝s ❛r❡ ❛s②♠♣t♦t✐❝❛❧❧② ♥♦r♠❛❧ ✭❍♦❡✛❞✐♥❣✱ ✶✾✹✽✮✳ ▲❡t s❤♦✇ t❤❛t I n ❜❡❧♦♥❣s t♦ t❤✐s ❝❧❛ss✳ ❚❤❡ st❛t✐st✐❝ I n ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ U ✲st❛t✐st✐❝ ♦❢ ❞❡❣r❡❡ ✸ ✇✐t❤ ❝❡♥t❡r❡❞ ❦❡r♥❡❧ Ψ( x, y, z ) = 1 3 ( ✶ { k ( x, y ) < z } + ✶ { k ( x, z ) < y } + ✶ { k ( y, z ) < x } ) − 1 / 2 . ✭✶✮ ▲❡t ❝❛❧❝✉❧❛t❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ ✭✶✮ ψ ( x ) := E (Ψ( X, Y, Z ) | X = x ) = = 1 3 ( P ( k ( x, Y ) < Z ) + P ( k ( x, Z ) < Y ) + P ( k ( Y, Z ) < x )) − 1 / 2 . ❆❢t❡r s♦♠❡ ❝❛❧❝✉❧❛t✐♦♥s ✇❡ ♦❜t❛✐♥ t❤❛t t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤❡ ❦❡r♥❡❧ Ψ ✐s 2 ) ✇✐t❤ t❤❡ ✈❛r✐❛♥❝❡ σ 2 = ψ ( x ) = 1 3 (Φ( x ) − 1 ❘ R 1 ψ 2 ( x ) d Φ( x ) = 1 108 . ❍❡♥❝❡ t❤❡ ❦❡r♥❡❧ ✭✶✮ ✐s ♥♦♥✲❞❡❣❡♥❡r❛t❡✳ ❇② ❍♦❡✛❞✐♥❣✬s t❤❡♦r❡♠ √ nI n → N (0 , 1 d − 12) .

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳ ▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ st❛t✐st✐❝ D n ❚❤❡ r✈ H n ( t ) − F n ( t ) ❢♦r ✜①❡❞ t ✐s ❛s②♠♣t♦t✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ U ✲st❛t✐st✐❝ ✇✐t❤ t❤❡ ❦❡r♥❡❧ ❞❡♣❡♥❞✐♥❣ ♦♥ t ∈ R 1 Ξ( x, y ; t ) = ✶ { k ( x, y ) < t } − 1 2( ✶ { x < t } + ✶ { y < t } ) . ❚❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ t❤✐s ❦❡r♥❡❧ ❢♦r ✜①❡❞ t ✐s ❡q✉❛❧ t♦ ✽ − 1 2 ( ✶ { x < t } + Φ( t )) , ✐❢ t < − 2 | x | ; ❃ ❃ ❃ ❃ ❃ ❃ ❃ ✒ ✓ ❁ t | x | √ − 1 ξ ( x ; t ) = Φ 2 ( ✶ { x < t } + Φ( t )) , ✐❢ − 2 | x | ≤ t ≤ 2 | x | ; 4 x 2 − t 2 ❃ ❃ ❃ ❃ ❃ ❃ ❃ 1 − 1 ✿ 2 ( ✶ { x < t } + Φ( t )) , ✐❢ t > 2 | x | .

❇❛❤❛❞✉r ❡✣❝✐❡♥❝② ♦❢ ❛ ❣♦♦❞♥❡ss✲♦❢✲✜t t❡sts ❜❛s❡❞ ♦♥ ❞✐str✐❜✉t✐♦♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s✳ ❚❡sts ♦❢ ♥♦r♠❛❧✐t② ❜❛s❡❞ ♦♥ ❙❤❡♣♣ ♣r♦♣❡rt②✱ ❛♥❞ t❤❡✐r ❡✣❝✐❡♥❝✐❡s✳ ▲✐♠✐t✐♥❣ ❞✐str✐❜✉t✐♦♥ ♦❢ st❛t✐st✐❝ D n ❚❤❡ ✈❛r✐❛♥❝❡ ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥ ❤❛s t❤❡ ❢♦r♠ ✒ ✓ ✒ ✓ ✽ ❘ ∞ ❘ t tx tx t/ 2 Φ 2 √ √ 2 d Φ( x ) − t/ 2 Φ d Φ( x ) − ❃ ❃ ❃ 4 x 2 − t 2 4 x 2 − t 2 ❃ ❃ ❃ − 1 4 Φ( t ) − 1 4 Φ 2 ( t ) + Φ( t/ 2) − 1 2 , ✐❢ t ≥ 0; ❃ ❃ ❃ ❁ σ 2 ξ ( t ) = Eξ 2 ( X ; t ) = ✒ ✓ ✒ ✓ ❃ ❘ ∞ ❘ ∞ ❃ − t/ 2 Φ 2 √ tx √ tx 2 d Φ( x ) − − t Φ d Φ( x )+ ❃ ❃ ❃ 4 x 2 − t 2 4 x 2 − t 2 ❃ ❃ ❃ ❃ + 1 4 Φ( t ) − 1 4 Φ 2 ( t ) , ✐❢ t < 0 . ✿ ❚❛❦✐♥❣ ✐♥ ❝♦♥s✐❞❡r❛t✐♦♥ t❤❡ s②♠♠❡tr② ♦❢ t❤❡ ❢✉♥❝t✐♦♥ σ 2 ξ ( t ) ✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t sup t ∈ R σ 2 ξ ( t ) = σ 2 1 ξ (0) = 16 . ❚❤✐s ✈❛❧✉❡ ✐s ✐♠♣♦rt❛♥t ✇❤❡♥ ❝❛❧❝✉❧❛t✐♥❣ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥ ❛s②♠♣t♦t✐❝s✳