9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d

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1 9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((, ) S). For Eh( ), the emprcal estmator s asymptotcally optmal. Now, U-statstcs geeralze the dea. The advatage of usg U-statstcs s ubasedess. Notato. Let t (P) = R R h(x,..., x m )d F (x ) d F (x m ) where h s a kow fucto; h s called kerel. Assume that, the followg, wthout loss of geeralty, h s symmetrc,.e. h(x, x )=h(x, x ). If h s ot symmetrc, we ca replace t wth h (x,..., x m )= (m!) P h(x (),..., x (m ) ) where : N 7! N ad s set of all possble permutatos of {,..., m}. Note that h s also ubased, Eh =(m!) Eh( (),..., (m ) )..d =(m!) Eh(,..., m )=t(p). Defto 9.. Suppose h : R m 7! R s symmetrc ts argumets. The U-statstc for estmatg t (P)=Eh(,..., m ) s a symmetrc average u (,..., m )= m apple < < < m apple h(,..., m ). Example. Suppose m =, the u (, )= h(, )= h(, j ) () ( ) < <j = h(, j ) () ( ) 6=j Remark. E m apple < < < m apple Eh(,..., m )=Eh(,..., m )=t (P). Example 9.. (a) (b) Suppose t (P)=E = R x d F (x ). The, h(x )=x ad u (,..., m )= h( )= =. Suppose t (P)=(E ) = ÄR x d F (x ) ä. The u = from (a) s based sce 3 E( )= 6 4 E j + E( ) 7 5 6=(E ). = j 6= =(E ) = E 6=(E ) 35

2 Now, wrte t (P)= RR x x d F (x )d F (x ) ad h(x, x )=x x. The u (,..., )= j. ( ) (c) Suppose t (P)=P( apple t 0 )=F (t 0 )= R h(x )d F (x ). The h(x )= (,t 0 ](x ) ad whch s just the emprcal CDF. <j { apple t 0 } = ˆF (t 0 ) (d) Suppose t (P)=Var = RR x +x x x d F (x )d F (x ). The h(x, x )=(x +x x x )/ = (x x ) /. Note that Eh(x, x )={Var( )+[E( )] + Var( )+(E ) E( )}/..d. = Var( ). Hece, h(, j ) ( ) <j = + j j ( ) = 6= ˆ = 6=j ( ) = ( ). = Note that ˆ s a based estmator ad u s a ubased estmator. (e) Suppose t (P)=E = RR x x d F (x )d F (x ) (measure of dsperso). The j. ( ) u s called G s Mea Dfferece. (f ) Suppose t (P) =P( + apple 0) = RR {x + x apple 0}d F (x )d F (x ). The, h(x, x )= {x + x apple 0} ad {x + x j apple 0}. ( ) <j <j 36

3 Remark 9.3 (Prelmary Remark). Wrte () =( (),..., () ) as the order statstc. U-statstc ca be regarded as codtoal expectato gve (). For m =, h( )= h( ( ) )=EˆF [h( ) () ]. For m=, <j h(, j )= h( ( ), (j ) )=EˆF [h(, ) () ]. <j For arbtrary m, E ˆF [h(,,..., m ) () ]. Now: ay ubased estmator S = S(,..., ) ca be mproved by ts U-statstc verso (or S f S s ot symmetrc) E ˆF [S () ]. For example, s ubased for E. Now, E ˆF ( () )= P = ( ) = = P = = E ˆF ( ). Theorem 9.4. Let S = S(,..., ) be a ubased estmator of t (P) wth correspodg U- statstc u. The, u s ubased as well ad Varu apple VarS wth the equalty holdg f P( S)=. Proof. u s ubased: E(u )=E[EˆF (S () )] = t (P). E(S) Sce both u ad S are ubased we show Eu apple ES, Eu = E[E ˆF (S Jese equ. ())] apple E[E ˆF (S () )] = E(S ) wth "=" f the dstrbuto of E ˆF (S () ) s degeerate wth E ˆF (S () )=S almost surely. Note: Ths also follows from the Rao-Blackwell Theorem: Takg codtoal expectato of a ubased statstc codtoal o a suffcet statstc (eg. () here) wll gve us a estmator whch s as least as good the sese of lower rsk/varace. Remark 9.5 (The Varace for m apple (heurstcs)). m =. Var Var! h( ) = Varh( )=O(/). = 37

4 m =. Sce = h(, j )= h(, j ) ( ) <j <j [h(, )+h(, 3 )+ + h(, )], ot depedet the 0 B Var Var@ C h(, j ) A ( ) j > = Cov[h(, j ), h( k, l )]. ( ) j > k l >k =0 f,j,k,l dfferet The secod largest term (three sums): e.g. = k but k, j, l are dfferet (' u 3 ) Varu 3 3 ( ) = same order as m =. 4 Thus, we expect geeral, Var O(/). Notato. h (x,..., x )=E[h(,..., m ) = x,..., = x ] ad = Varh (,..., ) Lemma 9.6. (a) Eh (,..., )=t(p)(= Eh(,..., m )) for all apple apple m. (b) Cov[h(,...,, +,..., m ), h(,...,, 0 +,..., 0 )] = m Proof. We oly cosder m = here. (a) (b) =. Eh (, )=E{E[h(, ), ]} = E[h(, )] = t (P) Eh (, )=E{E[h(, ) ]} = E[h(, )] = t (P) Cov[h(, ), h(, )] = Var[h(, )] = Varh (, )=. The secod equalty s because h (, )=E[h(, ), ]=h(, ). =. Cov[h(, ), h(, 0 )] = E[h(, )h(, 0 )] E[h(, )]E[h(, 0 )]. ={E[h(, )]} =[t (P)] 38

5 Note that frst term o the rght had sde ca be computed by E[h(, )h(, 0 )] = E {E [h(, )h(, 0 ) ]} = E {E [h(, ) ]E [h(, 0 ) ]} = Eh ( ). Theorem 9.7 (Hoeffdg). (a) The varace of the U-statstc s Var m m m = m m. Note that oe ca compute from Lemma 9.6. (b) If > 0 ad < for =,,..., m, the Var( p u )! m. Proof. (a) For geeral proof, see Lee U-statstcs (990). We oly prove the case of m =. We wat to show that Var = [( ) ] wth = Cov[h(, ), h(, 0 )] ad = Cov[h(, ), h(, )] = Var[h(, )]. From Remark 9.5 we have Ç å Var Cov[h(, j ), h( k, l )] = j > k j > k l >k l >k Case, j, k, l are all dfferet. The Cov = 0. Case = k ad j = l : Cov[h(, j ), h( k, l )] =( ) + Cov[h(, j ), h( k, l )] = Cov[h(, j ), h(, j )] = Var[h(, j )] =.. Note that the umber of ways to choose, j out of {,,..., } s / =.. Ths gves us 39

6 Case 3 ( = k ad j 6= l ) or ( 6= k ad j = l ). Cov[h(, j ), h( k, l )] =. (b) Note that the umber of ways to choose, j, k, l s {z} ( ) {z} {z} = ( ). j 6= l or Ths gves ( ) = ( ) Cosder the varace formula from (a) m = k k!. ( m )( m ) ( m k + ) m k k! s large f k s large. Ths mples that s large f m s large,.e. =. Thus the m terms of the sum are domated by the = term,.e., by m m m m m (m )! m /m! = m. Hece, Var( p u )! m. m Theorem 9.8. p (u t (P)) D! N (0, m ). Proof. For m =, For geeral proof, see p. 78 of Serflg. m m u = c c c= apple < < c apple g c (,..., c )+o p ( / ). T = / (u t (P)) = / [h ( ) t (P)] + o p () := T. We wat to show that E(T T )! 0 () T T = o p()). E(T T ) = Var(T T ) = Var(T )+Var(T ) Cov(T,T ) = Var[ p (u t (P))] + Var p [h ( ) t (P)] p(u Cov t (P)), p [h ( ) t (P) = Var( p u ) + 4!m Var[h ( )] 4 4 Cov(u, h ( )) ( ) =!! 0. 40

7 For ( ), Cov(u, h ( )) = Cov[h( l, k ), h ( )] = ( ) k l 6=k ( ) =, f k = or l = The umber of o-zero terms s ( ).. For ( ), (9.6) = Cov[h(, ), h(, 0 )] = E[h(, )h(, 0 )] [t (P)] = E{E[h(, )h(, 0 ), ]} [t (P)] = E{h(, )E[h(, 0 ), ]} [t (P)] =E(h(, ) )=h ( ) = Cov[h(, ), h ( )]. Example 9.9. Suppose t (P)=P( + > 0), m =, h(, )=( + > 0). The ( + j > 0) ( ) <j ad, by Theorem 9.8, p (u P( + > 0)) D! N (0, 4 ). The ext thg s to calculate. (9.6) = Cov[h(, ), h(, 0 )] = E[h(, )h(, 0 )] [P( + > 0)] = E[( + > 0)( + 0 > 0)] [P( + > 0)] = E[( + > 0, + 0 > 0)] [P( + > 0)]. To obta a explct form we eed some assumptos. Suppose, for example, F s symmetrc aroud zero,.e. P( < a )=P( > a )=P( < a ) whch mples ad have the same dstrbuto. Moreover, suppose F s cotuous. The P( + > 0)=P( > 0)=P( + < 0) wth P( + = 0 = 0), ad therefore = P( + < 0)+P( + = 0)+P( + > 0)=P( + > 0) ) P( + > 0)=. O the other had, sce P( + > 0)=P( > )=P( > ), P( + > 0, + 0 > 0)=P( = max{,, 0 })=P( = max{,, 3 }, =,, 3)=/3. I sum, = /3 (/) = /. 4

8 Remark 9.0 (Geeralzato: Two-Sample Problems). Suppose,..., are..d. wth cdf F, Y,..., Y are..d. wth cdf G, ad F ad G are ukow, ad Y are depedet. Let h(,..., {z m, Y },..., Y my ) symmetrc symmetrc be a fucto of m +m Y argumets, wth m apple ad m Y apple Y. We wat to estmate t (P)= Eh(,..., m, Y,..., Y my ). For example, t (P)=P( < Y )=Eh(, Y ) where h(, Y )=( < Y ); or t (P)=E Y. Note also that h(,..., m, Y,..., Y my ) s trvally ubased for t (P)= Eh. Thus h(,..., m, Y j,..., Y j my ) wth apple apple... apple m apple ad apple j apple... apple j my apple Y s ubased as well. The umber of combatos s. Our ubased estamtor s m Y m Y m Y m Y h Ä ä,..., m, Y j,..., Y j my Example (m = m Y = ). Y h(, k, Y j, Y l ) <k j <l where h(,, Y, Y )=( < Y Y ). The Y ( k < Y j Y l ) <k j <l =the umber of (,j,k,l ) where y dstace s larger ad t (P) =P( Y Y > ). Note that ths U-statstc ca be used to test Y s more dspersed tha. Remark 9. (Propertes of the Two-Sample U-statstc). (a) Formula for Var u, see Lee or Serflg. (b) Asymptotc varace ad ormalty. Let = + Y ad /! c where c (0, ). Suppose Var[h(,,..., m, Y, Y,..., Y my )] > 0. The Var( p u )! = m c + m Y 0 c 0 ad p (u t (P)) D! N (0, ) where = Cov[h( 0,,..., m, Y, Y,..., Y my ), h(, 0,..., 0 m, Y 0, Y 0,..., Y 0 m Y ) = Cov[h( 0,,..., m, Y, Y,..., Y my ), h( 0, 0,..., 0 m, Y, Y 0,..., Y 0 m Y )] 4

9 Example 9.. Suppose t (P)=P( < Y )=E[( < Y )] (thus m = m Y = ). The ad ( < Y j ) Y = Cov[( 0 < Y ), ( < Y 0 )] = E[(( < Y )( < Y 0 )] {E[( < Y )]} = P( < Y, < Y 0 ) [P( < Y )] = Cov[( 0 < Y ), ( 0 < Y )] = E[(( < Y )( 0 < Y j )] {E[( < Y )]} = P( < Y, 0 < Y ) [P( < Y )]. Now suppose F = G s cotuous. The P( < Y )= P( Y )= P( > Y )= P(Y > ) ad hece P( < Y )= /. Furthermore, P( < Y, < Y 0)= P( = m{, Y, Y 0})=/3 ad P( < Y, 0 < Y )=P(Y = max{, 0, Y }) =/3. The = 0 = / ad the asymptotc varace s 0 j = m c + m Y 0 c = 0 c( c). I sum, p D (u P( < Y ))! N 0, c( c) Note: Y P P ( j < Y j ) s the Ma-Whtey test statstc wth H 0 : F = G (ad equvalet to a Wlcoxo rak sum statstc). Defto 9.3. Cosder a symmetrc fucto h : R m 7! R wth m apple. The V-statstc for estmatg t (P)=Eh(,..., m ) s V = V (,..., m )= m = h(,..., m ) m = Remark 9.4 (Comparg U- ad V-Statstcs). m =. P h( )=v m =. Frst ote that h(, j )= h(, j ). ( ) ( ) <j 6=j 43

10 O the other had, 3 v = h(, j )= h(, j )+ h(, ) 5 Moreover, Eh(, )=t (P). j j 6= = h(, j )+ h(, ) j 6= ( ) = u + h( {z }, ).! P!0 Ev = Eu + Eh(, ) = t (P) t (P)+ Eh(, ) 3 = t (P) Eh(, ) t (P) 5 =costat {z } =bas!0 Theorem 9.5. Let m =, = Varh (,..., ) (see 9.6), ad suppose 0 < <, <. The U- ad V-statstcs have the same asymptotc dstrbuto, Proof. From Remark 9.4, p p (V t (P)) = u + = p = =! p (V t (P)) D! N (0, 4 ).! h(, ) t (P) + (u t (P)) + h(, ) p (u t (P)) + p (u t (P)) + D!N (0,4 ) t (P) p h (h(, ) t (P)) D p (h(, ) t (P))! N (0, 4 ). {z } p!e[h(, ) t (P)]=costat Cocluso: U- ad V-statstcs are asymptotcally equvalet. The V-statstc s a more tutve estmator, the U-statstc s more coveet for proofs (ad ubased). 44

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