✵✳✶✳ ❖r❞❡r ❙t❛t✐st✐❝s
❘❛♥❞♦♠ s❛♠♣❧✐♥❣ ❛ss✉♠❡s t❤❛t t❤❡ s❛♠♣❧❡ ✐s t❛❦❡♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ❢♦r ❡❛❝❤ tr✐❛❧ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ❢♦❧❧♦✇ t❤❡ ❝♦♠♠♦♥ ♣♦♣✉❧❛t✐♦♥ ❞❡♥s✐t② ❢✉♥❝t✐♦♥✳ ■♥ t❤✐s ❝❛s❡✱ t❤❡ ❥♦✐♥t ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ✐s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ ❝♦♠♠♦♥ ♠❛r❣✐♥❛❧ ❞❡♥s✐t✐❡s✳
❉❡❢✐♥✐t✐♦♥ ✶✳ ❚❤❡ s❡t r❛♥❞♦♠ ✈❛r✐❛❜❧❡sX1, X2, ..., Xn✐s s❛✐❞ t♦ ❜❡ ❛ r❛♥❞♦♠
s❛♠♣❧❡ ♦❢ s✐③❡n❢r♦♠ ❛ ♣♦♣✉❧❛t✐♦♥ ✇✐t❤ ❞❡♥s✐t② ❢✉♥❝t✐♦♥f(x)✐❢ t❤❡ ❥♦✐♥t ♣❞❢ ❤❛s
t❤❡ ❢♦r♠
f(x1, x2, ..., xn) =f(x1)f(x2)...f(xn)
❋♦r ❡①❛♠♣❧❡✱ ✶✵✵ ♠♦♥t❤s ✇❡r❡ r❡q✉✐r❡❞ ❜❡❢♦r❡ ❛❧❧ ✜✈❡ ❧✐❣❤t ❜✉❧❜s ❢❛✐❧❡❞✱ ❜✉t t❤❡ ✜rst ❢♦✉r ❢❛✐❧❡❞ ✐♥ ✶✼ ♠♦♥t❤s✳ ■♥ s♦♠❡ ❝❛s❡s ♦♥❡ ♠❛② ❞❡s✐r❡ t♦ st♦♣ ❛❢t❡r t❤❡ r s♠❛❧❧❡st ♦r❞❡r❡❞ ♦❜s❡r✈❛t✐♦♥s ♦✉t ♦❢ n ❤❛✈❡ ❜❡❡♥ ♦❜s❡r✈❡❞✱ ❜❡❝❛✉s❡ t❤✐s ❝♦✉❧❞ r❡s✉❧t ✐♥ ❛ ❣r❡❛t s❛✈✐♥❣ ♦❢ t✐♠❡✳ ❚❤❡ ❥♦✐♥t ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ ♦r❞❡r❡❞ ✈❛r✐❛❜❧❡s ✐s ♥♦t t❤❡ s❛♠❡ ❛s t❤❡ ❥♦✐♥t ❞❡♥s✐t② ♦❢ t❤❡ ✉♥♦r❞❡r❡❞ ✈❛r✐❛❜❧❡s✳
❈♦♥s✐❞❡r ❛ tr❛♥s❢♦r♠❛t✐♦♥ t❤❛t ♦r❞❡rs t❤❡ ✈❛❧✉❡s ♦❢x1, x2, ..., xn✳ ▲❡t
y1=u1(x1, x2, ..., xn) = min (x1, x2, ..., xn) =x1:n
y2=u2(x1, x2, ..., xn) = x2:n
✳✳✳ ✳✳✳ ✳✳✳
yn =un(x1, x2, ..., xn) = max (x1, x2, ..., xn) =xn:n
❲❤❡♥ t❤✐s tr❛♥s❢♦r♠❛t✐♦♥ ✐s ❛♣♣❧✐❡❞ t♦ ❛ r❛♥❞♦♠ s❛♠♣❧❡X1, X2, ..., Xn✱ ❛ s❡t
♦❢ ♦r❞❡r❡❞ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✇✐❧❧ ❜❡ ♦❜t❛✐♥❡❞✱ ❝❛❧❧❡❞ t❤❡ ♦r❞❡r st❛t✐st✐❝s✱ ❛♥❞ ❞❡♥♦t❡❞ ❜② ❡✐t❤❡rX1:n, X2:n, ..., Xn:n ♦rY1, Y2, ..., Yn✳
❚❤❡♦r❡♠ ✷✳ ■❢X1, X2, ..., Xn ✐s ❛ r❛♥❞♦♠ s❛♠♣❧❡ ❢r♦♠ ❛ ♣♦♣✉❧❛t✐♦♥ ✇✐t❤ ❝♦♥✲
t✐♥✉♦✉s ♣❞❢ f(x)✱ t❤❡♥ t❤❡ ❥♦✐♥t ♣❞❢ ♦❢ t❤❡ ♦r❞❡r st❛t✐st✐❝sY1, Y2, ..., Yn ✐s
g(y1, y2, ..., yn) =n!f(y1)f(y2)...f(yn)
✐❢ y1< y2< ... < yn✱ ❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳
❊①❛♠♣❧❡ ✸✳ ❙✉♣♣♦s❡X1, X2 ❛♥❞ X3 t❤❛t r❡♣r❡s❡♥t ❛ r❛♥❞♦♠ s❛♠♣❧❡ ♦❢ s✐③❡ ✸ ❢r♦♠ ❛ ♣♦♣✉❧❛t✐♦♥ ✇✐t❤ ♣❞❢f(x) = 2x,0< x <1✳ ❋✐♥❞ t❤❡ ❥♦✐♥t ♣❞❢ ♦❢ t❤❡ ♦r❞❡r
st❛t✐st✐❝sY1, Y2 ❛♥❞Y3✳
❙♦❧✉t✐♦♥✳ ■t ❢♦❧❧♦✇s t❤❛t t❤❡ ❥♦✐♥t ♣❞❢ ♦❢ t❤❡ ♦r❞❡r st❛t✐st✐❝sY1, Y2 ❛♥❞Y3 ✐s g(y1, y2, y3) = 3! (2y1) (2y2) (2y3) = 48y1y2y3,0< y1< y2< y3<1
❛♥❞ ③❡r♦ ♦t❤❡r✇✐s❡✳
❆♥❞ ♠❛r❣✐♥❛❧ ♣❞❢ ♦❢Y1✐s
gY1(y) =
... ˆ
... ... ˆ
...
48y1y2y3d...d...=...
■t ✐s ♣♦ss✐❜❧❡ t♦ ❞❡r✐✈❡ ❛♥ ❡①♣❧✐❝✐t ❣❡♥❡r❛❧❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❞✐str✐❜✉t✐♦♥ ♦❢ t❤❡ kt❤ ♦r❞❡r st❛t✐st✐❝ ✐♥ t❡r♠s ♦❢ ♣❞❢✱ f(x)✱ ❈❉❋✱ F(x)✱ ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥ r❛♥❞♦♠
✈❛r✐❛❜❧❡✳
✵✳✶✳ ❖❘❉❊❘ ❙❚❆❚■❙❚■❈❙ ✸
❊①❛♠♣❧❡ ✺✳ ❋r♦♠ ❊①❛♠♣❧❡ ✸✱ ✜♥❞ t❤❡ ♣❞❢ ♦❢X1:3✱X2:3 ❛♥❞X3:3✳
❙♦❧✉t✐♦♥✳ X1:3✱X2:3 ❛♥❞X3:3 ❛ r❛♥❞♦♠ s❛♠♣❧❡ ♦❢ s✐③❡ ✸ ❢r♦♠ ❛ ♣♦♣✉❧❛t✐♦♥ ✇✐t❤ ❈❉❋FX(y) =
´y
0 f(y)dx=
´y
0 2xdx=y2✳ ❋r♦♠ ❚❤❡♦r❡♠ ✹✱ fY1(y) =
( 3!
(1−1)!(3−1)! y
21−12
y 1−y23−1= 6
y 1−y22 ,0< y <1
0 , otherwise
fY2(y) =
( 3!
(2−1)!(3−2)! y
22−1
2y 1−y23−2
= 12y3 1−y2
,0< y <1
0 , otherwise
fY3(y) =
( 3!
(3−1)!(3−3)! y
23−1
2y 1−y23−3
= 6y5 ,0< y <1
0 , otherwise
❚❤❡♦r❡♠ ✻✳ ❋♦r ❛ r❛♥❞♦♠ s❛♠♣❧❡ ♦❢ s✐③❡ n ❢r♦♠ ❛ ❞✐s❝r❡t❡ ♦r ❝♦♥t✐♥✉♦✉s ❈❉❋✱ t❤❡ ♠❛r❣✐♥❛❧ ❈❉❋ ♦❢ t❤❡kt❤ ♦r❞❡r st❛t✐st✐❝ ✐s ❣✐✈❡♥ ❜②
Gk(yk) = n
X
j=k
n j
(F(yk)) j
(1−F(yk)) n−j
❊①❛♠♣❧❡ ✼✳ ❈♦♥s✐❞❡r ❛ r❛♥❞♦♠ s❛♠♣❧❡ ♦❢ s✐③❡n❢r♦♠ ❛ ❞✐str✐❜✉t✐♦♥ ✇✐t❤ ♣❞❢ ❛♥❞ ❈❉❋ ❣✐✈❡♥ ❜② f(x) = 2x ❛♥❞ F(x) =x2; 0< x <1✳ ❋✐♥❞ ♣❞❢ ❛♥❞ ❈❉❋ ♦❢
X1:n ❛♥❞Xn:n✳
❙♦❧✉t✐♦♥✳
❊①❛♠♣❧❡ ✽✳ ❋r♦♠ ❊①❛♠♣❧❡ ✼✱ ✇❤❛t ✐s t❤❡ ❞❡♥s✐t② ♦❢ t❤❡ r❛♥❣❡ ♦❢ t❤❡ s❛♠♣❧❡✱ R=Yn−Y1✳
❙♦❧✉t✐♦♥✳ ❋r♦♠ ❡q✉❛t✐♦♥ ✵✳✶✳✶✱ gin(y1, yn) = n!
(n−2)!2y1 y
2
n−y
2 1
n−2
2yn,0< y1< yn<1
▼❛❦✐♥❣ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥R=Yn−Y1 ❛♥❞S =Y1✱ ②✐❡❧❞s t❤❡ ✐♥✈❡rs❡ tr❛♥s✲
❢♦r♠❛t✐♦♥y1=s✱yn=r+s❛♥❞|J|= 1✳ ❚❤✉s✱ t❤❡ ❥♦✐♥t ♣❞❢ ♦❢R❛♥❞S ✐s
h(r, s) =gin(s, r+s)|J|= n!
(n−2)!2s (r+s)
2− s2n−2
